\begin{thm}\label{mainthm}
    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.
\end{thm}

\par In slightly more detail,
given a boundaryless surface with negative Euler characteristic
and a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\colon S \to S$, one can construct the veering triangulation of the mapping torus of $f$, whose
flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.

\subsection{Connections to literature}\label{discuss}

\par Work of Penner-Papadopoulos \cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.
\par In the setting of \cite[Theorem 4.1]{PP},
the support map $\sigma$ produced for the $f$-invariant train track $\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\tau$ could e.g. be mapped by $\sigma \circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\sigma \circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\sigma \circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\tau) \prec \tau$ has irreducible transition matrix while $f(\tau) \prec \tau$ does not.

\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \textit{\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \Cref{mainthm} necessarily produces train tracks which are not generic.
This motivates the following question:

\begin{qn}
    For all $f\colon S \to S$ pseudo-Anosov, does there exist a \textit{generic} invariant train track for $f$ with irreducible transition matrix?
\end{qn}

\par \textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.

\section{Background}\label{background}

\par A \textit{\textbf{strongly connected component}} of a directed graph $G$ is a maximal subset $C$ of the vertices of $G$ such that for any two vertices in $C$, there is a path in $G$ from the first vertex to the second.
If $G$ has only one strongly connected component, it is called \textit{\textbf{strongly connected}}.
A nonnegative square matrix $A$ is called \textit{\textbf{irreducible}} if for every pair of indices $i$, $j$ there is an $n>0$ such that the $(i,j)$-th entry of $A^n$ is positive. $A$ is irreducible if and only if it is the adjacency matrix of a strongly connected graph.
\par
    Given an orientable finite-type surface $S$ without boundary and negative Euler characteristic, a homeomorphism $f\colon S \to S$ is called \textit{\textbf{pseudo-Anosov}} if there is a pair of transverse measured singular foliations $(\Lambda^u,\mu_u)$ and $(\Lambda^s,\mu_s)$, called the \textit{\textbf{unstable}} and \textit{\textbf{stable}} foliations, and a constant $\lambda > 1$, called the \textit{\textbf{stretch factor}}, such that $f(\Lambda^u)=\Lambda^u$, $f(\Lambda^s)=\Lambda^s$ and $f(\mu^u) = \lambda \mu^u$, $f(\mu^s) = \lambda^{-1} \mu^s$. No nontrivial power of a pseudo-Anosov map fixes the homotopy class of any essential closed curve in $S$. A pseudo-Anosov map is a diffeomorphism except at the singular points of $\Lambda^u$.

    \par The \textit{\textbf{mapping torus of $f$}} is the 3-manifold
    $$ M_f = \faktor{(S \times [0,1])}{ (f(p),0) \sim (p,1)} $$ which fibers over the circle with fiber $S$ and monodromy $f$.

    \par If all of the singularities of the foliations are at the punctures of $S$, we say $f$ is \textit{\textbf{fully punctured}}. We will study general pseudo-Anosovs by deleting the singularities of the foliations to obtain a new surface $S^\circ$ on which the restriction of $f$ is fully punctured.
For full background on pseudo-Anosov homeomorphisms, see Fathi--Laudenbach--Poénaru \cite{FLP}.

\subsection{Train tracks}\label{tts}
\par A \textit{\textbf{train track}} $\tau$ is a closed 1-complex embedded in a surface $S$ with a ``smoothing" at each vertex so that $\tau$ has a well-defined tangent space at each vertex, and $S \setminus \tau$ contains no nullgons, unpunctured monogons or unpunctured bigons. See Penner-Harer \cite{PH}. The vertices of $\tau$ are called \textit{\textbf{switches}} and the edges are called \textit{\textbf{branches}}. If the switches of $\tau$ all have degree three, $\tau$ is called \textit{\textbf{generic}}.

\par Given two train tracks $\tau_1$ and $\tau_2$ on $S$, $\tau_2$ is said to \textit{\textbf{carry}} $\tau_1$ (or $\tau_1$ \textit{\textbf{is carried by}} $\tau_2$), denoted $\tau_1 \prec \tau_2$, if there is a $C^1$ map $\sigma\colon S \to S$ called the \textit{\textbf{support map}}
such that
    \begin{enumerate}
        \item $\sigma$ is homotopic to the identity,
        \item $\sigma(\tau_1) \subseteq \tau_2$,
        \item for all points $x$ in $\tau_1$,
    $d_x \sigma_{|\tau_1}\colon T_x \tau_1 \to T_{h(x)}\tau_2$ is an isomorphism of tangent spaces,
    \end{enumerate}
Note that carrying is transitive: if $\tau_1 \prec \tau_2$ and $\tau_2 \prec \tau_3$, then
$\tau_1 \prec \tau_3$.
We also write $\tau_1 \prec_\sigma \tau_2$ to indicate that $\sigma$ is the support map for $\tau_1 \prec \tau_2$. A carrying map $\sigma$ for $\tau_1 \prec \tau_2$  is called \textit{\textbf{combinatorial}} if for every switch $v$ in $\tau_1$, $\sigma(v)$ is a switch in $\tau_2$.
    \par
    Given $\tau_1$ with branches $\{b^1_j\}_{j=1\ldots n}$ and $\tau_2$ with branches $\{b^2_i\}_{i=1\ldots m}$ such that $\tau_1$ is carried by $\tau_2$ with combinatorial support map $\sigma$,
    the \textit{\textbf{transition matrix}} is the $m \times n$ matrix $T = [t_{i,j}]$ where
    $t_{i,j}$ is the number of times the image of $b^1_j$ under $\sigma$ passes over $b^2_i$.
\par
    A \textit{\textbf{train path}} is a $C^1$ immersion $p\colon [0,1] \to \tau$ so that the endpoints $\{0,1\} = \partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\tau$. For $t \in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\colon [0,t] \to \tau$, $p_1 = p_{|[t,1]}\colon [t,1] \to \tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\partial_t)$ and $d_0 p_1(\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\colon [0,2] \to \tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\ldots p_{n-1})$, where $p_i\colon\left[\frac{i}{n},\frac{i+1}{n}\right] \to \tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \textit{\textbf{embedded}}.
\par
    A \textit{\textbf{fold}} is a particular combinatorial support map $\phi$ for $\tau_1 \prec \tau_2$ which induces a bijection between the branches of $\tau_1$ and $\tau_2$
    except on a particular connected subgraph where it is defined according to either of the two pictures in \Cref{fig:fold}. The three branches of $\tau_1$ which are mapped over the small branch $e'$ in the domain of \Cref{fig:fold} are said to \textit{\textbf{fold to}} $e'$.
    If there is a $\tau_1 \prec_\phi \tau_2$
    we say $\tau_1$ \textit{\textbf{folds to}} $\tau_2$, denoted $\tau_1 \leftharpoonup \tau_2$. If $\tau_1$ folds to $\tau_2$, $\tau_1$ is carried by $\tau_2$. If $\tau_1 \prec \tau_2$ with support map given by a sequence of folds we also say $\tau_1$ folds to $\tau_2$.
    \begin{figure}[h]
    \centering
    \includegraphics[width=0.66\textwidth]{foldgood.pdf}
    \caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}
    \label{fig:fold}
\end{figure}
    \par
    A train track $\tau$ is an \textit{\textbf{invariant}} track for a map $f: S \to S$ with support map $\sigma$ if $f(\tau)$ is carried by $\tau$ with support map $\sigma$.
    If the transition matrix for $f(\tau) \prec_\sigma \tau$ is irreducible, we also say $\tau$ is \textit{\textbf{irreducible}}.
By smoothly isotoping $\tau$ if necessary, we may assume it is disjoint from the singularities of $\Lambda^u$. This will be convenient later when we remove the singularities of $\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:

\begin{thm}[\protect{\cite[Theorem 3.5]{A}}]\label{ttfs} Given $f\colon S \to S$ pseudo-Anosov, there exists a generic invariant train track $\tau$ for $f$ such that $f(\tau)$ folds to $\tau$, i.e. there exist a finite sequence of train tracks
$$f(\tau) = \tau_0 \leftharpoonup \tau_1 \leftharpoonup \ldots \leftharpoonup \tau_n = \tau,$$
where $\tau_i$ is carried by $\tau_{i+1}$ with support map given by a single fold.
\end{thm}

