\begin{thm} \label{thm:main}
	Let $1 \le \ell \le k-1$ be such that $k \ge 3$ and $(k,\ell) \neq (3,1)$, let $t = \floor{\frac{k}{k-\ell}}(k - \ell)$, and let $n$ be sufficiently large and divisible by $k-\ell$.
	Then every $k$-uniform $n$-vertex hypergraph without isolated vertices and having minimum supported co-degree at least $(1 - 1/t)n-(k-3)$ will contain a Hamilton $\ell$-cycle.
\end{thm}

\label{sec:intro}
    A widespread research theme in extremal graph theory is to determine sufficient conditions that ensure the existence of specified spanning structures in graphs and hypergraphs. A classic result in this vein, due to Dirac~\cite{dirac1952some}, states that any graph on $n\geq 3$ vertices with minimum degree at least $n/2$ contains a Hamilton cycle, and it is not too difficult to see that the degree condition is the best possible. Dirac's theorem has been generalised to various distinct settings over the years (see the surveys \cite{gould2014recent,kuhn2014hamilton,simonovits2020embedding,kuhn2012survey,frieze2019hamilton,rodl2010dirac}) and this has contributed to the development of several powerful techniques, such as regularity, absorption, and rotation-extension methods. In this paper, we will focus on hypergraph extensions of Dirac's theorem.

	A \emph{$k$-uniform hypergraph} or simply a \emph{$k$-graph} $G$ consists of a set of vertices $V(G)$ and a set of edges $E(G)$, where each edge consists of exactly $k$ vertices. The most common generalisation of a minimum degree condition from graphs to hypergraphs is accomplished via the notion of a \emph{minimum co-degree} -- which is the minimum $d$ such that every set of $k-1$ vertices in $G$ is contained in at least $d$ edges -- and we denote it here by $\delta(G)$. The larger uniformity $k$ allows for multiple distinct ways to define a cycle in a $k$-graph. The cyclic structures we search for are \emph{$\ell$-cycles}, which are perhaps the most common extensions of graph cycles. Intuitively, we think of an $\ell$-cycle as a spanning path formed by a cyclic set of edges such that each edge has exactly $\ell$ vertices in common with the preceding edge (it could intersect other edges as well). More formally, given an $n$-vertex $k$-graph $G$ and any $\ell \in [k-1]$ such that $k - \ell$ divides $n$, we define a \emph{Hamilton $\ell$-cycle} in $G$ to be an ordering of the vertices of $G$, say $v_1 \dots v_n$, such that the vertices of the subsequence $v_{i(k - \ell)+1} \dots v_{i(k - \ell) +k}$ form an edge for all $i \ge 0$, where we view the indices cyclically modulo $n$. Since each edge of an $\ell$-cycle contains $k-\ell$ vertices that were not in the previous edge, we trivially require that $k - \ell$ divides $n$ so that when we ``cycle around'' the vertices of $G$, the original sequence of edges gets repeated. We call $(k-1)$-cycles and $1$-cycles \emph{tight} and \emph{loose} cycles respectively.

The question of determining sufficient minimum co-degree conditions to ensure a tight Hamilton cycle was first raised by Katona and Kierstead~\cite{katona1999hamiltonian} who conjectured that if $\delta(G)\ge (n-k+2)/2$ then $G$ has a tight Hamilton cycle, and they showed that this bound is the best possible. R\"{o}dl, Ruci\'nski and Szemer\'edi~\cite{rodl2006dirac,rodl2008approximate} first proved this asymptotically for $k\geq 3$ and $\delta(G)\geq n/2+o(n)$, and later obtained an exact result for $k=3$~\cite{rodl2011dirac}. Additionally, Liu and Liu \cite{liu2022hamiltonian} made progress towards an exact result for $k = 4$. In fact, for any $\ell \in [k-1]$ such that $k - \ell$ divides $k$, a tight Hamilton cycle will contain such a Hamilton $\ell$-cycle (provided the necessary divisibility condition holds). If we further suppose that $k$ divides $n$, then this Hamilton $\ell$-cycle will imply the existence of a perfect matching which necessitates $\delta(G) \ge n/2 - k$ as shown in \cite{kuhn2010hamilton,rodl2009perfect}, and hence the aforementioned tight cycle bound of $\delta(G) \ge n/2 + o(n)$ is tight. If $k$ does not divide $n$, then R\"{o}dl, Ruci\'nski and Szemer\'edi~\cite{rodl2009perfect} have determined the minimum codegree threshold for a near-perfect matching of size $\left\lfloor n/k \right\rfloor$ to be roughly $n/k$. It is therefore plausible that the correct minimum codegree threshold for a spanning $\ell$-cycle, when $k-\ell$ divides $k$ but is not $1$ and $n$ is not divisible by $k$, is significantly lower than $n/2$. Conversely, if $k-\ell$ does not divide $k$, a series of works by K\"uhn and Osthus~\cite{kuhn2006loose}, Keevash, K\"uhn, Mycroft and Osthus~\cite{keevash2011loose}, H\`an and Schacht~\cite{han2010dirac} and K\"uhn, Mycroft and Osthus~\cite{kuhn2010hamilton} have established that
\[
\delta(G)\geq \frac{n}{\ceil{\frac{k}{k-\ell}}(k-\ell)}+o(n)
\]
suffices and is optimal up to the $o(n)$ term, yielding a much lower threshold. Exact bounds, however, have proven notoriously difficult to obtain and are known only in a few special cases. To the best of our knowledge, these are $(k,\ell)=(3,2)$ by R\"{o}dl, Ruci\'nski and Szemer\'edi~\cite{rodl2011dirac}, $(k,\ell)=(3,1)$ by Czygrinow and Molla~\cite{czygrinow2014tight}, $(k,\ell)=(4,2)$ by Garbe and Mycroft~\cite{garbe2018hamilton}, and $k\ge 3$ and $\ell<k/2$ by Han and Zhao~\cite{han2015minimum}. We refer the reader to the surveys~\cite{kuhn2014hamilton,zhao2016recent,rodl2010dirac} for more detailed discussions.

A drawback of a minimum co-degree condition is that it tends to be a fairly strong requirement. For instance, if we start with a complete $k$-graph and remove all edges that contain a fixed pair of vertices, it is immediate that the resulting hypergraph has minimum co-degree equal to zero. However, if $k\ge 3$, it is not hard to see that even fairly small hypergraphs of this form are quite dense, and trivially contain Hamilton cycles. In fact, there are several natural hypergraph classes that are quite dense, but have minimum co-degree equal to zero (multipartite hypergraphs, for instance), and hence the previous theorems are not immediately applicable. This motivates the notion of the \emph{minimum supported co-degree} of a $k$-graph $G$, which we denote $\delta^*(G)$, and define as the maximum integer $d$ such that every $(k-1)$-set that is contained in at least one edge is contained in at least $d$ edges. A minimum supported co-degree requirement immediately handles the previous issues of zero co-degree sets, although it does not rule out isolated vertices, and hence we will usually impose the (fairly weak) requirement of all vertices being non-isolated.

Multiple recent studies have investigated questions that arise from replacing a minimum co-degree with a minimum supported co-degree requirement in classical extremal hypergraph theoretic problems. The notion of supported co-degrees was introduced by Balogh, Lemons and Palmer~\cite{balogh2021maximum} (they use the term ``positive co-degree'' instead) to formulate a generalisation of the Erd\H{o}s-Ko-Rado theorem, providing bounds on the sizes of intersecting families satisfying a minimum supported co-degree condition. Subsequently, Frankl and Wang~\cite{frankl2025intersecting} improved these bounds in almost all cases, and Spiro~\cite{spiro2023t} has extended these studies to the more general case of $t$-intersecting families. Other papers have worked on generalisations of the Andr\'asfai-Erd\H{o}s-S\'os theorem~\cite{liu2024positive} and unique colourability of hypergraphs~\cite{liu2024uniquely}.

Halfpap, Lemons and Palmer~\cite{halfpap2025positive} investigated variants of hypergraph Tur\'an problems, and established the asymptotic supported co-degree threshold for a host $3$-graph to contain a copy of a fixed graph $F$ for several distinct $3$-graphs $F$, and Wu~\cite{wu2025positive} has consequently addressed this question for other examples. Halfpap, Lemons and Palmer~\cite{halfpap2025positive} also study ``jumps'' in this threshold, which has further been tackled by Balogh, Halfpap, Lidick\'y, and Palmer~\cite{balogh2024positive}. Pikhurko~\cite{pikhurko2023limit} recently proved that these problems are well-defined for a generalisation to the ``minimum positive/supported $\ell$-co-degree'' for any $\ell \in [k-1]$, providing analogous results to those Lo and Markstr\"om~\cite{lo2014degree} regarding the usual notion of $\ell$-co-degree.

The question of finding minimum supported co-degree conditions that ensure the existence of spanning structures in hypergraphs was first raised by Halfpap and Magnan~\cite{halfpap2024positive}. They prove an exact optimal lower bound on the minimum supported co-degree required to guarantee a perfect matching in $3$-graphs and a slightly weaker bound for all higher uniformities, which was later improved to an exact tight bound by Mycroft and Z\'arate-Guer\'en~\cite{mycroft2025matching}. Furthermore, Halfpap and Magnan also establish an exact best-possible minimum supported co-degree condition for Hamilton Berge cycles and an asymptotic one for $3$-uniform loose cycles. In a different direction, Illingworth, Lang, M\"uyesser, Parczyk and Sgueglia~\cite{illingworth2025spanning} showed that an $n$-vertex $k$-graph with $\delta^*(G)\ge n/2 + o(n)$ contains a spanning $k$-sphere, asymptotically confirming a conjecture of Georgakopoulos, Haslegrave, Montgomery and Narayanan~\cite{georgakopoulos2022spanning}.

The problem of determining the optimal minimum supported co-degree for $k$-uniform $\ell$-cycles was initially suggested by Halfpap and Magnan in~\cite[Section 5]{halfpap2024positive}, and reiterated by Illingworth, Lang, M\"uyesser, Parczyk and Sgueglia~\cite{illingworth2025spanning} for the special case of tight cycles. This question was recently tackled by Mycroft and Z\'arate-Guer\'en~\cite{mycroft2025positive}, who prove an asymptotically optimal result for all $k\ge 3$ and $\ell\in [k-1]$.

Our main result provides an improved sufficient minimum supported co-degree condition for $k$-uniform $\ell$-cycles for all $k\ge 3$ and $\ell\in [k-1]$ except $(k,\ell)=(3,1)$, which is the best possible for infinitely many values of $k$ and $\ell$, and is off by at most $1$ for all values of $k$ and $\ell$. We discuss the optimality of our result in \Cref{sec:lower bound}. Importantly, we emphasise that our results are (essentially) exact and not asymptotic, a rather stark difference from the minimum co-degree version of the problem where very few cases have been resolved exactly.

\label{sec:intro}
 A widespread research theme in extremal graph theory is to determine sufficient conditions that ensure the existence of specified spanning structures in graphs and hypergraphs. A classic result in this vein, due to Dirac~\cite{dirac1952some}, states that any graph on $n\geq 3$ vertices with minimum degree at least $n/2$ contains a Hamilton cycle, and it is not too difficult to see that the degree condition is the best possible. Dirac's theorem has been generalised to various distinct settings over the years (see the surveys \cite{gould2014recent,kuhn2014hamilton,simonovits2020embedding,kuhn2012survey,frieze2019hamilton,rodl2010dirac}) and this has contributed to the development of several powerful techniques, such as regularity, absorption, and rotation-extension methods. In this paper, we will focus on hypergraph extensions of Dirac's theorem.