\label{intro}
Given a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.
The existence of invariant train tracks with irreducible transition matrix
allows the application of Perron-Frobenius theory to the study of the properties of these maps.
For example, when an invariant train track is irreducible it can be used to build a Markov partition for the map.
The existence of irreducible invariant train tracks is often taken for granted
(See e.g. \cite{FR}, \cite{F}, \cite{J}, \cite{EF}, \cite{M}, \cite{Pe})
but to our knowledge no complete proof is currently in the literature. See \Cref{discuss} for more details.

\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.
One important datum attached to a veering triangulation is its \textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \cite{LMT1}.
When the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with
irreducible transition matrices.
\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component" from which there are paths to every vertex, and some number of ``infinitesimal cycles" which do not have any paths back to the rest of the graph \cite{AT}.
By studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:

\par In slightly more detail,
given a boundaryless surface with negative Euler characteristic
and a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\colon S \to S$, one can construct the veering triangulation of the mapping torus of $f$, whose
flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.

\begin{qn}
 For all $f\colon S \to S$ pseudo-Anosov, does there exist a \textit{generic} invariant train track for $f$ with irreducible transition matrix?
\end{qn}

\par Given two train tracks $\tau_1$ and $\tau_2$ on $S$, $\tau_2$ is said to \textit{\textbf{carry}} $\tau_1$ (or $\tau_1$ \textit{\textbf{is carried by}} $\tau_2$), denoted $\tau_1 \prec \tau_2$, if there is a $C^1$ map $\sigma\colon S \to S$ called the \textit{\textbf{support map}}
such that
 \begin{enumerate}
 \item $\sigma$ is homotopic to the identity,
 \item $\sigma(\tau_1) \subseteq \tau_2$,
 \item for all points $x$ in $\tau_1$,
 $d_x \sigma_{|\tau_1}\colon T_x \tau_1 \to T_{h(x)}\tau_2$ is an isomorphism of tangent spaces,
 \end{enumerate}
Note that carrying is transitive: if $\tau_1 \prec \tau_2$ and $\tau_2 \prec \tau_3$, then
$\tau_1 \prec \tau_3$.
We also write $\tau_1 \prec_\sigma \tau_2$ to indicate that $\sigma$ is the support map for $\tau_1 \prec \tau_2$. A carrying map $\sigma$ for $\tau_1 \prec \tau_2$ is called \textit{\textbf{combinatorial}} if for every switch $v$ in $\tau_1$, $\sigma(v)$ is a switch in $\tau_2$.
 \par
 Given $\tau_1$ with branches $\{b^1_j\}_{j=1\ldots n}$ and $\tau_2$ with branches $\{b^2_i\}_{i=1\ldots m}$ such that $\tau_1$ is carried by $\tau_2$ with combinatorial support map $\sigma$,
 the \textit{\textbf{transition matrix}} is the $m \times n$ matrix $T = [t_{i,j}]$ where
 $t_{i,j}$ is the number of times the image of $b^1_j$ under $\sigma$ passes over $b^2_i$.
\par
 A \textit{\textbf{train path}} is a $C^1$ immersion $p\colon [0,1] \to \tau$ so that the endpoints $\{0,1\} = \partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\tau$. For $t \in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\colon [0,t] \to \tau$, $p_1 = p_{|[t,1]}\colon [t,1] \to \tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\partial_t)$ and $d_0 p_1(\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\colon [0,2] \to \tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\ldots p_{n-1})$, where $p_i\colon\left[\frac{i}{n},\frac{i+1}{n}\right] \to \tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \textit{\textbf{embedded}}.
\par
 A \textit{\textbf{fold}} is a particular combinatorial support map $\phi$ for $\tau_1 \prec \tau_2$ which induces a bijection between the branches of $\tau_1$ and $\tau_2$
 except on a particular connected subgraph where it is defined according to either of the two pictures in \Cref{fig:fold}. The three branches of $\tau_1$ which are mapped over the small branch $e'$ in the domain of \Cref{fig:fold} are said to \textit{\textbf{fold to}} $e'$.
 If there is a $\tau_1 \prec_\phi \tau_2$
 we say $\tau_1$ \textit{\textbf{folds to}} $\tau_2$, denoted $\tau_1 \leftharpoonup \tau_2$. If $\tau_1$ folds to $\tau_2$, $\tau_1$ is carried by $\tau_2$. If $\tau_1 \prec \tau_2$ with support map given by a sequence of folds we also say $\tau_1$ folds to $\tau_2$.
 \begin{figure}[h]
 \centering
 \includegraphics[width=0.66\textwidth]{foldgood.pdf}
 \caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}
 \label{fig:fold}
\end{figure}
 \par
 A train track $\tau$ is an \textit{\textbf{invariant}} track for a map $f: S \to S$ with support map $\sigma$ if $f(\tau)$ is carried by $\tau$ with support map $\sigma$.
 If the transition matrix for $f(\tau) \prec_\sigma \tau$ is irreducible, we also say $\tau$ is \textit{\textbf{irreducible}}.
By smoothly isotoping $\tau$ if necessary, we may assume it is disjoint from the singularities of $\Lambda^u$. This will be convenient later when we remove the singularities of $\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:

\par Note that the referenced theorem is stated in terms of \textit{splits} rather than folds, which are the combinatorial inverse of folds.
 Additionally, the referenced theorem is stronger than what we state here because the splitting sequence is shown to be canonical up to ``commuting maximal splits": at each step in the sequence, the train track is split at a branch carrying maximal measure of the invariant lamination, and the splits of maximal branches commute if there is not a unique branch carrying maximal measure.
 This theorem motivates the following definition:
\begin{defn}
 An invariant train track $\tau$ for a pseudo-Anosov map $f$ is called \textit{\textbf{veering}} if it
 is generic and $\tau$ can be obtained from $f(\tau)$ by a sequence of folds, in which case there is a sequence of train tracks
 $$f(\tau) = \tau_0 \leftharpoonup \tau_1 \leftharpoonup \ldots \leftharpoonup \tau_n = \tau$$
 where $\tau_i$ is carried by $\tau_{i+1}$ with support map consisting of a single fold.
\end{defn}
\par In \Cref{clvts} we will see that a veering train track as defined here is sufficient to produce the veering triangulation of the fully punctured mapping torus of $f$. While veering train tracks are not necessarily irreducible, they will be the beginning of our construction of irreducible tracks.
\par
 Given two train tracks $\tau_1$, $\tau_2$ on $S$, a (surface) \textit{\textbf{train track map}} is a map $t: S \to S$ so that $t(\tau_1)$ is contained in $\tau_2$, and the restriction of $t$ to $\tau_1$ is
 such that for any train path $p\colon [0,1] \to \tau_1$, $t \circ p\colon [0,1] \to \tau_2$ is also a train path. In other words, a train track map is like a combinatorial support map except that it does not have to be homotopic to the identity map.
 \par The following lemma will be useful for producing support maps:
\begin{lemma}\label{supplemma}
 Let $t\colon S \to S$ be a train track map taking $\tau_1$ to $\tau_2$, and $g\colon S \to S$ be a diffeomorphism.
 If $t$ is homotopic to $g$, then $g(\tau_1) \prec \tau_2$ with support map given by $t \circ g^{-1}$.
\end{lemma}
\begin{proof}
 Since $t$ and $g$ are homotopic, $t \circ g^{-1}$ is homotopic to the identity. Since $t(\tau_1) \subseteq \tau_2$, $(t \circ g^{-1})(g(\tau_1)) = t(\tau_1) \subseteq \tau_2$. Let $p \in g(\tau_1)$. Since $g$ is a diffeomorphism, the restriction of the differential
 $$d_p g^{-1}_{|g(\tau_1)}\colon T_p g(\tau_1) \to T_{g^{-1}(p)} \tau_1$$
 is an isomorphism. Since $t$ is a train track map, $d_{g^{-1}(p)} t_{| \tau_1}$ is an isomorphism, so the composition
 $$d_p (t \circ g^{-1})_{|g(\tau_1)} = (d_{g^{-1}(p)} t_{| \tau_1}) \circ (d_p g^{-1}_{|g(\tau_1)})\colon T_p g(\tau_1) \to T_{(t \circ g^{-1})(p)} \tau_2$$
 is also an isomorphism.
\end{proof}

Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\colon S\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.

\label{mainthm}
    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.,

Question: Let \(f\colon S\to S\) be a pseudo-Anosov homeomorphism. A train track \(\tau\) on \(S\) is called invariant for \(f\) if \(f(\tau)\) is carried by \(\tau\) via a combinatorial support map. If the branches of \(f(\tau)\) are \(\{b^1_j\}\) and the branches of \(\tau\) are \(\{b^2_i\}\), the associated transition matrix is the nonnegative matrix \(T=[t_{i,j}]\), where \(t_{i,j}\) is the number of times the image of \(b^1_j\) under the support map passes over \(b^2_i\). A nonnegative matrix is irreducible if for every \(i,j\) there exists \(n\ge 1\) such that \((T^n)_{i,j}>0\). Under these assumptions, which existence statement holds?

For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that the transition matrix associated to \(f(\tau)\prec \tau\) is irreducible.

Choices:
B. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), every invariant train track \(\tau\) for \(f\) has the property that the transition matrix associated to \(f(\tau)\prec \tau\) is irreducible.
C. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that the transition matrix associated to \(f(\tau)\prec \tau\) is nonzero.
D. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that for every pair of branches \(b^2_i,b^2_j\) of \(\tau\), one has \(t_{i,j}>0\) in the transition matrix associated to \(f(\tau)\prec \tau\).
E. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that each strongly connected component of the transition matrix associated to \(f(\tau)\prec \tau\) is irreducible.

existential choice of modified track after contracting obstructing branches

irreducibility requirement dropped to mere existence of a nontrivial transition matrix

irreducible replaced by strictly positive one-step transition matrix

infinitesimal-cycle obstruction bypassed only after contracting branches, not for the original component decomposition


\begin{Definition}\label{def:phi homogenization}
Fix $\lambda = \lambda(n) > 0$ sufficiently small such that $\sigma = C_1(n)\lambda$ satisfies $4^{n+1}\sigma < c(n)$, where $C_1(n)$ is the one in Lemma \ref{lem:covering Lemma} and $c(n)$ is the one in Proposition \ref{prop:basic}. For $\Lambda\ge \max\{\Lambda_0,\Lambda_1,\Lambda_2\}$, and any nonnegative functions $v \in C(\Omega)$, we define its $\phi$-homogenization $M_{\phi,\Lambda}v(x):=M_{\phi,\lambda,\Lambda}v(x)$ in $\frac{7}{8}\Omega$ through the following maximal averaging type operator:
\begin{equation}\label{eq:phi homogenization}
\begin{split}
M_{\phi,\Lambda}v(x)=  \sup\left\{ t\in\R:\; \exists \ \check{h}(x)<h<  \hat{h}(x),\ s.t., \
\frac{\mu ( \{v> t\} \cap S_h (x)  )}{\mu(S_h (x))} > \lambda \right\}.
\end{split}
\end{equation}
\end{Definition}

\begin{Theorem}\label{thm:uniqueness within periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ be a quadratic function with $A\in \mathcal{S}_+^{n\times n}$ satisfying \eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \eqref{eq:periodic equation ma} with quadratic component $P$.
\end{Theorem}

In this paper, we study convex solutions to the Monge-Amp\`ere equation
\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \mathbb{R}^n,
\end{equation}
where $\mu\not\equiv 0$ is a nonnegative locally finite Borel measure on $\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\mu $ is periodic with respect to the integer lattice $\mathbb{Z}^n := \{(k_1,\dots,k_n) \mid k_i \in \mathbb{Z}\} \subset \mathbb{R}^n$, i.e., for any Borel set $E \subset \mathbb{R}^n$,
\begin{equation}\label{eq:periodic f}
\mu (E+z)=\mu (E),\quad \forall \ z \in \mathbb{Z}^n.
\end{equation}

Let $\mathcal{S}_+^{n\times n}$ denote the set of positive definite symmetric $n\times n$ matrices.
\begin{Definition}[Periodic solution]\label{def:periodic solution}
A \emph{periodic solution} to \eqref{eq:periodic equation ma} is a convex function $u \in C(\mathbb{R}^n)$ satisfying \eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as
\begin{equation}\label{eq:periodic solution}
u(x) = v(x) +P(x),
\end{equation}
where $v$ is periodic with respect to $\mathbb{Z}^n$ and $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ is a quadratic function with $A\in \mathcal{S}_+^{n\times n}$, $b\in\mathbb{R}^n$ and $c\in\mathbb{R}$.
\end{Definition}

Note that if $u$ is a periodic solution to \eqref{eq:periodic equation ma} and is expressed as \eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition
\begin{equation}\label{eq:A compatible}
 \det A =\mu (\mathbb{T}^n),
\end{equation}
where $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$.

Our first result is the uniqueness of periodic solutions.

\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \R^n,
\end{equation}

Let $\mu \not\equiv 0$ be a nonnegative locally finite periodic Borel measure on $\R^n$. We show that any convex solution to the Monge-Amp\`ere equation
\[
\det D^2 u = \mu \quad \text{in } \R^n
\]
admits a unique decomposition (up to addition of constants) as the sum of a quadratic polynomial and a periodic function. This result extends,
in full generality, the earlier works for the case $\mu=f(x)\,\ud x$: when $\log f \in C^\alpha$, it was established by Caffarelli and Li; and
when $\log f$ is merely bounded, it was proved by Li and Lu. Our result thus answers a question raised by Li and Lu. A key ingredient in the proof is a new dichotomous Harnack-type inequality for linearized Monge-Amp\`ere equations with nonnegative periodic measures.

Let $\mathcal{S}_+^{n\times n}$ denote the set of positive definite symmetric $n\times n$ matrices.
\begin{Definition}[Periodic solution]\label{def:periodic solution}
A \emph{periodic solution} to \eqref{eq:periodic equation ma} is a convex function $u \in C(\mathbb{R}^n)$ satisfying \eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as
\begin{equation}\label{eq:periodic solution}
u(x) = v(x) +P(x),
\end{equation}
where $v$ is periodic with respect to $\mathbb{Z}^n$ and $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ is a quadratic function with $A\in \mathcal{S}_+^{n\times n}$, $b\in\R^n$ and $c\in\R$.
\end{Definition}

Our first result is the uniqueness of periodic solutions.