The question of determining sufficient minimum co-degree conditions to ensure a tight Hamilton cycle was first raised by Katona and Kierstead~\cite{katona1999hamiltonian} who conjectured that if $\delta(G)\ge (n-k+2)/2$ then $G$ has a tight Hamilton cycle, and they showed that this bound is the best possible. R\"{o}dl, Ruci\'nski and Szemer\'edi~\cite{rodl2006dirac,rodl2008approximate} first proved this asymptotically for $k\geq 3$ and $\delta(G)\geq n/2+o(n)$, and later obtained an exact result for $k=3$~\cite{rodl2011dirac}. Additionally, Liu and Liu \cite{liu2022hamiltonian} made progress towards an exact result for $k = 4$. In fact, for any $\ell \in [k-1]$ such that $k - \ell$ divides $k$, a tight Hamilton cycle will contain such a Hamilton $\ell$-cycle (provided the necessary divisibility condition holds). If we further suppose that $k$ divides $n$, then this Hamilton $\ell$-cycle will imply the existence of a perfect matching which necessitates $\delta(G) \ge n/2 - k$ as shown in \cite{kuhn2010hamilton,rodl2009perfect}, and hence the aforementioned tight cycle bound of $\delta(G) \ge n/2 + o(n)$ is tight. If $k$ does not divide $n$, then R\"{o}dl, Ruci\'nski and Szemer\'edi~\cite{rodl2009perfect} have determined the minimum codegree threshold for a near-perfect matching of size $\left\lfloor n/k \right\rfloor$ to be roughly $n/k$. It is therefore plausible that the correct minimum codegree threshold for a spanning $\ell$-cycle, when $k-\ell$ divides $k$ but is not $1$ and $n$ is not divisible by $k$, is significantly lower than $n/2$. Conversely, if $k-\ell$ does not divide $k$, a series of works by K\"uhn and Osthus~\cite{kuhn2006loose}, Keevash, K\"uhn, Mycroft and Osthus~\cite{keevash2011loose}, H\`an and Schacht~\cite{han2010dirac} and K\"uhn, Mycroft and Osthus~\cite{kuhn2010hamilton} have established that
\[
\delta(G)\geq \frac{n}{\ceil{\frac{k}{k-\ell}}(k-\ell)}+o(n)
\]
suffices and is optimal up to the $o(n)$ term, yielding a much lower threshold. Exact bounds, however, have proven notoriously difficult to obtain and are known only in a few special cases. To the best of our knowledge, these are $(k,\ell)=(3,2)$ by R\"{o}dl, Ruci\'nski and Szemer\'edi~\cite{rodl2011dirac}, $(k,\ell)=(3,1)$ by Czygrinow and Molla~\cite{czygrinow2014tight}, $(k,\ell)=(4,2)$ by Garbe and Mycroft~\cite{garbe2018hamilton}, and $k\ge 3$ and $\ell<k/2$ by Han and Zhao~\cite{han2015minimum}. We refer the reader to the surveys~\cite{kuhn2014hamilton,zhao2016recent,rodl2010dirac} for more detailed discussions.

A drawback of a minimum co-degree condition is that it tends to be a fairly strong requirement. For instance, if we start with a complete $k$-graph and remove all edges that contain a fixed pair of vertices, it is immediate that the resulting hypergraph has minimum co-degree equal to zero. However, if $k\ge 3$, it is not hard to see that even fairly small hypergraphs of this form are quite dense, and trivially contain Hamilton cycles. In fact, there are several natural hypergraph classes that are quite dense, but have minimum co-degree equal to zero (multipartite hypergraphs, for instance), and hence the previous theorems are not immediately applicable. This motivates the notion of the \emph{minimum supported co-degree} of a $k$-graph $G$, which we denote $\delta^*(G)$, and define as the maximum integer $d$ such that every $(k-1)$-set that is contained in at least one edge is contained in at least $d$ edges. A minimum supported co-degree requirement immediately handles the previous issues of zero co-degree sets, although it does not rule out isolated vertices, and hence we will usually impose the (fairly weak) requirement of all vertices being non-isolated.

The problem of determining the optimal minimum supported co-degree for $k$-uniform $\ell$-cycles was initially suggested by Halfpap and Magnan in~\cite[Section 5]{halfpap2024positive}, and reiterated by Illingworth, Lang, M\"uyesser, Parczyk and Sgueglia~\cite{illingworth2025spanning} for the special case of tight cycles. This question was recently tackled by Mycroft and Z\'arate-Guer\'en~\cite{mycroft2025positive}, who prove an asymptotically optimal result for all $k\ge 3$ and $\ell\in [k-1]$.

Our main result provides an improved sufficient minimum supported co-degree condition for $k$-uniform $\ell$-cycles for all $k\ge 3$ and $\ell\in [k-1]$ except $(k,\ell)=(3,1)$, which is the best possible for infinitely many values of $k$ and $\ell$, and is off by at most $1$ for all values of $k$ and $\ell$. We discuss the optimality of our result in \Cref{sec:lower bound}. Importantly, we emphasise that our results are (essentially) exact and not asymptotic, a rather stark difference from the minimum co-degree version of the problem where very few cases have been resolved exactly.

\begin{ex}\label{ex:lower bd weak}
 Let $k\ge 3$, $\ell \in [k-1]$ and $t = \floor{\frac{k}{k-\ell}}(k - \ell)$. Let $n$ be chosen such that $k-\ell$ divides $n$. Now, consider an $n$-vertex $k$-graph $G$ with a partition $A\sqcup B$ of its vertex set such that $|A| = \floor{n/t} + 1$ and $E(G)$ consists of all $k$-sets $e \in \binom{V(G)}{k}$ that satisfy $|e \cap A| \le 1$ (and so $A$ is a strong independent set). Suppose $S$ is any supported $(k-1)$-set. If $|S\cap A| = 0$, then $S$ supports all vertices outside itself, and so $d_G^1(S) = n-k+1$. If $|S\cap A| = 1$, then it supports all vertices in $B\setminus S$, meaning that $d_G^1(S) = n - \floor{n/t} - (k-2)$, and this is equal to $\delta^*(G)$ since there is no supported set with $|S\cap A| \ge 2$. The following observation shows that $G$ cannot contains a Hamilton $\ell$-cycle.
 \end{ex}

\begin{ex}\label{ex:lower bd strong}
 Next, we provide a specialised modification of the previous example. Let $k\ge 3$ be odd, $\ell = (k-1)/2$, and consequently $t = \floor{\frac{k}{k-\ell}}(k - \ell) = (k+1)/2 = \ell+1 = k - \ell$. Let $n$ be such that $t$ divides $n$ and $n/t+1$ is even. Yet again, we consider an $n$-vertex $k$-graph $G$ with a partition $A\sqcup B$ of its vertex set with $|A| = n/t + 1$. Write $A = \{a_1, \dots, a_{|A|}\}$ and let $E(G)$ consist of all $e\in \binom{V(G)}{k}$ such that $e\cap A \subseteq \{a_{2i-1}, a_{2i}\}$ for some $i\in [|A|/2]$ (so the graph $\partial^2[A]$ is a perfect matching). Clearly $|e\cap A| \le 2$ for all $e \in E(G)$. If $S$ is any supported $(k-1)$-set, then, similar to the previous example, it is not too hard to see that
 \[d_G^1(S) =
 \left\{
 \begin{array}{ll}
 n - (k-1) & \textnormal{if} \ |e\cap A|=0,\\
 |B| - (k - 2) +1 = |B| - (k-3) & \textnormal{if} \ |e\cap A|=1,\\
 |B| - (k - 3) & \textnormal{if} \ |e\cap A|=2,\\
 \end{array}
 \right.
 \]
 and so $\delta^*(G) = |B| - (k - 3) = n - n/t - (k - 2)$. We now argue that $G$ cannot contain a Hamilton $\ell$-cycle, which shows that the bound in \cref{thm:main} cannot be improved.

\begin{restatable}{thm}{thmHamiltonCycleNonExtremal}
 \label{thm:Hamilton cycle non-extremal}
 Let $k \ge 3$ and $\ell \in [k-1]$ be such that $(k,\ell) \neq (3,1)$, and set $t = \floor{\frac{k}{k-\ell}}(k-\ell)$. Let $1/n \ll \eps \ll \mu \ll 1/k$ such that $n$ is divisible by $k-\ell$. If $G$ is an $n$-vertex $k$-graph with no isolated vertices and $\delta^*(G)\geq (1-1/t-\eps)n$ such that every set of $\floor{n/t}$ vertices of $G$ contains at least $\mu n^2$ supported pairs, then $G$ has a Hamilton $\ell$-cycle.
 \end{restatable}

\begin{restatable}{thm}{thmHamiltonCycleExtremal}
 \label{thm:Hamilton cycle extremal}
 Let $k \ge 3$ and $\ell \in [k-1]$ be such that $(k,\ell) \neq (3,1)$, and set $t = \floor{\frac{k}{k-\ell}}(k-\ell)$.
 Let $1/n\ll \eps \ll 1/k\leq 1/3$ be such that $n$ is divisible by $k-\ell$.
 Suppose $G$ is an $n$-vertex $k$-graph with no isolated vertices and $\delta^*(G)\geq (1-1/t)n-(k-3)$. If there is a subset $A \subseteq V(G)$ with $|A|=\floor{n/t}$ that contains at most $\eps n^2$ supported pairs, then $G$ has a Hamilton $\ell$-cycle.
 \end{restatable}

\begin{prop} \label{prop:supported-l-paths}
 Let $G$ be a $k$-graph, where $k \ge 3$, $\ell \in [k-1]$ and $t = \floor{\frac{k}{k-\ell}}(k-\ell)$.
 \begin{enumerate}[label = \rm(\roman*)]
 \item \label{itm:extended-path-1}
 Let $P$ and $Q$ be two sequences of vertices in $G$ of length at least $t$, both of which support an $\ell$-path, and whose vertices are disjoint. If the sequence consisting of the last $t$ vertices in $P$ followed by the first $t-1$ vertices in $Q$ is a tight path, then $PQ$ supports an $\ell$-path.
 \item \label{itm:extended-path-2}
 Let $P_1, \ldots, P_m$ be sequences of vertices in $G$ of length at least $t$, each of which supports an $\ell$-path, and whose vertices are pairwise disjoint. If $P_iP_{i+1}$ supports an $\ell$-path for every $i \in [m]$ (addition of indices taken modulo $m$) then $P_1 \ldots P_m$ is an $\ell$-cycle.
 \item \label{itm:extended-path-3}
 Let $P = v_0 \ldots v_r$ be a sequence of distinct vertices where $r$ is divisible by $t$. If every subsequence of at most $k$ consecutive vertices that contains at most one vertex $v_i$ with $i$ divisible by $t$ is supported by an edge, then $P$ supports an extended $\ell$-path.
 \item \label{itm:extended-path-4}
 Let $P$ be a sequence of distinct vertices of length $r$ such that $r = -1 \pmod{k-\ell}$ and $r \ge t-1$, and let $u,v$ be two vertices that do not appear in $P$, such that $uP$ and $Pv$ are tight paths (or supported sets if $r+1 \le k$). Then $uPv$ supports an extended $\ell$-path.
 \end{enumerate}
 \end{prop}
 \begin{proof}
 For \ref{itm:extended-path-1}, say that the length of $P$ is $p$ and that of $Q$ is $q$. Let $PQ = v_1 \dots v_{p+q}$, where $P = v_1 \dots v_p$ and $Q = v_{p+1} \dots v_{p+q}$. Since $P$ and $Q$ both support $\ell$-paths, we know that $p$ and $q$ are divisible by $k-\ell$, implying the same is true of $p+q$. Furthermore, as $q\ge t$, we also know that the last $t$ vertices of $PQ$ correspond to the last $t$ vertices of $Q$, and hence are supported. All that is left to show is that $v_{i(k-\ell) + 1} \ldots v_{i(k-\ell)+k}$ is an edge for all $i\le \frac{p + q - t}{k-\ell} - 1$. Since $P$ and $Q$ support $\ell$-paths (and in particular their length is divisible by $k-\ell$), we only need to restrict our attention to $i \in [\frac{p - t}{k - \ell}, \frac{p}{k - \ell}-1]$. For this, it would suffice to argue that the subsequence $v_{i_1} \dots v_{i_2}$ forms a tight path, where $i_1 = p - t + 1$ and $i_2 = p + \ell$. By assumption, we know that $v_{p - t + 1} \dots v_{p+t-1}$ is a tight path, and we know $t-1\ge \ell$ (see \Cref{obs:useful t info}), which completes the proof.

These results contribute to the broader research theme of finding the best possible minimum supported co-degree to ensure a specified spanning substructure, an area which poses some tantalising questions. Our original motivation for working on this problem arose from Illingworth, Lang, M\"uyesser, Parczyk and Sgueglia~\cite[Conjecture $1.4$]{illingworth2025spanning}, who conjectured that $\delta^*(G) \ge (1-1/k)n$ guarantees a tight Hamilton cycle. The main focus of their paper is the minimum supported co-degree required to ensure a spanning $(k-1)$-sphere -- a subset of edges in a $k$-graph that induces a homogeneous simplicial complex homeomorphic to the $(k-1)$-sphere $\mathbb{S}^{k-1}$, and whose $0$-skeleton spans the entire vertex set. They tackle the following conjecture of Georgakopoulos, Haslegrave, Montgomery and Narayanan~\cite{georgakopoulos2022spanning} (paraphrased).
 \begin{conj}\label{conj:spheres}
 Every $k$-graph $G$ of order $n>k \ge 2$ with $\delta^*(G) \ge n/2$ and with no isolated vertices contains a spanning copy of $\mathbb{S}^{k-1}$.\footnote{In the original phrasing of the conjecture $G$ is required to be tightly connected -- meaning that any two edges can be joined by a tight walk -- but this follows from the other assumptions.}
 \end{conj}
 The chief result in~\cite{georgakopoulos2022spanning} provides an asymptotically optimal minimum co-degree condition for a $3$-graph to contain a vertex-spanning copy of any \emph{surface}, meaning an arbitrary connected, closed $2$-manifold. In particular, this addresses the minimum co-degree necessary to guarantee a spanning $2$-sphere in a $3$-graph.
 \begin{thm}[{\cite[Theorem 1.4]{georgakopoulos2022spanning}}]\label{thm:spheres-codegree}
 Let $\cS$ be an arbitrary surface and $\eps>0$. Then any sufficiently large $n$-vertex $3$-graph with $\delta(G) \ge n/3 + \eps n$ contains a spanning copy of $\cS$. Moreover, for any $n \in \mathbb{N}$, there exists an $n$-vertex $3$-graph $H$ with $\delta(H) = \floor{\frac{n}{3}} - 1$ such that there are at most $2\ceil{\frac{n}{3}}$ vertices in the $0$-skeleton of a copy of any surface in $H$.
 \end{thm}

This improves the aforementioned asymptotic results of Mycroft and Z\'arate-Guer\'en~\cite{mycroft2025positive}, who showed that $\delta^*(G) \ge (1 - 1/t)n + o(n)$ suffices. Our methods are largely different from theirs. Our analysis splits into two regimes: the extremal and non-extremal (see \Cref{sec:stability} for details). The extremal regime does not feature in \cite{mycroft2025positive} at all, and for this we use ad-hoc structural analysis with lemmas about random matchings in bipartite graphs and about finding almost spanning subhypergraphs with large minimum supported codegree in almost complete hypergraphs. For the non-extremal case, while Mycroft and Z\'arate-Guer\'en rely on a version of the regularity lemma for hypergraphs (called the weak regularity lemma) along with the absorption method, our approach stems from Lang's breakthrough work \cite{lang2023tiling} on hypergraph tilings, which allows us to completely avoid the regularity lemma and greatly simplifies the use of the absorption method. For the latter part, however, we use a novel idea from \cite{mycroft2025positive} about weighted fractional matchings that, coupled with Farkas' linear-algebraic lemma, allows one to find spanning $\ell$-cycles in certain $k$-partite $k$-graphs, which is one of the steps in our proof.

We point out that Illingworth, Lang, M\"uyesser, Parczyk and Sgueglia~\cite{illingworth2025spanning} conjectured that for all $k\ge 3$, any sufficiently large $k$-graph with $\delta^*(G) \ge (1-1/k)n$ contains a tight Hamilton cycle, which is a special case of \cref{thm:main} (up to the $k-3$ term). Observe that if $k$ divides $n$, then a tight Hamilton cycle in $G$ will contain a perfect matching. Thus, our result implies that $\delta^*(G) \ge (1 - 1/k)n - (k-3)$ ensures a perfect matching, which recovers a result of Mycroft and Z\'arate-Guer\'en~\cite{mycroft2025matching} up to an additive constant of one, who show that a bound of $(1 - 1/k)n - (k - 2)$ suffices.

As alluded to above, the bound in \cref{thm:main} is off by one in most cases. In fact, in many cases (namely whenever $t \ge \ell+2$ or when $\ell = k-1$, that is, we are seeking a tight Hamilton cycle), our techniques can be adapted to prove \cref{thm:main} with the improved bound of $\delta^*(G) \ge (1 - 1/t)n - (k - 2)$ instead (which, for instance, would immediately imply the previously discussed optimal perfect matching bound).
However, since this will require some technical modifications, we choose to present a single unified proof for $\delta^*(G) \ge (1 - 1/t)n - (k - 3)$ that will work for all cases instead. We point out the part of our proof that requires the exact degree condition in \Cref{rem:exact degree}, and briefly discuss how to suitably alter our proof and remove the extra one in the bound in all relevant cases in \Cref{sec:improving the bound}.

Finally, we remark that in a recent personal communication, Mycroft and Z\'arate-Guer\'en informed us that they are preparing a manuscript where they prove that any sufficiently large $3$-graph with no isolated vertices and $\delta^*(G) \ge n/2$ contains a loose Hamilton cycle. This improves the previous asymptotic bound of $n/2 + o(n)$ established by them~\cite{mycroft2025positive} and Halfpap and Magnan~\cite{halfpap2024positive} (which our methods can recover as well) and extends \cref{thm:main} to include $(k,\ell) = (3,1)$, the only case we do not handle.

\paragraph{Organisation of the paper.} In \Cref{sec:overview}, we first provide lower bound constructions to prove the optimality of \cref{thm:main}. We then split the proof of \cref{thm:main} into two complementary cases and handle them separately in \cref{thm:Hamilton cycle non-extremal,thm:Hamilton cycle extremal}. We also provide brief proof overviews for both these theorems in \Cref{sec:proof-overviews}. Then in \Cref{sec:prelims} we introduce some notation and definitions and provide some basic tools.

\Crefrange{sec:non-extremal}{sec:proof Hamilton paths} are dedicated to proving \cref{thm:Hamilton cycle non-extremal}, and \Crefrange{sec:extremal hypergraphs}{sec:Hamilton cycle extremal} prove \cref{thm:Hamilton cycle extremal}. These two parts are treated largely independently, and can be read as such. We conclude with some open problems in \Cref{sec:conclusion}.

The proof of \cref{thm:main} is split into two regimes: the \emph{extremal} and \emph{non-extremal} cases (see \Cref{sec:stability}). The two cases are then handled separately in \cref{thm:Hamilton cycle non-extremal,thm:Hamilton cycle extremal}.

In the extremal regime, the authors use “ad-hoc structural analysis” together with “lemmas about random matchings in bipartite graphs” and lemmas “about finding almost spanning subhypergraphs with large minimum supported codegree in almost complete hypergraphs.”

In the non-extremal regime, the approach “stems from Lang's breakthrough work \cite{lang2023tiling} on hypergraph tilings,” which “allows us to completely avoid the regularity lemma and greatly simplifies the use of the absorption method.” In addition, they use “a novel idea … about weighted fractional matchings that, coupled with Farkas' linear-algebraic lemma, allows one to find spanning $\ell$-cycles in certain $k$-partite $k$-graphs,” described as “one of the steps in our proof.”

They also note that the degree bound is “off by one in most cases,” and indicate they “point out the part of our proof that requires the exact degree condition in \Cref{rem:exact degree},” and “briefly discuss how to suitably alter our proof and remove the extra one … in \Cref{sec:improving the bound}.”

The proof of the main theorem is split into two regimes: the \emph{extremal} and \emph{non-extremal} cases (see \Cref{sec:stability}). The two cases are then handled separately in \cref{thm:Hamilton cycle non-extremal,thm:Hamilton cycle extremal}.

In the extremal regime, the authors use “ad-hoc structural analysis” together with “lemmas about random matchings in bipartite graphs” and lemmas “about finding almost spanning subhypergraphs with large minimum supported codegree in almost complete hypergraphs.”

In the non-extremal regime, the approach “stems from Lang's breakthrough work Lang, \emph{tiling} (2023) on hypergraph tilings,” which “allows us to completely avoid the regularity lemma and greatly simplifies the use of the absorption method.” In addition, they use “a novel idea … about weighted fractional matchings that, coupled with Farkas' linear-algebraic lemma, allows one to find spanning $\ell$-cycles in certain $k$-partite $k$-graphs,” described as “one of the steps in our proof.”

They also note that the degree bound is “off by one in most cases,” and indicate they “point out the part of our proof that requires the exact degree condition in the following remark. \begin{rem}\label{rem:exact degree}
    A critical point to note is that this is the only part of the entire proof that utilises the exact bound $\delta^*(G) \ge n - \floor{\frac{n}{t}} - (k - 3)$ via \cref{obs:min 1-degree,lem:disjoint edges cherries}. As mentioned in \Cref{sec:intro}, we can prove \cref{thm:Hamilton cycle extremal} and consequently \cref{thm:main} with the weaker bound of $\delta^*(G) \ge n - \floor{\frac{n}{t}} - (k - 2)$ instead, for most values of $k$ and $\ell$ (namely when $t \ge \ell+2$ or when $\ell = k-1$) which is achieved by modifying \cref{lem:disjoint edges cherries} to potentially allow for an additional cherry or some edges. We discuss how to suitably adapt the rest of the proof in \Cref{sec:improving the bound}.
\end{rem}” and “briefly discuss how to suitably alter our proof and remove the extra one … in \Cref{sec:improving the bound}.”

\label{thm:main}
	Let $1 \le \ell \le k-1$ be such that $k \ge 3$ and $(k,\ell) \neq (3,1)$, let $t = \floor{\frac{k}{k-\ell}}(k - \ell)$, and let $n$ be sufficiently large and divisible by $k-\ell$.
	Then every $k$-uniform $n$-vertex hypergraph without isolated vertices and having minimum supported co-degree at least $(1 - 1/t)n-(k-3)$ will contain a Hamilton $\ell$-cycle.,

Question:
Let $1\le \ell\le k-1$ with $k\ge 3$ and $(k,\ell)\neq (3,1)$, and define
\[
t=\left\lfloor \frac{k}{k-\ell}\right\rfloor (k-\ell).
\]
Assume $n$ is sufficiently large and divisible by $k-\ell$. Let $G$ be a $k$-uniform hypergraph on $n$ vertices with no isolated vertices. Define the minimum supported co-degree $\delta^*(G)$ to be the maximum integer $d$ such that every $(k-1)$-subset of vertices that is contained in at least one edge of $G$ is contained in at least $d$ edges of $G$. Suppose that
\[
\delta^*(G)\ge \left(1-\frac1t\right)n-(k-3).
\]
A Hamilton $\ell$-cycle means a spanning $\ell$-cycle, i.e. a cyclic ordering of all $n$ vertices such that each edge consists of $k$ consecutive vertices in the ordering and consecutive edges intersect in exactly $\ell$ vertices. Under these hypotheses, which conclusion holds?

The hypergraph $G$ contains a Hamilton $\ell$-cycle.

Choices:
B. The hypergraph $G$ contains a Hamilton $\ell$-cycle provided, in addition, that $k-\ell$ divides $k$.
C. The hypergraph $G$ contains an $\ell$-cycle spanning all but at most $k-\ell$ vertices.
D. There exists a constant $n_0=n_0(k,\ell)$ such that for every integer $n\ge n_0$ divisible by $k-\ell$, every $k$-uniform $n$-vertex hypergraph $G$ without isolated vertices and with minimum supported co-degree at least $\left(1-\frac1t\right)n-(k-2)$ contains a Hamilton $\ell$-cycle.
E. The hypergraph $G$ contains a Hamilton $\ell$-path.

unnecessary divisibility restriction $k-\ell\mid k$

exact degree offset $(k-3)$ replaced by $(k-2)$ uniformly

cycle conclusion weakened to path despite spanning cyclic closure being the hard step

The stem does not explicitly reveal the correct option. It states the hypotheses and asks for the resulting conclusion, without directly naming the theorem's conclusion in advance.

This is essentially a direct theorem-recall item: the stem gives the full hypothesis set and the correct choice restates the theorem's exact conclusion.

Some precision is needed to distinguish the exact conclusion from nearby variants such as a weaker spanning claim, an added unnecessary divisibility condition, or a shifted degree bound. However, the item mainly tests recall of the exact theorem rather than substantial generative reasoning.

The distractors are plausible and mathematically targeted: one adds an unnecessary condition, one weakens the conclusion, one perturbs the threshold, and one replaces cycle by path. These reflect realistic failure modes and are clearly distinct.

A mathematically precise MCQ with strong distractors, but it is largely a theorem restatement and therefore only moderately effective at testing genuine generative reasoning.