The existence of periodic solutions follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof. For the uniqueness, when $\mu=f(x)\,\ud x$ is a positive measure, the case $\log f \in C^\alpha$ was proved by Li \cite{li1990existence}, and the case $\log f\in L^\infty$ was shown by Li-Lu \cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.

\begin{Theorem}\label{thm:solutions are periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $u$ be a convex solution of \eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \ref{def:periodic solution}.
\end{Theorem}

\begin{Proposition}\label{prop:semiconcave}
Let $u$ be a convex solution to \eqref{eq:periodic equation ma} with $\mu$ being a $\mathbb{Z}^n/8n$-periodic measure (where $\mathbb{Z}^n := \Gamma_{\{e_1,\cdots,e_n\}}$) satisfying $\mu(\mathbb{Z}^n) = 1$.
Under the nondegeneracy condition
\begin{equation}\label{eq:nondegenerate}
\Delta^2_{i} u(0):=\Delta^2_{e_i} u(0) > 0 \quad \text{for all } 1 \leq i \leq n,
\end{equation}
there exists a unique compatible matrix $A$ satisfying:
\[
\sup_{ \R^n}\Delta_z^2 u=z^{\top}Az ,\quad \forall z \in \mathbb{Z}^n.
\]
\end{Proposition}
\begin{proof}
Without loss of generality, we may assume $u(0) = 0$ and $u \geq 0$.
Let us define $S_h := \{x \in \mathbb{R}^n : u(x) \leq h\}$, and select $\Lambda_u>0$ such that the section $S_{\Lambda_u}$ satisfies $\kappa_{\mathbb{Z}^n}(S_{\Lambda_u}) \geq \Lambda_0^2$.
Let us now normalize $S_h$ such that $B_1(0) \subset T_h^{-1} S_h \subset B_n(0)$ and consider the normalized function
\[
u_{h}(x) =\frac{u(T_h x)}{(\det T_h)^{\frac{2}{n}}} \quad \text{with } \det D^2 u_h= \mu_h=\mu \circ T_h .
\]
Due to the periodicity of $\mu$ and large $h$, we know that
\[
c(n)h^{\frac{n}{2}} \leq \det T_h \leq C(n)h^{\frac{n}{2}},
\]
and thus, $c(n)h^{\frac{n}{2}} \leq |S_h| \leq C(n)h^{\frac{n}{2}}$. Applying Proposition \ref{prop:basic} to $\phi=u_h-h(\det T_h)^{-\frac{2}{n}}$, we obtain that for sufficiently large $h$,
\[
c(n) (h/\Lambda_u)^{\frac{1}{3}} S_{\Lambda_u} \subset S_h \subset C(n) (h/\Lambda_u)^{\frac{2}{3}} S_{\Lambda_u},
\]
and consequently, $B_{c(n) h^{\frac{1}{4}}}\subset S_h\subset B_{C(n) h^{\frac{3}{4}}}$ for all large $h$. That is, for sufficiently large $h$, we now have
\[
\quad c(n)h^{\frac{1}{4}} I\leq T_h \leq C(n)h^{\frac{3}{4}} I.
\]
The transformation law \eqref{eq:transformation law} then yields the following estimates at corresponding points
\begin{equation}\label{eq:transformh}
c(n)h^{-\frac{1}{2}}\Delta^2_{e_i} u \leq \Delta^2_{T_h^{-1} e_i} u_h \leq C(n)h^{\frac{1}{2}}\Delta^2_{e_i} u.
\end{equation}

\textbf{Step 4.} Suppose for some sequence $h \to \infty$, the rescaled functions $\tilde{u}_h$ locally converge to a quadratic form $P_A(x) = \frac{1}{2}x^{\top}A x$ for $A \in \mathcal{S}_+^{n\times n}$ with $\det A = 1$.
For each fixed direction $i$, we denote
\[
\alpha=\sup_{\R^n}\Delta^2_{e_i} u,\quad \beta= e_i^{\top} A e_i.
\]
Note that the estimate \eqref{eq:w21 estimate} continues to hold for $\tilde{u}_{h}$, from which we derive $\alpha \geq \beta$.
We claim
\begin{equation}\label{eq:w2infty sup}
\alpha=\beta.
\end{equation}
Assume by contradiction that $\alpha = \beta + 4s$ for some $s > 0$.
Through rescaling from infinity, we may assume without loss of generality that there exist points $x_h \in B_{1/8}$ satisfying
\[
\alpha- \Delta^2_{h^{\frac{1}{2}}e_i} \tilde{u}_{h}(x_h)= \inf_{B_{1/8}} (\alpha- \Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_{h} ) \leq a_h,
\]
where $a_h \to 0 $ as $h \to \infty$. Define the normalized function $v = (\alpha - \Delta^2_{h^{\frac{1}{2}}e_i} \tilde{u}_{h})/a_h$ and set $\delta = s/[C(\beta + 2s)]$. Applying Lemma \ref{lem:Theorem 7.3.1 modify} yields with $\phi=\tilde{u}_h-1$ and $\Omega=S_{1,h}=\{\tilde{u}_h \leq 1\} $, we obtain that
\[
\tilde{\mu}_h ( \{v\leq M_1(n,\delta) \} \cap S_{1,h} ) \geq (1-\delta) \tilde{\mu}_h(S_{1,h} ),
\]
where $\tilde{\mu}_h=\tilde{f}_{h}\,\ud x$. For sufficiently small $a_h$, we consequently obtain
\[
\tilde{\mu}_h\left(\left\{\Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_h>\beta +2 s\right\} \cap S_{1,h}\right)= \tilde{\mu}_h\left(\left\{v <\frac{2s}{a_h}\right\} \cap S_{1,h}\right)\geq (1-\delta) \tilde{\mu}_h (S_{1,h}).
\]
This leads to the integral lower bound
\[
\int_{S_{1,h}} \Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_h \ud \tilde{\mu}_h \geq (1-\delta) (\beta+2s) \tilde{\mu}_h (S_{1,h})\geq
(\beta+s) \tilde{\mu}_h (S_{1,h}).
\]
This contradicts the uniform estimate \eqref{eq:w21 estimate} after performing a suitable rescaling.

\begin{proof}[Proof of Theorems \ref{thm:uniqueness within periodic} and \ref{thm:solutions are periodic}]
Let $u$ be a global solution of \eqref{eq:periodic equation ma}.
Clearly, $u$ cannot be linear on any ray segment. Indeed, if it were linear on a ray, after subtracting its supporting hyperplane on this ray and by using the convexity of $u$, $u$ would be bounded within any cylinder having that ray as its axis. This would imply that the associated measure $\mu$ is finite on any strictly smaller cylinder, contradicting the assumptions that $\mu$ is periodic and $\mu \not\equiv 0$. Therefore, there exists a sufficiently large integer $R$ such that $S_1(0) \subset B_R(0)$.
We now rescale $u$ by defining $u(8nRx)/64n^2R^2$. For simplicity, we continue to denote this rescaled function as $u$. Under this scaling, one can verify that condition \eqref{eq:nondegenerate} in Proposition \ref{prop:semiconcave} is satisfied. Hence, by Proposition \ref{prop:semiconcave}, there exists a compatible matrix $A \in \mathcal{S}_+^{n \times n}$ such that
\[
\sup_{ \R^n}\Delta_z^2 u=z^{\top}Az,\quad \forall z \in \mathbb{Z}^n.
\]
Let $w$ be a periodic solution to \eqref{eq:periodic equation ma} in the sense of Definition \ref{def:periodic solution} with quadratic part $\frac{1}{2}z^{\top} A z $. Then $\Delta_z^2 w=z^{\top}Az$. Let
\[
v(x)=u(x)-w(x).
\]
By subtracting a suitable linear function, we now assume that $v(0)=0$ and
\[
v(e_i)=v(-e_i),\quad 1\leq i\leq n.
\]
Noting that $\sup_{\R^n} \Delta_z^2 v = \sup_{\R^n} \Delta_z^2 (u - w) = 0$, we then find that
\[
v(\pm ke_i)\leq 0 \quad \text{for all } k\in \mathbb{Z},\ 1\leq i\leq n.
\]

The existence of periodic solutions follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof.  For the uniqueness, when $\mu=f(x)\,\mathrm{d} x$ is a positive measure, the case $\log f \in C^\alpha$ was proved by Li \cite{li1990existence}, and the case  $\log f\in L^\infty$ was shown by Li-Lu \cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.