\label{THM:main}
Let $u(t,x)$ be solutions to \eqref{weakNLS1} with initial data $u_0^\omega$ given in \eqref{random_data}.
For any $t = \mathcal O ( \varepsilon^{-1} |\log \varepsilon|)$ and $z_0 > 0$, we have that
\begin{align}
\label{THM_LDP4}
\lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \|u(t,x) \|_{\mathcal{F} L^1 (\mathbb{T})} > z_0\varepsilon^{-1/2} \Big) = - \frac{z_0^2}{\sum_k c_k^2},
\end{align}

\noindent
holds if and only if $(c_k) \in \ell^1$.

\begin{proposition}
\label{PROP:linear2}
Consider the linear Schr\"odinger equation on the torus $\T = [0,2\pi]$ as in \eqref{linear}, with random initial data $u_0^\o$ given by \eqref{random_data}.
Then,
\begin{align}
\label{LDP_linear2}
\lim_{\eps\to 0^+} \eps \log \PP \big( \|u(t)\|_{L^\infty (\T)} \ge z_0 \eps^{-1/2} \big) = - \frac{z_0^2}{\sum_{k \in \Z} c_k^2},
\end{align}

\noi
provided $(c_k) \in \l^1$.
\end{proposition}

\begin{theorem}
\label{THM:main}
Let $u(t,x)$ be solutions to \eqref{weakNLS1} with initial data $u_0^\o$ given in \eqref{random_data}.
For any $t = \mathcal O ( \eps^{-1} |\log \eps|)$ and $z_0 > 0$, we have that
\begin{align}
\label{THM_LDP4}
\lim_{\eps \to 0^+} \eps \log \PP \Big( \|u(t,x) \|_{\FL^1 (\T)} > z_0\eps^{-1/2} \Big) = - \frac{z_0^2}{\sum_k c_k^2},
\end{align}

\noi
holds if and only if $(c_k) \in \l^1$.
\end{theorem}

In this paper, we consider the following Cauchy problem of the one-dimensional cubic nonlinear Schr\"odinger equation (NLS) on the torus $\mathbb{T}= \mathbb{R} / (2 \pi \mathbb{Z})$:
\begin{equation}
\label{NLS}
\begin{cases}
i\pa_t u + \partial_x^2 u = \pm \,|u|^2\, u,\\
u(t,x)|_{t=0}=u_0.
\end{cases}
\end{equation}

\noindent
The equation \eqref{NLS} is fundamental in theoretical physics and applied mathematics, for describing the evolution of wave packets in various physical systems, such as nonlinear optics, fluids, and plasmas; see \cite{SS99} for a review.
Bourgain \cite{BO93} proved \eqref{NLS} is deterministically global well-posed in $L^2 (\mathbb{T})$.
The probabilistic study of \eqref{NLS} has emerging importance since the seminal work of Bourgain on invariant measures \cite{BO94,BO96}.

More recently, probabilistic analyses of \eqref{NLS} have also found applications in oceanography, particularly in modelling rare extreme events such as the formation of {\it rogue waves} in deep-sea dynamics~\cite{DGV18,DGV19,GGKS21}. In particular, Garrido et al.~\cite{GGKS21} established a large deviations principle (LDP) describing the probability of observing a wave of unusually large height in the weakly nonlinear regime. Their result relied on a strong exponential decay assumption on the Fourier coefficients of the random initial data, which plays a crucial role in their analysis.
In this paper, we improve the exponential decay condition on coefficients of initial data in \cite{GGKS21} to a $\ell^1$ decay condition.

We consider
random initial data of the form:
\begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\mathbb{Z}} c_k g_k e^{ikx},
\end{equation}
where $c_k \in \mathcal{C}$ and $\{g_k\}_{k\in\mathbb{Z}}$ are independent, identically distributed, complex Gaussian random variables with $\mathbb{E} g_k=0$, $\mathbb{E} g_k g_j=0$ and $\mathbb{E} g_k \overline{g}_j =\delta_{kj}$ for $k,j\in\mathbb{Z}$.
Without loss of any generality, we only consider nonnegative real coefficients $c_k \ge 0$, due to the rotation invariance of complex Gaussian random variables.

The study of the large deviation principle for solutions to \eqref{NLS} was initiated by \cite{DGV18,DGV19}, where the authors conjectured that the existence of an LDP can be used to predict the formation of rogue waves.
In particular, they considered \eqref{NLS} with random initial data $u_0^N$ of the form
\begin{align}
\label{random1}
u_0^N (x) = \sum_{|k| \le N} \theta_k e^{ikx},
\end{align}

\noindent
where the initial data is parametrized by a random vector $\theta = (\theta_k)_{|k| \le N} \in \mathbb C^{2N+1}$.
Here $(\theta_k)$ are independent complex Gaussian random variables with $\mathbb{E} \theta_k = \mathbb{E} \theta_k^2 = 0$ and $\mathbb{E} |\theta_k|^2 = c_k^2$, for some fast-decaying $c_k > 0$.
Then, the set of initial data that generates a rouge wave of height at least $z>0$ at time $t>0$ is given by
\begin{align}
\label{Dtz}
\mathcal D (t,z) : = \bigg\{ (\theta_k)_{|k| \le N} \in \mathcal{C}^{2N+1} \Big| \sup_x |u (t,x| \theta)| > z\bigg\},
\end{align}

\noindent
where $u$ is the solution to NLS \eqref{NLS} with the truncated initial data \eqref{random1}.
Dematteis et al, then proposed a theoretical framework of a large deviations principle (LDP) to quantify the likelihood of $\mathcal D (t,z)$.
In particular, consider the minimization problem
\begin{align}\label{minp}
\theta^* (z) = {\rm argmin}\,\, \mathcal I (\theta)
\end{align}

\noindent
where
\[
\mathcal I (\theta) = \max_{y \in \mathcal{C}^{2N+1}} [\langle y,\theta \rangle - S(y)] \quad \textup{ and } \quad S(y) = \log \mathbb{E} e^{\langle y, \theta \rangle}.
\]

\noindent
Then Dematteis et al. claim that for $t >0$
\begin{align}
\label{LDP1}
\log \mathbb{P} (\mathcal D(t,z)) = - \mathcal I (\theta^* (z)) + o(1),
\end{align}

\noindent
as $z \to \infty$, provided
the minimization problem \eqref{minp} has a unique solution.

Garrido et al.~\cite{GGKS21} rigorously proved the conjectured LDP~\eqref{LDP1} locally in time for the nonlinear Schr\"odinger equation with weak nonlinearity.
A major difficulty in analysing the LDP problem~\eqref{LDP1} lies in estimating the gradient
$\nabla_{\theta} \sup_{x} |u(t,x|\theta)|$
and in verifying the convexity of the set $\mathcal{D}(t,z)$,
both of which are extremely challenging to establish in practice.
To overcome these obstacles, Garrido et al.~\cite{GGKS21} introduced a new formulation and considered the following LDP problem:
\begin{align}
\label{LDP2}
\log \mathbb{P} \Big( \sup_{x} |u(t,x)| > z \Big) = - I(z) + o(1), \quad z \to \infty,
\end{align}
where $I$ denotes the corresponding rate function.
The advantage of this reformulation is that it circumvents the need to solve the minimization problem in~\eqref{minp}.
Within this framework, they established~\eqref{LDP2} for initial data of the form~\eqref{random1} with infinitely many Fourier modes ($N = \infty$) satisfying a \emph{strong decay condition} on the coefficients.

In particular, consider the weakly nonlinear Schr\"odinger equation on the circle $\mathbb{T} = [0,2\pi]$:
\begin{align}
\label{weakNLS1}
i\pa_t u + \Delta u = \varepsilon^{2} |u|^2 u.
\end{align}
It was proved in~\cite{GGKS21} that
\begin{align}
\label{THM_LDP3}
\lim_{\varepsilon \to 0^+}
\varepsilon \log \mathbb{P} \Big( \sup_{x} |u(t,x)| > z_0 \varepsilon^{-1/2} \Big)
= - \frac{z_0^2}{\sum_k c_k^2},
\end{align}
for $0 < t \lesssim \varepsilon^{-1}$ and $z_0 > 0$,
where $u(t,x)$ denotes the solution to~\eqref{weakNLS1} with random initial data $u_0^\omega$ given by~\eqref{random_data} and coefficients satisfying
\begin{align}\label{decay}
c_k = a e^{-b|k|}
\quad \text{or} \quad
c_k = a e^{-b|k|^2},
\end{align}
for some fixed $a,b > 0$.

The proof of~\eqref{THM_LDP3} is based on the G\"artner--Ellis theorem combined with resonant approximation techniques.
In their analysis, the rapid decay condition~\eqref{decay} plays a crucial role in both the linear and nonlinear estimates.
They further observed that alternative families of coefficients $(c_k)$ might also yield~\eqref{LDP2}, and posed an \textit{open question} concerning the identification of the optimal decay condition under which~\eqref{LDP2} holds; see~\cite[Remark~1.3]{GGKS21}.
One of the principal aims of the present work is to address this question.

To this purpose, we consider a LDP under Fourier-Lebesgue norm, which is stronger than the $L^\infty (\mathbb{T})$ norm used in \eqref{THM_LDP3}.
Our conventions for the Fourier transform are as follows:
\[
\widehat f (n) = \frac1{{2\pi}} \int_0^{2\pi} f(x) e^{ix n}dx \quad \textup{ and } \quad f(x) = \sum_{n\in \mathbb{Z}} \widehat f(n) e^{ixn}
\]

\noindent
for functions on the circle $\mathbb{T} = \mathbb{R}/(2\pi \mathbb{Z})$.
With the above notations, we define the Fourier-Lebesgue norm by
\[
\begin{split}
\|f\|_{\mathcal{F} L^1 (\mathbb{T})} = \sum_{n \in \mathbb{Z}} |\widehat f(n)|.
\end{split}
\]

\noindent
It is easy to see that
\[
\sup_{x \in \mathbb{T}} |f(x)| \le \|f\|_{\mathcal{F} L^1 (\mathbb{T})}.
\]

\noindent
We note that $L^\infty$ and $\mathcal{F} L^1$ scales the same, and furthermore, the LDPs under $L^\infty$ and $\mathcal{F} L^1$ share the same rate function.
In particular, for $t \lesssim \varepsilon^{-1}$, we have
\begin{align}\label{obser}
\lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \sup_{x} |u(t,x) | > z_0\varepsilon^{-1/2} \Big) = \lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \|u(t,x) \|_{\mathcal{F} L^1} > z_0\varepsilon^{-1/2} \Big),
\end{align}

\noindent
where $u(t,x)$ is the solution to \eqref{weakNLS1} with initial data \eqref{random1} in $\mathcal{F}L^1$.
This motivates us to consider the LDP under the $\mathcal{F} L^1$ norm.
We are ready to state our first main result.

We consider
random initial data of the form:
\begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\mathbb{Z}} c_k g_k e^{ikx},
\end{equation}
where $c_k \in \mathcal{C}$ and $\{g_k\}_{k\in\mathbb{Z}}$ are independent, identically distributed, complex Gaussian random variables with $\mathbb{E} g_k=0$, $\mathbb{E} g_k g_j=0$ and $\mathbb{E} g_k \overline{g}_j =\delta_{kj}$ for $k,j\in\mathbb{Z}$.
Without loss of any generality, we only consider nonnegative real coefficients $c_k \ge 0$, due to the rotation invariance of complex Gaussian random variables.

\noindent
where the initial data is parametrized by a random vector $\theta = (\theta_k)_{|k| \le N} \in \mathbb C^{2N+1}$.
Here $(\theta_k)$ are independent complex Gaussian random variables with $\mathbb{E} \theta_k = \mathbb{E} \theta_k^2 = 0$ and $\mathbb{E} |\theta_k|^2 = c_k^2$, for some fast-decaying $c_k > 0$.
Then, the set of initial data that generates a rouge wave of height at least $z>0$ at time $t>0$ is given by
\begin{align}
\label{Dtz}
\mathcal D (t,z) : = \bigg\{ (\theta_k)_{|k| \le N} \in \mathcal{C}^{2N+1} \Big| \sup_x |u (t,x| \theta)| > z\bigg\},
\end{align}

Garrido et al.~\cite{GGKS21} rigorously proved the conjectured LDP~\eqref{LDP1} locally in time for the nonlinear Schr\"odinger equation with weak nonlinearity.
A major difficulty in analysing the LDP problem~\eqref{LDP1} lies in estimating the gradient
$\nabla_{\theta} \sup_{x} |u(t,x|\theta)|$
and in verifying the convexity of the set $\mathcal{D}(t,z)$,
both of which are extremely challenging to establish in practice.
To overcome these obstacles, Garrido et al.~\cite{GGKS21} introduced a new formulation and considered the following LDP problem:
\begin{align}
\label{LDP2}
\log \mathbb{P} \Big( \sup_{x} |u(t,x)| > z \Big) = - I(z) + o(1), \quad z \to \infty,
\end{align}
where $I$ denotes the corresponding rate function.
The advantage of this reformulation is that it circumvents the need to solve the minimization problem in~\eqref{minp}.
Within this framework, they established~\eqref{LDP2} for initial data of the form~\eqref{random1} with infinitely many Fourier modes ($N = \infty$) satisfying a \emph{strong decay condition} on the coefficients.