Our second result is the classification below, which extends the main results  of Caffarelli-Li \cite{caffarelli2004liouville} and Li-Lu \cite{li2019monge} to the full generality. In particular, it provides a complete answer to Question 1.1 in \cite{li2019monge}.

\begin{Theorem}\label{thm:solutions are periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $u$ be a convex solution of \eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \ref{def:periodic solution}.
\end{Theorem}

Caffarelli-Li \cite{caffarelli2004liouville} first proved Theorem \ref{thm:solutions are periodic} for positive measures $\mu=f(x)\,\mathrm{d} x$ with $\log f \in C^\alpha$, and conjectured the extension to merely bounded $\log f$. This conjecture was resolved by Li-Lu \cite{li2019monge}, who further asked whether the result holds for the nonnegative case $f \geq 0$ with $f \not\equiv 0$. Theorem \ref{thm:solutions are periodic} affirmatively answers this question -- and indeed establishes it in full generality.

\begin{Remark}
When $\mu \equiv 0$, equation \eqref{eq:periodic equation ma} reduces to the homogeneous Monge-Amp\`ere equation $\det D^2u = 0$. The solutions are precisely those functions that are linear along certain directions (see, e.g., Caffarelli-Nirenberg-Spruck \cite{caffarelli1986dirichlet}).
In particular, writing $x = (x',x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}$, any convex function of the form $u(x) = w(x')$ is a solution, and such solutions are not necessarily periodic.
\end{Remark}

Since $\mu$ is assumed to be nonnegative,  equation \eqref{eq:periodic equation ma} represents a possibly degenerate Monge-Amp\`ere equation, for which many arguments in  \cite{caffarelli2004liouville,li2019monge} do not apply.  The principal challenge stems from the behavior of $u$ on small sections: the associated measure $\mu$ may not satisfy the doubling condition. This failure implies that sections lose the engulfing property, consequently invalidating the Vitali or  Besicovitch covering lemma which is essential for the Calder\'on-Zygmund decomposition on sections.
Therefore, we cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez \cite{caffarelli1997properties} for subsolutions or supersolutions of linearized Monge-Amp\`ere equations.

Nonetheless, by exploiting the equation's periodic structure, we derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator (see Definition \ref{def:phi homogenization}). This constitutes one of the key innovations of the present work. For supersolutions, we address the challenge by excluding contributions from small sections. Theorem \ref{thm:harnack super} establishes decay estimates for homogenized level-sets of supersolutions within our framework. For subsolutions, for which standard $L^{\infty}$ bounds fail, Theorem \ref{thm:harnack sub} establishes an alternative dichotomy regarding their growth behavior: either the $L^{\infty}$ bound holds or the $L^{\infty}$ norm grows exponentially. Finally, we obtain  in Theorem \ref{thm:harnack} a Harnack inequality with a quantitatively controlled error term.

After overcoming this critical difficulty arising from the degeneracy of equation \eqref{eq:periodic equation ma}, the rest proof focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart. As observed in \cite{caffarelli2004liouville}, these quantities naturally arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations. This is another point where the periodicity of the measure plays a crucial role. We then establish the quadratic behavior of solutions near infinity, extending the arguments in \cite{caffarelli2004liouville,li2019monge} for positive measures $\mu=f(x)\mathrm{d} x$ to  the degenerate case considered here.  In this step, we must also overcome additional difficulties due to the degeneracy of
 \eqref{eq:periodic equation ma}. Consequently, when Theorem \ref{thm:harnack} is applied to the difference of two solutions, exponential growth is precluded. Hence, the classical Harnack inequality holds for this difference, and our theorems would follow.

This paper is organized as follows. In Section \ref{sec:harnack}, we study convex functions whose Monge-Amp\`ere measure has small period, establishing a dichotomous Harnack type  inequality for linearized Monge-Amp\`ere equations. Section \ref{sec:semiconvavity} investigates  the uniqueness of compatible quadratic component of each solution in the sense of Proposition \ref{prop:semiconcave}. Finally, in Section \ref{sec:main theorem}, we prove  Theorems \ref{thm:uniqueness within periodic} and \ref{thm:solutions are periodic}.

The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof,” while Theorem~\ref{thm:uniqueness within periodic} “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including Theorem~\ref{thm:uniqueness within periodic}) in the nonnegative/degenerate setting: “the associated measure $\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez.”

To overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” For supersolutions they do so “by excluding contributions from small sections,” obtaining “decay estimates for homogenized level-sets” (Theorem~\ref{thm:harnack super}); for subsolutions (where “standard $L^{\infty}$ bounds fail”), they prove a “dichotomy” (Theorem~\ref{thm:harnack sub}): “either the $L^{\infty}$ bound holds or the $L^{\infty}$ norm grows exponentially”; combining these yields “a Harnack inequality with a quantitatively controlled error term” (Theorem~\ref{thm:harnack}).

“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying Theorem~\ref{thm:harnack} “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and “our theorems would follow.”

The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof,” while in establishing the main theorem it “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including the main theorem) in the nonnegative/degenerate setting: “the associated measure $\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez.”

To overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” We first use the following theorem for supersolutions:
\begin{Theorem}\label{thm:harnack super}
Assume that $v \in C^2(\Omega)$ is a nonnegative supersolution of \eqref{eq:linearized ma phi} in $\Omega$.
Then there exist positive constants  $\Lambda_6(n)$, $\epsilon(n) $ and $C(n) $ such that if
\[
\inf_{B_{1/4}} v \leq 1,
\]
then for $\Lambda\ge \Lambda_6(n)$, we have
\begin{equation}\label{eq:harnack super}
\mu(\{M_{\phi,\Lambda}v >  t\} \cap B_{1/2})< Ct^{-\epsilon} .
\end{equation}
\end{Theorem}
For subsolutions (where “standard $L^{\infty}$ bounds fail”), they prove the following “dichotomy”:
\begin{Theorem}\label{thm:harnack sub}
Assume that $v \in C^2(\Omega)$ with $v \geq 0$ is a subsolution of \eqref{eq:linearized ma phi} in $\Omega$.
Then, there exist positive constants $\Lambda_7(n)$ and $\beta(n)$ such that for any $\epsilon > 0$, there is a constant $C_2(n, \epsilon)$ with the following property: If $\Lambda\ge \Lambda_7(n)$ and
\[
\min\{\mu(\{v >  t\}  \cap B_{1/2}), \mu(\{M_{\phi,\Lambda}v >  t\} \cap B_{1/2})  \} \leq t^{-\epsilon}   \quad  \forall t>0,
\]
then either
\[
\sup_{B_{1/4} } v  \leq C_2(n,\epsilon),
\]
or for $\kappa =\kappa(B_1)\geq \Lambda^2$ that
\[
\sup_{B_{1/2} } v  \geq e^{\beta \kappa^{\frac{1}{3}} } \sup_{B_{1/4} }v.
\]
\end{Theorem}
Combining these yields “a Harnack inequality with a quantitatively controlled error term,” namely the following theorem:
\begin{Theorem}\label{thm:harnack}
Let $\Omega$ and $\widetilde{\Omega}$ be open convex subsets of $\mathbb{R}^n$ satisfying $B_1\subset \Omega \subset B_n, B_1\subset \widetilde{\Omega} \subset B_n$, $n\geq 2$, and let $\phi\in C^2(\overline{\Omega})$ and $\tilde{\phi}\in C^2(\overline{\widetilde{\Omega}})$ be convex functions satisfying,
\[
\mu_\phi =\mu \quad \text{in } \Omega, \quad \phi=0 \quad \text{on } \partial \Omega,
\]
\[
\mu_{\tilde{\phi}} =\mu \quad \text{in }  \widetilde{\Omega}, \quad \tilde{\phi}=0 \quad \text{on } \partial \widetilde{\Omega},
\]
where $\mu$ is a periodic Borel measure defined on $\R^n$ satisfying \eqref{eq:period gamma},  \eqref{eq:normalized assumption}, and \eqref{eq:small period assumption}. Assume that $v\in C^2(\Omega) \cap C^2(\widetilde{\Omega})$ with $v\geq 0$ satisfies
\[
L_{\phi}v \leq 0   \quad  \text{in } \Omega, \quad L_{\tilde{\phi}}v \geq 0   \quad  \text{in }\widetilde{\Omega}.
\]

Then there exist positive constants $\beta(n)$, $\Lambda_9(n)$,  and $C(n)$ such that if $\Lambda>\Lambda_9(n)$, then either
\[
\sup_{B_{1/4}}v\leq C\inf_{B_{1/4}}v,
\]
or for $\kappa =\kappa(B_1)\geq \Lambda^2$ that
\[
\sup_{B_{1/2}} v  \geq e^{\beta \kappa^{\frac{1}{6}}}\sup_{B_{1/4}}v.
\]
Consequently, we have
\[
\sup_{B_{1/4}}v \leq C\inf_{B_{1/4}}v+e^{-\beta \kappa^{\frac{1}{6}}}\sup_{B_{1/2}} v.
\]
\end{Theorem}

“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying the preceding theorem (the Harnack inequality with controlled error term) “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and this completes the proof strategy for the main theorem.

\label{thm:uniqueness within periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying
\begin{equation}\label{eq:periodic f}
\mu (E+z)=\mu (E),\quad \forall \ z \in \mathbb{Z}^n.
\end{equation}
and let $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ be a quadratic function with $A\in \mathcal{S}_+^{n\times n}$ satisfying
\begin{equation}\label{eq:A compatible}
 \det A =\mu (\mathbb{T}^n),
\end{equation}
Then, to prove the main theorem, we show that there exists a unique periodic solution (up to addition of constants) to
\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \R^n,
\end{equation}
with quadratic component $P$.

Question: Let $\mu\not\equiv 0$ be a nonnegative locally finite Borel measure on $\mathbb{R}^n$ that is $\mathbb{Z}^n$-periodic in the sense that for every Borel set $E\subset \mathbb{R}^n$ and every $z\in \mathbb{Z}^n$,\n\[\n\mu(E+z)=\mu(E).\n\]\nLet\n\[\nP(x)=\tfrac12 x^\top A x+b\cdot x+c,\n\]\nwhere $A\in \mathcal S_+^{n\times n}$ is positive definite symmetric, $b\in \mathbb{R}^n$, $c\in \mathbb{R}$, and\n\[\n\det A=\mu(\mathbb{T}^n),\qquad \mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n.\n\]\nA periodic solution of\n\[\n\det D^2u=\mu\quad\text{in }\mathbb{R}^n\n\]\nmeans a convex function $u\in C(\mathbb{R}^n)$ satisfying the equation in the Alexandrov sense and admitting a decomposition $u(x)=v(x)+P(x)$ with $v$ $\mathbb{Z}^n$-periodic. Under these assumptions, which statement holds?

There exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, and it is unique up to addition of constants; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1-u_2$ is constant on $\mathbb{R}^n$.

Choices:
B. There exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, and it is unique without any ambiguity; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1=u_2$ on $\mathbb{R}^n$.
C. There exists at most one periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$ up to addition of constants; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1-u_2$ is constant on $\mathbb{R}^n$.
D. For every quadratic function $P(x)=\tfrac12 x^\top A x+b\cdot x+c$ with $A\in \mathcal S_+^{n\times n}$ satisfying $\det A\ge \mu(\mathbb{T}^n)$, there exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, unique up to addition of constants.
E. There exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, and for any two periodic solutions $u_1,u_2$ with quadratic components having the same matrix part $A$, the difference $u_1-u_2$ is an affine function on $\mathbb{R}^n$.

constant-shift ambiguity in comparing two solutions

dropped existence while retaining uniqueness up to constants

exact compatibility condition $\det A=\mu(\mathbb{T}^n)$

normalization removing linear parts before applying Harnack to the difference


\begin{prop}[{cf.~\cite[Corollary~3.4]{S78}}]\label{prop:ci}
Let $R$ be a standard graded $\kk$-algebra.
If $R$ is a complete intersection, then the $h$-polynomial of $R$ is of type~CI, that is, $h_R(t)=\prod_{i=1}^r\Psi_{m_i}(t)$ for some non-negative integer $r$ and positive integers $m_1, \dots, m_r$.
\end{prop}

Throughout this paper, let $\Bbbk$ denote a field. We say that $R$ is a \emph{standard graded} $\Bbbk$-algebra
if $R = \bigoplus_{n \ge 0} R_n$ is a commutative graded $\Bbbk$-algebra with $R_0 = \Bbbk$ and finitely generated in degree one.
(A standard graded $\Bbbk$-algebra is also called a \emph{homogeneous} $\Bbbk$-algebra in other literature.)

\subsection{Background}
The Hilbert series of a (not necessarily commutative) graded $\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\Bbbk$-algebras.
It provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\Bbbk$-algebra as a $\Bbbk$-vector space.
It is a classical and well-known fact that if a standard graded $\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \textit{$h$-polynomial}, has all positive coefficients.
This fact is one of the cornerstones in graded commutative algebra.
Furthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series.
A prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\cite{S78}),
which has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness,
such as almost Gorensteinness (\cite{H}), nearly Gorensteinness (\cite{M}), and Artin-Schelter Gorensteinness (\cite{JZ}).

It is also known that when a standard graded $\Bbbk$-algebra is a complete intersection, its Hilbert series takes a particularly simple form.
In fact, for a positive integer $m$, let \[\Psi_m(t)=1+t+\cdots+t^{m-1}.\]
Then the $h$-polynomial of a standard graded complete intersection is of the form of a product of a finite collection of $\Psi_m(t)$'s (see Proposition~\ref{prop:ci}).
In addition, the Hilbert series also serves as an essential tool in investigating noncommutative graded complete intersections (see \cite{KKZ, KKZ2}).

\subsection{Cyclotomic standard graded $\Bbbk$-algebras and polynomials of type CI}

In recent years, increasing attention has been directed to the finer analytic behavior of the Hilbert series of standard graded $\Bbbk$-algebras, particularly the roots of their $h$-polynomials.
These roots encode subtle structural information and sometimes admit striking combinatorial interpretations
(see, e.g., the survey \cite{Br} for combinatorial consequences of the real-rootedness of the numerator polynomials of generating series).