In particular, consider the weakly nonlinear Schr\"odinger equation on the circle $\mathbb{T} = [0,2\pi]$:
\begin{align}
\label{weakNLS1}
i\pa_t u + \Delta u = \varepsilon^{2} |u|^2 u.
\end{align}
It was proved in~\cite{GGKS21} that
\begin{align}
\label{THM_LDP3}
\lim_{\varepsilon \to 0^+}
\varepsilon \log \mathbb{P} \Big( \sup_{x} |u(t,x)| > z_0 \varepsilon^{-1/2} \Big)
= - \frac{z_0^2}{\sum_k c_k^2},
\end{align}
for $0 < t \lesssim \varepsilon^{-1}$ and $z_0 > 0$,
where $u(t,x)$ denotes the solution to~\eqref{weakNLS1} with random initial data $u_0^\omega$ given by~\eqref{random_data} and coefficients satisfying
\begin{align}\label{decay}
c_k = a e^{-b|k|}
\quad \text{or} \quad
c_k = a e^{-b|k|^2},
\end{align}
for some fixed $a,b > 0$.

\noindent
We note that $L^\infty$ and $\mathcal{F} L^1$ scales the same, and furthermore, the LDPs under $L^\infty$ and $\mathcal{F} L^1$ share the same rate function.
In particular, for $t \lesssim \varepsilon^{-1}$, we have
\begin{align}\label{obser}
\lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \sup_{x} |u(t,x) | > z_0\varepsilon^{-1/2} \Big) = \lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \|u(t,x) \|_{\mathcal{F} L^1} > z_0\varepsilon^{-1/2} \Big),
\end{align}

\noindent
where $u(t,x)$ is the solution to \eqref{weakNLS1} with initial data \eqref{random1} in $\mathcal{F}L^1$.
This motivates us to consider the LDP under the $\mathcal{F} L^1$ norm.
We are ready to state our first main result.

\begin{align}
\label{LDP1}
\log \PP (\mathcal D(t,z)) = - \mathcal I (\theta^* (z)) + o(1),
\end{align}

\begin{align}
\label{LDP2}
\log \PP \Big( \sup_{x} |u(t,x)| > z \Big) = - I(z) + o(1), \quad z \to \infty,
\end{align}

\begin{align}\label{minp}
\theta^* (z) = {\rm argmin}\,\, \mathcal I (\theta)
\end{align}

\begin{align}
\label{random1}
u_0^N (x) = \sum_{|k| \le N} \theta_k e^{ikx},
\end{align}

\begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\Z} c_k g_k e^{ikx},
\end{equation}

In particular, consider the weakly nonlinear Schr\"odinger equation on the circle $\T = [0,2\pi]$:
\begin{align}
\label{weakNLS1}
i\pa_t u + \Delta u = \varepsilon^{2} |u|^2 u.
\end{align}
It was proved in~\cite{GGKS21} that
\begin{align}
\label{THM_LDP3}
\lim_{\varepsilon \to 0^+}
\varepsilon \log \PP \Big( \sup_{x} |u(t,x)| > z_0 \varepsilon^{-1/2} \Big)
= - \frac{z_0^2}{\sum_k c_k^2},
\end{align}
for $0 < t \lesssim \varepsilon^{-1}$ and $z_0 > 0$,
where $u(t,x)$ denotes the solution to~\eqref{weakNLS1} with random initial data $u_0^\omega$ given by~\eqref{random_data} and coefficients satisfying
\begin{align}\label{decay}
c_k = a e^{-b|k|}
\quad \text{or} \quad
c_k = a e^{-b|k|^2},
\end{align}
for some fixed $a,b > 0$.

\noi
We note that $L^\infty$ and $\FL^1$ scales the same, and furthermore, the LDPs under $L^\infty$ and $\FL^1$ share the same rate function.
In particular, for $t \les \eps^{-1}$, we have
\begin{align}\label{obser}
\lim_{\eps \to 0^+} \eps \log \PP \Big( \sup_{x} |u(t,x) | > z_0\eps^{-1/2} \Big) = \lim_{\eps \to 0^+} \eps \log \PP \Big( \|u(t,x) \|_{\FL^1} > z_0\eps^{-1/2} \Big),
\end{align}

\begin{remark}
\rm
We remark that the time scale $t = \mathcal O ( \eps^{-1} |\log \eps|)$ is critical;
see \cite[Remark 1.7]{GGKS21} for more discussion on critical time.
In the limit $\eps \to 0^+$, the large deviation principle \eqref{THM_LDP4} may hold for any fixed time $t > 0$.
\end{remark}

\begin{corollary}\label{COR:main}
Let $u(t,x)$ be solutions to \eqref{weakNLS1} with initial data $u_0^\o$ given in \eqref{random_data}.
For any $t = \mathcal O ( \eps^{-1} |\log \eps|)$ and $z_0 > 0$, we have that \eqref{THM_LDP3} holds provided $(c_k) \in \l^1$.
\end{corollary}

\begin{proposition}
\label{PROP:linear1}
Consider the linear Schr\"odinger equation on the torus $\T = [0,2\pi]$ as in \eqref{linear}, with random initial data $u_0^\o$ given by \eqref{random_data}.
Then,
\begin{align}
\label{LDP_linear1}
\lim_{\eps\to 0^+} \eps \log \P \big( \|u(t)\|_{\FL^1(\T)} \ge z_0 \eps^{-1/2} \big) = - \frac{z_0^2}{\sum_{k \in \Z} c_k^2},
\end{align}

We turn to the summation in \eqref{main} over $k \in B$.
We note that
\begin{align}
\label{main2}
\begin{split}
\eps & \sum_{k \in B} \log \left( 1+ \sqrt{\pi}\varepsilon^{-1/2}\, c_k\, \exp\left(\frac{\varepsilon^{-1} \, c_k^2}{4}\right) \right) \\
& \le \eps \sum_{k \in B} \log \left( \big( 1+ \sqrt{\pi}\varepsilon^{-1/2}\, c_k\, \big) \exp\left(\frac{\varepsilon^{-1} \, c_k^2}{4}\right) \right) \\
& \le \eps \sum_{k \in B} \log \big( 1+ \sqrt{\pi}\varepsilon^{-1/2}\, c_k\, \big) + \eps \sum_{k \in B} \left(\frac{\varepsilon^{-1} \, c_k^2}{4}\right) \\
& \le \sqrt{\pi}\varepsilon^{1/2} \sum_{k \in B} \, c_k\, + \frac14 \sum_{k \in B} c_k^2,
\end{split}
\end{align}
which is again uniformly bounded as $(c_k) \in \l^1 \subset \l^2$, where we used $\eps^{1/2} \le c_k$ for $k \in B$.
\end{proof}

\subsection{Linear LDP II - \texorpdfstring{$L^\infty$}{Lg} norm}
In this subsection,
we prove an LDP for the linear solution to \eqref{linear} under $L^\infty$ norm.
In particular, we shall prove
\begin{proposition}
\label{PROP:linear2}
Consider the linear Schr\"odinger equation on the torus $\T = [0,2\pi]$ as in \eqref{linear}, with random initial data $u_0^\o$ given by \eqref{random_data}.
Then,
\begin{align}
\label{LDP_linear2}
\lim_{\eps\to 0^+} \eps \log \PP \big( \|u(t)\|_{L^\infty (\T)} \ge z_0 \eps^{-1/2} \big) = - \frac{z_0^2}{\sum_{k \in \Z} c_k^2},
\end{align}

\noi
where $\mathcal B_\eps$ is defined in \eqref{Beps}.
We first note, see also \cite[Proposition 4.1]{GGKS21},\footnote{Please note that \cite[Proposition 4.1]{GGKS21} only requires that $\|c_k\|_{\l^1_k} < \infty$.} that
\begin{align}
\label{pp400}
\lim_{\eps \to 0^+} \eps \log \PP \big( \| u_{\rm app} (t) \|_{L^\infty (\T)} > z_0 (\eps^{-1/2} + \eps^{-1/2 + \dl}) \big) = - \frac{z_0^2}{\sum_{k \in \Z} c_k^2}.
\end{align}

\begin{align}
\label{THM_LDP4}
\lim_{\eps \to 0^+} \eps \log \PP \Big( \|u(t,x) \|_{\FL^1 (\T)} > z_0\eps^{-1/2} \Big) = - \frac{z_0^2}{\sum_k c_k^2},
\end{align}

\begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\Z} c_k g_k e^{ikx},
\end{equation}

\begin{remark}
\rm
We remark that the time scale $t = \mathcal O ( \varepsilon^{-1} |\log \varepsilon|)$ is critical;
see \cite[Remark 1.7]{GGKS21} for more discussion on critical time.
In the limit $\varepsilon \to 0^+$, the large deviation principle \eqref{THM_LDP4} may hold for any fixed time $t > 0$.
\end{remark}

\begin{remark}
\rm
To prove Theorem~\ref{THM:main}, we first develop a novel approach to establish
the LDP for the linear Schr\"odinger equation under the sole assumption that
the initial data lies in $\mathcal{F} L^1$. We then carry out the nonlinear analysis,
together with a perturbative argument, entirely within the $\mathcal{F} L^1$ framework.
\end{remark}

As a consequence of Theorem \ref{THM:main} and the observation \eqref{obser}, we can improve \eqref{THM_LDP3} in terms of the decay of $c_k$.

\begin{corollary}\label{COR:main}
Let $u(t,x)$ be solutions to \eqref{weakNLS1} with initial data $u_0^\omega$ given in \eqref{random_data}.
For any $t = \mathcal O ( \varepsilon^{-1} |\log \varepsilon|)$ and $z_0 > 0$, we have that \eqref{THM_LDP3} holds provided $(c_k) \in \ell^1$.
\end{corollary}

\begin{remark}
\rm
We note that Corollary \ref{COR:main} improves \cite[Theorem 1.1]{GGKS21} in the following sense.
The coefficients $(c_k)$ in \cite[Theorem 1.1]{GGKS21} is required to decay condition \eqref{decay}.
In Proposition \ref{PROP:linear2}, it requires only $(c_k) \in \ell^1$.
It is expected that \eqref{THM_LDP3} holds for initial data $u_0 \in L^2 \cap L^\infty$, where $\mathcal{F} L^1 \subset L^2 \cap L^\infty$.
However, our current argument can only handle $\mathcal{F} L^1$ data.
\end{remark}

\begin{remark}
\rm
Recently, in \cite{BGMS25}, the authors studied rogue waves and large deviations
for two-dimensional pure-gravity deep-water waves, assuming initial data with
exponentially decaying Fourier coefficients; see \cite[(1.14)]{BGMS25}. It is
expected that our argument also applies in their setting and allows one to
relax this assumption to merely $\ell^1$ decay of the Fourier coefficients.
We plan to pursue this in future work.
\end{remark}

To prove Theorem~\ref{THM:main}, the authors say they (i) “first develop a novel approach to establish the LDP for the linear Schr\"odinger equation under the sole assumption that the initial data lies in $\mathcal{F} L^1$,” and then (ii) “carry out the nonlinear analysis, together with a perturbative argument, entirely within the $\mathcal{F} L^1$ framework.”

To prove the main theorem, the authors say they (i) “first develop a novel approach to establish the LDP for the linear Schr\"odinger equation under the sole assumption that the initial data lies in $\mathcal{F} L^1$,” and then (ii) “carry out the nonlinear analysis, together with a perturbative argument, entirely within the $\mathcal{F} L^1$ framework.”,

\label{THM:main}
Let $u(t,x)$ be solutions to \begin{align}
\label{weakNLS1}
i\pa_t u + \Delta u = \varepsilon^{2} |u|^2 u.
\end{align} with initial data $u_0^\omega$ given in \begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\Z} c_k g_k e^{ikx},
\end{equation}.
For any $t = \mathcal O ( \varepsilon^{-1} |\log \varepsilon|)$ and $z_0 > 0$, we have that
\begin{align}
\label{THM_LDP4}
\lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \|u(t,x) \|_{\mathcal{F} L^1 (\mathbb{T})} > z_0\varepsilon^{-1/2} \Big) = - \frac{z_0^2}{\sum_k c_k^2},
\end{align}

\noindent
holds if and only if $(c_k) \in \ell^1.\,

Biconditional or Equivalence
Asymptotic or Limit

Question:
Let $u(t,x)$ solve the weakly nonlinear Schr\"odinger equation on the torus $\mathbb T$,
\[
i\partial_t u+\Delta u=\varepsilon^2|u|^2u,
\]
with random initial data
\[
u_0(x)=\sum_{k\in\mathbb Z} c_k g_k e^{ikx},
\]
where $\{g_k\}_{k\in\mathbb Z}$ are independent identically distributed complex Gaussian random variables and $(c_k)$ is a sequence of coefficients. For any $t=\mathcal O(\varepsilon^{-1}|\log\varepsilon|)$ and any $z_0>0$, suppose one considers the large-deviation asymptotic
\[
\lim_{\varepsilon\to0^+}\varepsilon\log\mathbb P\Big(\|u(t,\cdot)\|_{\mathcal F L^1(\mathbb T)}>z_0\varepsilon^{-1/2}\Big)
= -\frac{z_0^2}{\sum_k c_k^2}.
\]
Which of the following statements is equivalent to this asymptotic formula?

The coefficient sequence satisfies $(c_k)\in \ell^1$.

Choices:
B. The coefficient sequence satisfies $(c_k)\in \ell^2$.
C. The coefficient sequence satisfies $\sum_k c_k^2<\infty$.
D. For each fixed $z_0>0$, there exists a time range $t=\mathcal O(\varepsilon^{-1}|\log\varepsilon|)$ on which the asymptotic formula holds whenever $(c_k)\in \ell^1\cap \ell^2$.
E. The asymptotic formula holds if and only if the coefficient sequence satisfies $(c_k)\in \mathcal F L^1(\mathbb T)$.

sharp \ell^1 threshold replaced by weaker square-summability

dropped the full equivalence and kept only the weaker consequence implied by $(c_k)\in\ell^1$

replaced the iff criterion by a merely sufficient mixed assumption with altered quantifier structure

confused coefficient summability on $(c_k)$ with function-space membership notation

The stem does not explicitly reveal the conclusion or single out the correct option. It gives the PDE, random data, norm, and time scale, but the actual asymptotic statement and the sharp \(\ell^1\) condition appear only in the choices.

The item is close to a theorem-recall question: it asks directly which limiting statement holds for the exact setup of a specific result. However, it is not a pure restatement because the options vary in logical strength (especially 'if' versus 'if and only if') and in the rate formula.

Some reasoning is required to distinguish the strongest valid conclusion from a weaker true statement (choice C) and from nearby false variants. Still, success depends heavily on recalling the theorem rather than deriving the answer from first principles.

The distractors are mathematically plausible and target realistic failure modes: confusing \(\ell^1\) with \(\ell^2\), dropping the 'only if' direction, mishandling the time-scale quantifier, and using an incorrect denominator. They are distinct and nontrivial.

A solid, technically well-constructed MCQ with strong distractors and little answer leakage, but it remains fairly close to theorem recall rather than deeply generative reasoning.


The study of equilateral polygons within lattices combines elements of geometry, number theory, and discrete mathematics. In particular, the problem of characterizing which polygons can be embedded in given planar lattices has drawn sustained interest. A central focus has been on determining for which values of $n$ there exists a convex equilateral $n$-gon whose vertices lie in a specified lattice.

Recent results by Maehara \cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\nu(L)$, satisfies $\nu(L) \equiv 3 \pmod{4}$.

Building on this, Iino and Sakiyama \cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\Lambda(m) = L[(1,0),(0,\sqrt{m})]$ for square-free integers $m \equiv 3 \pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.

In this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\Lambda(m)$, with $m \equiv 3 \pmod{4}$.

\begin{abstract}
We study the existence of equilateral polygons in planar integer lattices. Maehara showed that it's sufficient to work with rectangular lattices $\Lambda(m) = L[(1,0),(0,\sqrt{m})]$ with $m \equiv 3 \pmod{4}$. Building on results of Maehara and of Iino and Sakiyama, we show that for every such $m$ there exists $N$ such that for all $n \geq N$, the lattice $\Lambda(m)$ contains an equilateral $n$-gon. This extends previous classifications of equilateral polygons in planar lattices.
\end{abstract}

Recent results by Maehara \cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\nu(L)$, satisfies $\nu(L) \equiv 3 \pmod{4}$.

Building on this, Iino and Sakiyama \cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\Lambda(m) = L[(1,0),(0,\sqrt{m})]$ for square-free integers $m \equiv 3 \pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.

In this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\Lambda(m)$, with $m \equiv 3 \pmod{4}$.

We will give an existence proof in \Cref{Section: Existence} followed by a explicit construction in \Cref{Section: Construction}.

For any $m \equiv 3 \pmod{4}$ does there exist $N$ such that for all $n \geq N$ there exists a \textbf{convex} equilateral polygon of size n?

\begin{open}
For any $m \equiv 3 \pmod{4}$ does there exist an equilateral polygon of size $n'$, where $n'$ is the largest prime factor of m?
\end{open}

For any $m \equiv 3 \pmod{4}$ does there exist a \textbf{convex} equilateral polygon of size $n'$, where $n'$ is the largest prime factor of m?

The post-theorem introduction indicates the proof strategy: it will present "an existence proof" (in \Cref{Section: Existence}) and then an "explicit construction" (in \Cref{Section: Construction}).

\label{thm: new theorem}
For any $m \equiv 3 \pmod{4}$ there exists $N$ such that for all $n \geq N$ there exists an equilateral polygon of size n.,

Question: Let \(\Lambda(m)=L[(1,0),(0,\sqrt m)]=\{(a,b\sqrt m):a,b\in \mathbb Z\}\) be the planar rectangular lattice associated to an integer \(m\equiv 3\pmod 4\). An equilateral polygon of size \(n\) means an \(n\)-gon whose vertices lie in \(\Lambda(m)\) and whose side lengths are all equal. Which statement holds for such lattices?

For every integer \(m\equiv 3\pmod 4\), there exists an integer \(N\) such that for every integer \(n\ge N\), the lattice \(\Lambda(m)\) contains an equilateral polygon of size \(n\).

Choices:
B. For every integer \(m\equiv 3\pmod 4\), there exists an integer \(N\) such that for every integer \(n\ge N\), the lattice \(\Lambda(m)\) contains a convex equilateral polygon of size \(n\).
C. For every integer \(m\equiv 3\pmod 4\), there exists an integer \(n\) such that the lattice \(\Lambda(m)\) contains an equilateral polygon of size \(n\).
D. There exists an integer \(N\) such that for every integer \(m\equiv 3\pmod 4\) and every integer \(n\ge N\), the lattice \(\Lambda(m)\) contains an equilateral polygon of size \(n\).
E. For every integer \(m\equiv 3\pmod 4\), there exists an integer \(N\) such that for every odd integer \(n\ge N\), the lattice \(\Lambda(m)\) contains an equilateral polygon of size \(n\), and one may take \(N\) to be the largest prime factor of \(m\).

The stem only defines the lattice and the notion of an equilateral polygon; it does not state or strongly hint at the eventual-for-all-large-n conclusion in choice A.

The item is close to theorem recognition: choice A appears to be the target theorem statement, while the other options are nearby quantifier or strengthening/weakening variants. This is better than a pure restatement, but still only mildly non-tautological.

There is some reasoning pressure because the options differ in subtle ways (eventual existence vs mere existence, dependence of N on m, convexity, odd n only). However, the item mainly tests precise recall/discrimination of the theorem statement rather than substantial mathematical generation.

The distractors are strong: B is a plausible overstrengthening, C is a weaker true-looking statement, D tests quantifier dependence, and E adds a tempting but unjustified explicit threshold. They are distinct and reflect common theorem-misreading errors.

A solid theorem-discrimination MCQ with good distractors and no answer leakage, but it leans more toward recognizing the exact statement than toward deep generative reasoning.


\begin{theorem}\label{thm:mainthm}
Let $S$ be a complex algebraic variety admitting a trim resolution. Then
\begin{itemize}
\item
For $s\in S$, the local Euler obstruction $\Eu_S(s)$ equals the Euler characteristic of
the fiber over $s$ in any trim resolution of $S$.
\item
The {\em stringy\/} Chern class of $S$ equals its Chern-Mather class:
$\cstr(S)=\cma(S)$.
\item
The characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.
\end{itemize}
\end{theorem}

\begin{equation}\label{eq:strMa}
f_*(c(TY)\cap [Y]) = \cma(S)
\end{equation}
in the Chow group of $S$. In general, stringy Chern classes are defined for
normal varieties with a $\Qbb$-Cartier canonical divisor and at worst log-terminal
singularities, and take value in the Chow group with rational coefficients. These
additional stipulations are not needed for the notion considered in this note; we can
adopt the left-hand side of~\eqref{eq:strMa} as the definition of stringy Chern class
in the (integral) Chow group of $S$, compatibly with the more general definition.
The fact that the class is independent of the choice of trim resolution is
also a consequence of Theorem~\ref{thm:mainthm}.

We note that Theorem~\ref{thm:mainthm} implies that Batyrev's {\em stringy Euler
number\/} (\cite{MR2001j:14018}) of a variety admitting a trim resolution equals
its Euler characteristic weighted by the local Euler obstruction.

Theorem~\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's
formalism of {\em conical Lagrangian cycles,\/} to which local Euler obstructions
and Chern-Mather classes relate directly (see~\S\ref{Lagrapf}). Concerning intersection
cohomology, recall that if $\pi:Y \to S$ is a small resolution, then the
intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant
sheaf on~$Y$ (\cite[\S6.2]{MR696691}). We evaluate the corresponding push-forward
at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.
More precisely, we prove (Proposition~\ref{prop:Sabpf}) that, for trim resolutions, the
Lagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle
$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this
more basic observation, as we show in~\S\ref{thmpf}.

There is considerable interest in conditions implying that the characteristic cycle of
the intersection cohomology sheaf is irreducible. Lusztig (\cite[13.7, p.~414]{MR1088333})
expressed the `hope' that this may be the case for Schubert varieties in flag manifolds of
type A, D, E. A counterexample was constructed by Kashiwara and Saito for type A in
$F\ell(8)$ (\cite{MR1458969}, \cite{MR1896039}), while it holds in $F\ell(n)$ for $n\le 7$.
Irreducibility is also known for all Schubert
varieties in the standard Grassmannian (\cite{MR1084458}), and more generally
for all Schubert varieties in cominuscule Grassmannians of
types A, D, and E, see~\cite{MR1451256, MihalceaSingh}.

We hope that Theorem~\ref{thm:mainthm} may help in streamlining such verifications.
For instance, since the Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic
curve is trim, the irreducibility of the $\IC$ characteristic cycle in this case (first established
in \cite{MR1642745}) follows directly from~Theorem~\ref{thm:mainthm}.

The connection between the irreducibility of the $\IC$ characteristic cycle and
the equality of Chern-Mather and stringy Chern classes was pointed out by B.~Jones
in~\cite[Remark 3.3.2]{MR2628830}, ultimately as an application of the microlocal
index formula of Dubson and Kashiwara.
We freely borrow ideas from~\cite{MR2628830}
in~\S\ref{thmpf}.

A condition on fibers of small resolutions is considered in~\cite{graham}, including a proof
that the condition implies the irreducibility of the IC characteristic cycles.\medskip

{\em Acknowledgments.} This work was supported in part by an award from the Simons
Foundation, SFI-MPS-TSM-00013681. The author also thanks David Massey and
Leonardo Mihalcea for useful conversations, and Caltech for hospitality as most of this
work was carried out.

\section{Lagrangian push-forward}\label{Lagrapf}
Let $X$ be a nonsingular variety and denote by $T^* X$ the cotangent bundle of $X$.
The {\em conormal variety\/} $T^*_W X$ of a closed subvariety $W\subseteq X$ is the
closure in $T^* X$ of the conormal variety to the nonsingular part $W^\circ$ of $V$:
$T^*_W X = \overline{T^*_{W^\circ} X}$. The conormal variety of $X$ itself is the
zero-section $T^*_XX$ of the cotangent bundle. All conormal varieties have dimension
$\dim X$; they determine conormal {\em cycles\/} in $Z_{\dim X} T^* X$.
It will also be convenient to take the projective completion of these constructions:
we will denote by $\Tbb^* X$ the projective completion $\Pbb(T^* X\oplus \one)$
of $T^* X$ (here, $\Pbb$ denotes the projective bundle of lines) and by~$\Tbb^*_WX$
the closure of the conormal variety in $\Tbb^* X$.