Motivated by this perspective, we focus on the following notion for standard graded $\Bbbk$-algebras:

\begin{dfn}[cf. \cite{BD, KKZ}]
A standard graded $\Bbbk$-algebra $R$ is called \textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane.
\end{dfn}

Monic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \textit{Kronecker polynomials}.
It is known that every such polynomial can be expressed as a product of cyclotomic polynomials.
Note that the $h$-polynomials of cyclotomic $\Bbbk$-algebras are \textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials.
We say that a polynomial is \textit{of type CI} if it can be written as a product of $\Psi_m(t)$'s.
In particular, the $h$-polynomials of standard graded complete intersections are of type CI.
However, the converse does not hold in general; see, for example, \cite[Example~3.7]{S78}.
Borzi and D'Al\`i showed that the converse does hold when $R$ is Koszul~\cite[Theorem~3.9]{BD}.
Moreover, it was proved that a cyclotomic polynomial $\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\cite[Theorem~4.3]{BD}.

Despite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial.
This naturally leads to the following question:
\begin{q}
Which polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\Bbbk$-algebras, other than those of type CI?
\end{q}
For example, \cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\Bbbk$-algebra.

\subsection{Main Results}

The aim of this paper is to make a contribution to this question.
Our main result is stated as follows.

\subsection{Background}
The Hilbert series of a (not necessarily commutative) graded $\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\Bbbk$-algebras.
It provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\Bbbk$-algebra as a $\Bbbk$-vector space.
It is a classical and well-known fact that if a standard graded $\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \textit{$h$-polynomial}, has all positive coefficients.
This fact is one of the cornerstones in graded commutative algebra.
Furthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series.
A prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\cite{S78}),
which has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness,
such as almost Gorensteinness (\cite{H}), nearly Gorensteinness (\cite{M}), and Artin-Schelter Gorensteinness (\cite{JZ}).

\begin{dfn}[cf. \cite{BD, KKZ}]
A standard graded $\Bbbk$-algebra $R$ is called \textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane.
\end{dfn}

Monic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \textit{Kronecker polynomials}.
It is known that every such polynomial can be expressed as a product of cyclotomic polynomials.
Note that the $h$-polynomials of cyclotomic $\Bbbk$-algebras are \textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials.
We say that a polynomial is \textit{of type CI} if it can be written as a product of $\Psi_m(t)$'s.
In particular, the $h$-polynomials of standard graded complete intersections are of type CI.
However, the converse does not hold in general; see, for example, \cite[Example~3.7]{S78}.
Borzi and D'Al\`i showed that the converse does hold when $R$ is Koszul~\cite[Theorem~3.9]{BD}.
Moreover, it was proved that a cyclotomic polynomial $\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\cite[Theorem~4.3]{BD}.

Despite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial.
This naturally leads to the following question:
\begin{q}
Which polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\Bbbk$-algebras, other than those of type CI?
\end{q}
For example, \cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\Bbbk$-algebra.

\subsection{Main Results}

The aim of this paper is to make a contribution to this question.
Our main result is stated as follows.

\begin{abstract}
We call a standard graded commutative $\kk$-algebra \textit{cyclotomic} if its $h$-polynomial has all its roots on the unit circle in the complex plane.
Complete intersections provide typical examples of cyclotomic algebras,
since the $h$-polynomial of any standard graded complete intersection is a product of polynomials of the form $1 + t + \cdots + t^{m-1}$.
We refer to such polynomials as being \textit{of type CI}.
A natural question is whether there exists a cyclotomic standard graded $\kk$-algebra whose $h$-polynomial is not of type CI.
In this paper, we give a partial answer to this question.
We show that the $h$-polynomial $h_R(t)$ of a cyclotomic standard graded $\kk$-algebra $R$ is of type CI whenever $h_R(1) \in \{1, 4, 6\}$ or $h_R(1)$ is prime.
On the other hand, if $n \ge 8$ and $n$ is not prime, then there exists a cyclotomic standard graded $\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1) = n$.
\end{abstract}

Monic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \textit{Kronecker polynomials}.
It is known that every such polynomial can be expressed as a product of cyclotomic polynomials.
Note that the $h$-polynomials of cyclotomic $\kk$-algebras are \textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials.
We say that a polynomial is \textit{of type CI} if it can be written as a product of $\Psi_m(t)$'s.
In particular, the $h$-polynomials of standard graded complete intersections are of type CI.
However, the converse does not hold in general; see, for example, \cite[Example~3.7]{S78}.
Borzi and D'Al\`i showed that the converse does hold when $R$ is Koszul~\cite[Theorem~3.9]{BD}.
Moreover, it was proved that a cyclotomic polynomial $\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\kk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\cite[Theorem~4.3]{BD}.

The aim of this paper is to make a contribution to this question.
Our main result is stated as follows.

Note that $h_R(1)$ is nothing but the multiplicity of $R$ (see \cite[Proposition 4.1.9]{BH}), which is one of the important invariants of graded modules.

\section{Proof of Theorem~\ref{thm:main} (2)}\label{sec:(2)}
The goal of this section is to give a proof of Theorem~\ref{thm:main} (2).
Specifically, for any non-prime $n$ with $n \geq 8$, we construct a cyclotomic standard graded $\kk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.
Such $\kk$-algebras are realized as Stanley-Reisner rings of certain simplicial complexes.

We prove the following theorem, which implies Theorem~\ref{thm:main} (2).
\begin{thm}\label{thm:p=3}
{\em (1)} Let $q \ge 5$ be an odd integer.
Then there exists a cyclotomic standard graded $\kk$-algebra $R$ such that $h_R(t)$ is not of type CI with $h_R(1)=2q$.

\noindent
{\em (3)} Let $5 \le p \le q$ be odd integers.
Then there exists a cyclotomic standard graded $\kk$-algebra $R$ such that $h_R(t)$ is not of type CI with $h_R(1)=pq$.
\end{thm}

We postpone the proof of this theorem. So far, assume Theorem~\ref{thm:p=3} holds. Then we can prove the desired result. In fact,
let $R$ be a cyclotomic standard graded $\kk$-algebra whose $h$-polynomial is not of type CI.
Then, for any positive integer $a$, the standard graded $\kk$-algebra $R'=R[x]/(x^a)$ has the $h$-polynomial $h_{R'}(t)$ which is not of type CI and $h_{R'}(1)=a\cdot h_R(1)$.
(In fact, we have $h_{R'}(t)=h_R(t)\Psi_a(t)$.)

\begin{thm}[Main Result]\label{thm:main}
{\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\kk$-algebra $R$ is of type CI if $h_R(1) \in \{1,4,6\}$ or $h_R(1)$ is prime.

\noindent
{\em (2)} There exists a cyclotomic standard graded $\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.
\end{thm}

Note that $h_R(1)$ is nothing but the multiplicity of $R$ (see \cite[Proposition 4.1.9]{BH}), which is one of the important invariants of graded modules.

\subsection{Structure of the paper}
We briefly describe the structure of this paper.
In Section~\ref{sec:pre}, we fix our notation (Hilbert series, $h$-polynomials, simplicial complexes, etc.),
review fundamental notions (cyclotomic polynomials, Stanley-Reisner theory, and related topics),
and recall classical results, namely Macaulay's theorem and the $g$-theorem, which will be used in the proof of Theorem~\ref{thm:main}.
Sections~\ref{sec:(1)} and~\ref{sec:(2)} are devoted to the proofs of parts~(1) and~(2) of Theorem~\ref{thm:main}, respectively.