For a nonsingular variety $X$, we denote by $\cL (X)$ the free abelian group of
{\em conical Lagrangian cycles\/} in the cotangent bundle $T^* X$.
Conormal cycles are conical Lagrangian, and in fact (cf.~\cite[Lemma~3]{MR1063344})
$\cL (X)$ may be realized as the free abelian group on conormal cycles.
For a closed (and possibly singular) subvariety $V\subseteq X$, we denote by
$\cL(V)$ the subgroup of $\cL(X)$ generated by the conormal cycles $T^*_W X$
with $W\subseteq V$. Thus, elements of~$\cL (V)$ may be viewed as finite integer linear
combinations $\sum_W m_W T^*_WX$ ranging over closed subvarieties $W$ of $V$.
Clearly $\cL(V)$ is isomorphic to the group of algebraic cycles of~$V$, and in particular
it is independent of the ambient nonsingular variety $X$.

Important invariants of $V$ may be expressed directly in terms of Lagrangian cycles
by means of intersection-theoretic operations, after taking the projective completion.
As above, realize $V$ as a closed subvariety of a
nonsingular variety $X$; the results will be independent of the choice of $X$.
Let $\pi: \Tbb^*_V X\to V$ be the natural projection; and let $\cO(1)$ be the
tautological line bundle on $\Tbb^* X=\Pbb(T^* X\oplus \one)$.
\begin{itemize}
\item[---]
The {\em local Euler obstruction\/} $\Eu_V: V\to \Zbb$ is
\[
\Eu_V(p)= (-1)^{\dim X-\dim V} \int c(\pi^* TX|_V) c(\cO(1))^{-1}\cap s(\pi^{-1}(p), \Tbb^*_V X)
\]
where $s(\pi^{-1}(p), \Tbb^*_V X)$ denotes the {\em Segre class\/} in the sense
of~\cite[Chapter~4]{85k:14004};
\item[---]
The {\em Chern-Mather class\/} $\cma(V)\in A_*V$ is
\[
\cma(V)=(-1)^{\dim X-\dim V} c(TX|_V)\cap \pi_* \left(c(\cO(1))^{-1}\cap [\Tbb^*_V X]\right)\saf.
\]
\end{itemize}
Equivalent results were established by C.~Sabbah (\cite[1.2.1, 1.2.2]{MR804052}); we also
address the reader to~\cite{MR1063344} and~\cite{MR2002g:14005} for clear treatments of these
formulas.

Let $\cF(V)$ denote the abelian group of constructible functions $V\to \Zbb$.
Following~\cite[\S1]{MR2002g:14005}, the relation between conormal cycles and local
Euler obstructions is recorded by the homomorphism
\[
\CC: \cF(V) \to \cL(V)
\]
defined by prescribing
\[
\Eu_W \mapsto (-1)^{\dim W} T^*_W X
\]
for all closed subvarieties $W$ of $V$.
In fact, $\CC$ is an {\em isomorphism:\/} it simply matches a basis of $\cF(V)$ with a
basis of $\cL(V)$.

\begin{defin}\label{def:CC}
The {\em characteristic cycle\/} $\CC(\alpha)$ of a constructible function $\alpha$ on $V$
is the image of $\alpha$ in $\cL(V)$ under this isomorphism.
\qede\end{defin}

Likewise, the formula for Chern-Mather classes motivates the introduction of a homomorphism
\[
c_*:\cL(V) \to A_*(V)
\]
defined on generators by
\[
T^*_W X \mapsto (-1)^{\dim W} \cma(W)\saf.
\]
Then the composition $c_*\circ \CC:\cF(V) \to A_*(V)$ agrees with the value at $V$
of MacPherson's natural transformation $\cF \leadsto A_*$, where $\cF$ is taken as
a functor with push-forward defined by Euler characteristics of fibers;
see~\cite{MR0361141} and~\cite[Example 19.1.7]{85k:14004}.
For instance, the {\em Chern-Schwartz-MacPherson class\/} of a (possibly singular)
algebraic variety $V$ is the image $c_*(\CC(\one_V))\in A_*V$ of the characteristic
cycle in~$\cL(V)$ of the constant function $\one_V$.
Sabbah provides an alternative proof of the naturality of this assignment, which is
the main result of~\cite{MR0361141}, by defining a covariant {\em push-forward\/}
\[
\varphi_*: \cL(V') \to \cL(V'')
\]
for every proper map $\varphi: V' \to V''$, making $\cL$ into a functor, in such a way
that the above homomorphisms define natural transformations
\[
\cF \leadsto \cL\quad, \quad \cL \leadsto A_*
\]
whose composition agrees with MacPherson's natural transformation.

We are interested in explicit formulas for this {\em Lagrangian push-forward.\/}
After embedding~$V''$ in a nonsingular variety $X$ and replacing $V'$ by a resolution $Y$,
we can reduce to the case of a proper morphism $f:Y \to X$ of nonsingular varieties.
We are specifically interested in the image $f_*(T^*_YY)$ of the zero-section in this
situation.

Theorem~\ref{thm:mainthm} will be a consequence of the following result. We recall the
definition of `trim' given in the introduction.

\begin{defin}\label{def:contained}
Let $\pi:Y \to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.
For $0\le d\le \dim Y$, let $Y_d$ denote the locus where the rank of the differential
$d\pi$ equals $d$. Then $\pi$ is a {\em trim resolution\/} if $\dim Y_d<d$
for all $0\le d<\dim Y$.
\qede\end{defin}

\begin{prop}\label{prop:Sabpf}
Let $f: Y\to X$ be a proper morphism of nonsingular varieties, such that $Y \to f(Y)$ is
a trim resolution. Then $f_* (T^*_Y Y)=T^*_{f(Y)} X$.
\end{prop}

\begin{remark}\label{rem:YtoS}
Let $\pi: Y\to S$ be a proper surjective morphism, with $Y$ nonsingular, and
$\iota:S\hookrightarrow X$ a closed embedding, with $X$ nonsingular; and let
$f=\iota\circ \pi: Y \to X$ be the composition.
As explained above, $T^*_{f(Y)} X=T^*_S X$ may be viewed as an element of $\cL(S)$;
Proposition~\ref{prop:Sabpf} states that if $\pi$ is trim, then this is the image $\pi_*(T^*_YY)$
under $\pi_*: \cL(Y)\to \cL(S)$.
\qede\end{remark}

The rest of this section is devoted to the proof of Proposition~\ref{prop:Sabpf}.\smallskip

{\em A priori,\/} $f_*(T^*_Y Y)$ is an integer linear combination of conormal cycles
$\sum_Z m_Z T^*_ZX$, with $Z\subseteq f(Y)$. One of the summands is $T^*_{f(Y)}X$.
The task is to show that if $Y\to f(Y)$ is trim and $Z\subsetneq f(Y)$ is a {\em proper\/}
subvariety of $f(Y)$, then the coefficient of $T^*_Z X$ in $f_* (T^*_Y Y)$ is~$0$.
Following Sabbah, we write $f$ as the composition
\[
\xymatrix{
Y \ar[r]^-\gamma & Y\times X \ar[r]^-\rho & X
}
\]
where $\gamma$ is the graph of $f$ and $\rho$ is the (smooth) projection.
Let $p: \rho^*(T^* X) \to T^* X$ be the natural mophism. Then  (cf.~\cite[\S2]{MR804052})
the components $T^*_ZX$ appearing in the decomposition of $f_*(T^*_YY)$ are the
irreducible components of the image
\begin{equation}\label{eq:pint}
p \left(\rho^*(T^* X) \cap T^*_{\gamma(Y)}(Y\times X)\right)
\end{equation}

\label{intro}
Recall that a proper surjective stratified map $\pi: Y\to S$ of complex algebraic varieties
of the same dimension is {\em small\/} if for all proper strata $Z$ of $S$, $2d(Z)< \codim_Z S$,
where $d(Z)$ denotes the (common) dimension of the fibers of $\pi$ over points of $Z$.
We say that $\pi$ is a {\em small resolution\/} if $Y$ is nonsingular and $\pi$ is birational
and small.

We introduce a strengthening of this condition, which we name `trim'
(Definition~\ref{def:contained}).
Assume $\pi:Y\to S$ is a proper birational map, with $Y$ nonsingular.
For $d\ge 0$, denote by $Y_d\subseteq Y$ the locally closed subset along which
the rank of the differential of $\pi$ is~$d$. We say that $\pi$ is {\em trim\/} if
$\dim Y_d<d$ for all $d<\dim Y$.

Trim maps are small (Proposition~\ref{prop:consma}).
If $\pi:Y\to S$ is small, and for all strata~$Z$ of~$S$ the restriction $\pi^{-1}(Z)\to Z$ is a smooth
morphism, then $\pi$ is trim (\S\ref{Furrem}).
For instance, the standard Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic
curve (which is small, cf.~\cite{MR1642745}) is trim.

Our main result is the following.

\label{Furrem}

1. In this note we have considered the notion of `trim' only for proper birational morphisms
because that is the case relevant to our application in Theorem~\ref{thm:mainthm}.
It co

\begin{defin}\label{def:contained}
Let $\pi:Y \to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.
For $0\le d\le \dim Y$, let $Y_d$ denote the locus where the rank of the differential
$d\pi$ equals $d$. Then $\pi$ is a {\em trim resolution\/} if $\dim Y_d<d$
for all $0\le d<\dim Y$.
\qede\end{defin}

\begin{prop}\label{prop:consma}
Let $\pi: Y \to S$ be a trim proper birational map. Then $\pi$ is small.
\end{prop}

\begin{abstract}
We introduce {\em trim\/} resolutions of complex algebraic varieties, a strengthening
of the notion of small resolution. We prove that the characteristic cycle of the intersection
cohomology sheaf of a variety admitting a trim resolution is irreducible and that for
such varieties the stringy and Chern-Mather classes coincide.
\end{abstract}

Our main result is the following.

The definition of characteristic cycle will be recalled below (\S\ref{Lagrapf}).
`Stringy' Chern classes were defined in~\cite{MR2183846, MR2304329}
(see~\cite{MR2280127} for a lean account). If $\pi: Y\to S$ is a crepant resolution,
$\cstr(S)$ equals the push-forward of the total Chern class of the tangent bundle
of~$Y$; the equality stated in the theorem is
\begin{equation}\label{eq:strMa}
f_*(c(TY)\cap [Y]) = \cma(S)
\end{equation}
in the Chow group of $S$. In general, stringy Chern classes are defined for
normal varieties with a $\Qbb$-Cartier canonical divisor and at worst log-terminal
singularities, and take value in the Chow group with rational coefficients. These
additional stipulations are not needed for the notion considered in this note; we can
adopt the left-hand side of~\eqref{eq:strMa} as the definition of stringy Chern class
in the (integral) Chow group of $S$, compatibly with the more general definition.
The fact that the class is independent of the choice of trim resolution is
also a consequence of Theorem~\ref{thm:mainthm}.

We note that Theorem~\ref{thm:mainthm} implies that Batyrev's {\em stringy Euler
number\/} (\cite{MR2001j:14018}) of a variety admitting a trim resolution equals
its Euler characteristic weighted by the local Euler obstruction.

Theorem~\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's
formalism of {\em conical Lagrangian cycles,\/} to which local Euler obstructions
and Chern-Mather classes relate directly (see~\S\ref{Lagrapf}). Concerning intersection
cohomology, recall that if $\pi:Y \to S$ is a small resolution, then the
intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant
sheaf on~$Y$ (\cite[\S6.2]{MR696691}). We evaluate the corresponding push-forward
at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.
More precisely, we prove (Proposition~\ref{prop:Sabpf}) that, for trim resolutions, the
Lagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle
$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this
more basic observation, as we show in~\S\ref{thmpf}.

Important invariants of $V$ may be expressed directly in terms of Lagrangian cycles
by means of intersection-theoretic operations, after taking the projective completion.
As above, realize $V$ as a closed subvariety of a
nonsingular variety $X$; the results will be independent of the choice of $X$.
Let $\pi: \Tbb^*_V X\to V$ be the natural projection; and let $\cO(1)$ be the
tautological line bundle on $\Tbb^* X=\Pbb(T^* X\oplus \one)$.
\begin{itemize}
\item[---]
The {\em local Euler obstruction\/} $\Eu_V: V\to \Zbb$ is
\[
\Eu_V(p)= (-1)^{\dim X-\dim V} \int c(\pi^* TX|_V) c(\cO(1))^{-1}\cap s(\pi^{-1}(p), \Tbb^*_V X)
\]
where $s(\pi^{-1}(p), \Tbb^*_V X)$ denotes the {\em Segre class\/} in the sense
of~\cite[Chapter~4]{85k:14004};
\item[---]
The {\em Chern-Mather class\/} $\cma(V)\in A_*V$ is
\[
\cma(V)=(-1)^{\dim X-\dim V} c(TX|_V)\cap \pi_* \left(c(\cO(1))^{-1}\cap [\Tbb^*_V X]\right)\saf.
\]
\end{itemize}
Equivalent results were established by C.~Sabbah (\cite[1.2.1, 1.2.2]{MR804052}); we also
address the reader to~\cite{MR1063344} and~\cite{MR2002g:14005} for clear treatments of these
formulas.

{\em A priori,\/} $f_*(T^*_Y Y)$ is an integer linear combination of conormal cycles
$\sum_Z m_Z T^*_ZX$, with $Z\subseteq f(Y)$. One of the summands is $T^*_{f(Y)}X$.
The task is to show that if $Y\to f(Y)$ is trim and $Z\subsetneq f(Y)$ is a {\em proper\/}
subvariety of $f(Y)$, then the coefficient of $T^*_Z X$ in $f_* (T^*_Y Y)$ is~$0$.
Following Sabbah, we write $f$ as the composition
\[
\xymatrix{
Y \ar[r]^-\gamma & Y\times X \ar[r]^-\rho & X
}
\]
where $\gamma$ is the graph of $f$ and $\rho$ is the (smooth) projection.
Let $p: \rho^*(T^* X) \to T^* X$ be the natural mophism. Then (cf.~\cite[\S2]{MR804052})
the components $T^*_ZX$ appearing in the decomposition of $f_*(T^*_YY)$ are the
irreducible components of the image
\begin{equation}\label{eq:pint}
p \left(\rho^*(T^* X) \cap T^*_{\gamma(Y)}(Y\times X)\right)
\end{equation}
where we view both $\rho^*(T^* X)$ and $T^*_{\gamma(Y)}(Y\times X)$
as subschemes of $T^* (Y\times X)$.
Let $T^*_ZX$ be one such component, and let $z$ be a general point of $Z$
and $\xi$ a general covector in the fiber $(T^*_ZX)_z$. For $(z,\xi)$ to be in
the image~\eqref{eq:pint}, there must be a point $(y,z)\in \gamma(Y)$ mapping to
$z$, such that
$
(0,\xi)\in T^*_y Y\oplus T^*_z X\cong T^*_{(y,z)} (Y\times X)
$
vanishes on $T_{(y,z)}\gamma(Y)$.

By Proposition~\ref{prop:consma}, $\pi$ is small. It follows (cf.~\cite[\S6.2]{MR696691})
that the intersection cohomology sheaf $\IC^\bullet_S$ is the direct image of the intersection
cohomology sheaf of $Y$. Since $Y$ is nonsingular, the latter is a shift of the constant
sheaf. Therefore,
\[
\IC^\bullet_S = R\pi_* \Qbb_Y[\dim Y]\saf.
\]
(Concerning the shift, we follow the modern convention as in e.g.,
\cite[Remark~4.2.4]{MR2525735}.)
The following formula for the stalk Euler characteristic of $\IC^\bullet_S$ at $z\in S$
is a consequence of standard properties of (derived) direct images:
\[
\chi_z(\IC^\bullet_S) =
\chi_z(R\pi_* \Qbb_Y[\dim Y]) =
(-1)^{\dim S} \sum_i (-1)^i \dim H^i(\pi^{-1}(z);\Qbb)
= (-1)^{\dim S}\chi(\pi^{-1}(z))\saf.
\]
By the first assertion in Theorem~\ref{thm:mainthm}, this implies
\[
\chi_z(\IC^\bullet_S) = (-1)^{\dim S}\Eu_S(z)\saf.
\]
Now embed $S$ in a nonsingular variety $X$.
The characteristic cycle of $\IC^\bullet_S$ is
\[
CC(\chi_z(\IC^\bullet_S)) = CC((-1)^{\dim S}\Eu_S(z)) = T^*_SX\saf,
\]
and this verifies it is irreducible, completing the proof of
Theorem~\ref{thm:mainthm}.

The definition of characteristic cycle will be recalled below (\S\ref{Lagrapf}).
`Stringy' Chern classes were defined in~\cite{MR2183846, MR2304329}
(see~\cite{MR2280127} for a lean account). If $\pi: Y\to S$ is a crepant resolution,
$\cstr(S)$ equals the push-forward of the total Chern class of the tangent bundle
of~$Y$; the equality stated in the theorem is
\begin{equation}\label{eq:strMa}
f_*(c(TY)\cap [Y]) = \cma(S)
\end{equation}
in the Chow group of $S$. In general, stringy Chern classes are defined for
normal varieties with a $\Qbb$-Cartier canonical divisor and at worst log-terminal
singularities, and take value in the Chow group with rational coefficients. These
additional stipulations are not needed for the notion considered in this note; we can
adopt the left-hand side of~\eqref{eq:strMa} as the definition of stringy Chern class
in the (integral) Chow group of $S$, compatibly with the more general definition.
The fact that the class is independent of the choice of trim resolution is
also a consequence of Theorem~\ref{thm:mainthm}.

We note that Theorem~\ref{thm:mainthm} implies that Batyrev's {\em stringy Euler
number\/} (\cite{MR2001j:14018}) of a variety admitting a trim resolution equals
its Euler characteristic weighted by the local Euler obstruction.

Theorem~\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's
formalism of {\em conical Lagrangian cycles,\/} to which local Euler obstructions
and Chern-Mather classes relate directly (see~\S\ref{Lagrapf}). Concerning intersection
cohomology, recall that if $\pi:Y \to S$ is a small resolution, then the
intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant
sheaf on~$Y$ (\cite[\S6.2]{MR696691}). We evaluate the corresponding push-forward
at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.
More precisely, we prove (Proposition~\ref{prop:Sabpf}) that, for trim resolutions, the
Lagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle
$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this
more basic observation, as we show in~\S\ref{thmpf}.

There is considerable interest in conditions implying that the characteristic cycle of
the intersection cohomology sheaf is irreducible. Lusztig (\cite[13.7, p.~414]{MR1088333})
expressed the `hope' that this may be the case for Schubert varieties in flag manifolds of
type A, D, E. A counterexample was constructed by Kashiwara and Saito for type A in
$F\ell(8)$ (\cite{MR1458969}, \cite{MR1896039}), while it holds in $F\ell(n)$ for $n\le 7$.
Irreducibility is also known for all Schubert
varieties in the standard Grassmannian (\cite{MR1084458}), and more generally
for all Schubert varieties in cominuscule Grassmannians of
types A, D, and E, see~\cite{MR1451256, MihalceaSingh}.

We hope that Theorem~\ref{thm:mainthm} may help in streamlining such verifications.
For instance, since the Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic
curve is trim, the irreducibility of the $\IC$ characteristic cycle in this case (first established
in \cite{MR1642745}) follows directly from~Theorem~\ref{thm:mainthm}.

The connection between the irreducibility of the $\IC$ characteristic cycle and
the equality of Chern-Mather and stringy Chern classes was pointed out by B.~Jones
in~\cite[Remark 3.3.2]{MR2628830}, ultimately as an application of the microlocal
index formula of Dubson and Kashiwara.
We freely borrow ideas from~\cite{MR2628830}
in~\S\ref{thmpf}.

A condition on fibers of small resolutions is considered in~\cite{graham}, including a proof
that the condition implies the irreducibility of the IC characteristic cycles.\medskip

{\em Acknowledgments.} This work was supported in part by an award from the Simons
Foundation, SFI-MPS-TSM-00013681. The author also thanks David Massey and
Leonardo Mihalcea for useful conversations, and Caltech for hospitality as most of this
work was carried out.

Theorem~\ref{thm:mainthm} is said to be “a consequence of considerations concerning Sabbah's formalism of {\em conical Lagrangian cycles,\/}” (to which “local Euler obstructions and Chern-Mather classes relate directly”). For intersection cohomology, using that for a small resolution $\pi:Y\to S$ “the intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant sheaf on~$Y$,” the paper “evaluate[s] the corresponding push-forward at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.” More precisely, it proves (Proposition~\ref{prop:Sabpf}) that for trim resolutions “the Lagrangian push-forward of the zero-section~$T^*_Y Y$ of the cotangent bundle $T^*Y$ equals the conormal cycle $T^*_S X$.” “The theorem follows from this more basic observation,” as shown in~\S\ref{thmpf}.

To prove the main theorem, it is said to be “a consequence of considerations concerning Sabbah's formalism of {\em conical Lagrangian cycles,\/}” (to which “local Euler obstructions and Chern-Mather classes relate directly”). For intersection cohomology, using that for a small resolution $\pi:Y\to S$ “the intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant sheaf on~$Y$,” the paper “evaluate[s] the corresponding push-forward at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.” More precisely, it proves the following proposition.

\begin{prop}\label{prop:Sabpf}
Let $f: Y\to X$ be a proper morphism of nonsingular varieties, such that $Y \to f(Y)$ is
a trim resolution. Then $f_* (T^*_Y Y)=T^*_{f(Y)} X$.
\end{prop}

That is, for trim resolutions “the Lagrangian push-forward of the zero-section~$T^*_Y Y$ of the cotangent bundle $T^*Y$ equals the conormal cycle $T^*_S X$.” “The theorem follows from this more basic observation,” as shown later.
\label{thmpf}

Let $S$ be a variety admitting a trim resolution $\pi: Y\to S$; let $\iota: S\hookrightarrow X$
be an embedding in a nonsingular variety, and let $f=\iota\circ \pi:Y \to X$ be the compo

Question: Let $S$ be a complex algebraic variety that admits a trim resolution, i.e. there exists a proper birational morphism $\pi:Y\to S$ with $Y$ nonsingular such that, for each $0\le d\le \dim Y$, if $Y_d\subseteq Y$ denotes the locus where the differential $d\pi$ has rank $d$, then $\dim Y_d<d$ for every $d<\dim Y$. In this setting, write $\Eu_S(s)$ for the local Euler obstruction at $s\in S$, write $\cma(S)$ for the Chern--Mather class of $S$, and write $\cstr(S)$ for the stringy Chern class (here, for a trim resolution $\pi$, one may take $\cstr(S)=\pi_*(c(TY)\cap [Y])$). Which statement holds for every such variety $S$?

For every trim resolution $\pi:Y\to S$ and every point $s\in S$, one has $\Eu_S(s)=\chi(\pi^{-1}(s))$; moreover $\cstr(S)=\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.

Choices:
B. For every trim resolution $\pi:Y\to S$ and every point $s\in S$, one has $\Eu_S(s)=\chi(\pi^{-1}(s))$; moreover $\cstr(S)=\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is the push-forward of the zero-section $T^*_Y Y$ for any trim resolution, hence may have several irreducible components.
C. For every trim resolution $\pi:Y\to S$ and every point $s\in S$, one has $\Eu_S(s)=\chi(\pi^{-1}(s))$; moreover $\cstr(S)=\cma(S)$ in the Chow group of $S$.
D. For every proper birational morphism $\pi:Y\to S$ with $Y$ nonsingular such that $\dim Y_d\le d$ for every $d<\dim Y$, and for every point $s\in S$, one has $\Eu_S(s)=\chi(\pi^{-1}(s))$; moreover $\cstr(S)=\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.
E. For every complex algebraic variety $S$ admitting a trim resolution, there exists a trim resolution $\pi:Y\to S$ such that for every $s\in S$ one has $\Eu_S(s)=\chi(\pi^{-1}(s))$, and for this choice of $\pi$ one has $\cstr(S)=\cma(S)$ in the Chow group of $S$; however the characteristic cycle of the intersection cohomology sheaf of $S$ need not be irreducible.

The stem sets up definitions and hypotheses but does not itself state the target conclusion. There is no direct textual leakage of the correct option from the stem alone.

The item is close to a theorem-recall question: it asks which packaged conclusion holds under the stated trim-resolution hypothesis. The alternatives add quantifier and strength variations, so it is not purely tautological, but it remains a mild reformulation of a likely source theorem.

Some reasoning is required to distinguish the strongest correct statement from a weaker true-looking option and from quantifier/condition tamperings. However, for someone who knows the theorem, the answer is mostly recognition rather than substantial generative derivation.

The distractors are mathematically plausible and target realistic failure modes: dropping a clause, weakening 'for every' to 'there exists', altering the trim inequality, and modifying the characteristic-cycle conclusion. They are distinct and nontrivial.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/recognition than deep generative reasoning.