The post-theorem introduction gives only a high-level roadmap: it says that Section~\ref{sec:pre} recalls classical results—"namely Macaulay's theorem and the $g$-theorem"—"which will be used in the proof of Theorem~\ref{thm:main}," and that Sections~\ref{sec:(1)} and~\ref{sec:(2)} contain the proofs of parts~(1) and~(2) of Theorem~\ref{thm:main}, respectively.

The post-theorem introduction gives only a high-level roadmap: it says that next it recalls classical results—"namely Macaulay's theorem and the $g$-theorem"—"which will be used in establishing the main theorem," and that later it contains the proofs of parts~(1) and~(2) of the main theorem, respectively.

[Main Result]\label{thm:main}
{\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\Bbbk$-algebra $R$ is of type CI if $h_R(1) \in \{1,4,6\}$ or $h_R(1)$ is prime.

\noindent
{\em (2)} There exists a cyclotomic standard graded $\Bbbk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.,

Question: Let $R$ be a standard graded commutative $\Bbbk$-algebra with Hilbert series of the form $H_R(t)=\dfrac{h_R(t)}{(1-t)^d}$, where $h_R(t)$ is its $h$-polynomial. Call $R$ cyclotomic if all roots of $h_R(t)$ lie on the unit circle in $\mathbb{C}$. A polynomial is said to be of type CI if it can be written as a product of polynomials $\Psi_m(t)=1+t+\cdots+t^{m-1}$. Which statement holds for such algebras and $h$-polynomials?

For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, for every non-prime integer $n\ge 8$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.

Choices:
B. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6,8\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, for every non-prime integer $n\ge 9$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.
C. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI.
D. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, there exists a non-prime integer $N\ge 8$ such that for every non-prime integer $n\ge N$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ with $h_R(t)$ not of type CI and $h_R(1)=n$.
E. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is composite, then $h_R(t)$ is of type CI. Moreover, for every prime integer $n\ge 8$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.

sharp exceptional values and threshold $n\ge 8$

dropped the existence clause in part (2)

uniform bound replaced exact range of all non-prime $n\ge 8$

prime/non-prime dichotomy in both directions


\begin{theorem}\label{thm_main}
Let $M$ be a closed topological manifold of dimension $2n$ and let $b = \dim H^2(M,\mathbb{C})$.

Then there exist non-zero elements $e_1,\ldots,e_q \in H^2(M,\mathbb{C})$ with $q \leq b+1$ such that if $X$ is a smooth complex projective variety of dimension $n$ with underlying topological space $M$ and $f\colon X \to Y$ is a divisorial contraction to a point with exceptional divisor $E$, then there exists $i \in \{1, \dots, q\}$ such that $[E] = e_i$.
\end{theorem}

This result provides a topological bound on the possible exceptional divisors arising from some of the steps of the Minimal Model Program, establishing a new link between topology and birational geometry. It is a consequence of Theorem \ref{thm:W_F}, where we prove the finiteness of rank one points of $\mathcal H_F$ where $F$ is a non-degenerate $n$-form, a result of independent interest.

\medskip

The structure of the paper is as follows. In Section \ref{s_preliminaries}, we introduce the necessary background on tensors and forms. Sections \ref{s_low} and \ref{s:intersection} are devoted to the study of the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$ with $n\ge 3$. We show, in particular, that, up to a non-zero scalar, there are at most finitely many  classes $e\in H^2(M,\mathbb C)$ such that
$e^n\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.
(cf. Theorem \ref{thm:W_F}). Section \ref{s_blowups} establishes our main theorem by examining the rank  of the associated  form in the presence of a divisorial contraction. In Section \ref{s_volume},  we present some related open problems.

\medskip

\textbf{Acknowledgements:}  The first and third authors are members of the GNSAGA - Istituto Nazionale di Alta Matematica.
The first author was also partially supported by the PRIN ``Vartietà reali e complesse: geometria, topologia e analisi armonica".
The second author is partially supported by a Simons collaboration grant  and would like to thank the National Center for Theoretical Sciences in Taiwan and Professor Jungkai Chen for
their  hospitality, where some of the work for this paper was completed.
The third author is partially supported by the PRIN2020 research grant ``2020KKWT53”.
The authors would also like to thank the referee for careful reading the paper and for several useful comments.

\section{Preliminaries}\label{s_preliminaries}
We work over the field of complex numbers $\mathbb C$. A \emph{form} over $\mathbb C$ is a homogeneous polynomial $F\in \mathbb C[x_0,\ldots,x_b]$.

\subsection{Tensors} Let $a_1,\dots,a_n$ be positive integers. A \emph{tensor} of type $a_1\times \ldots \times a_n$ is a multilinear map
$$T\colon \mathbb C^{a_1}\times \ldots \times \mathbb C^{a_n}\to \mathbb C.$$
For any positive integer $a$, denote $[a]:=\{ 1,2,\ldots, a \}$. Then a tensor $T$ of type  $a_1\times \ldots \times a_n$ is determined uniquely by a function
$$\tilde T\colon [a_1]\times \cdots \times [a_n] \to \mathbb{C}.$$
We will refer to $\tilde T$ as the \emph{hypermatrix associated to $T$}.
Given positive integers
$a_j^{'} \le a_j$ for $1 \le j \le n,$ and strictly increasing functions $f_j\colon [a_j^{'}] \to [a_j]$,
we define a \emph{sub-tensor} $T'$ of type $a'_1\times\ldots\times a'_n$ so that if $\tilde T'$ is the hypermatrix associated to $T'$ then
$$\tilde T'=\tilde T(f_1,\dots,f_n)\colon [a'_1]\times \cdots \times [a'_n] \to \mathbb{C}.$$
Note that if $v_i\in \mathbb C^{a_i}$ for $i=1,\dots,n$, then there exists a natural tensor of type $a_1\times \ldots \times a_n$ defined by
$v_1\otimes \ldots \otimes v_n$.

\begin{definition}
A nonzero tensor $T$ of type  $a_1\times \ldots \times a_n$ has {\it rank one} if there are nonzero vectors $v_i \in \mathbb{C}^{a_i}$ such that $T=v_1 \otimes \cdots \otimes v_n $. We define the
{\it rank} of a nonzero tensor $T$ ($=\rk(T)$) to be the minimum positive integer $r$
such that there exist $r$ tensors $T_1, \cdots , T_r$ of rank one with

$$T = T_1 + \cdots + T_r .$$
\end{definition}

Note that the rank of any sub-tensor of $T$ cannot be larger than the rank of $T$.
\begin{definition}\label{d_cubic}
A \emph{cubic tensor} $T$ is a tensor of type $d^{\times n} =d \times \cdots \times d$. Let $A$ be the hypermatrix associated to $T$.
We denote by $\det A$ the \emph{hyperdeterminant} of  $A$ (see \cite[Chapter 14]{GKZ} for the definition and some of its main properties). In particular, $\det A$  is a polynomial  in the entries of $A$.

\end{definition}

The following theorem is useful to compute the rank of a tensor:
\begin{theorem}{\cite[Thm. 3.1.1.1]{Landsberg-book}} \label{thm:slices}
Let $A_1,\dots,A_n$ be $l\times m$ matrices and consider the  tensor $A=[A_1,\ldots,A_n]$ of type $l\times m\times n$.

Then $\rk(A)$ is equal to  the minimum number of rank one matrices needed to span a vector space containing $\langle A_1,\ldots, A_n \rangle$.
\end{theorem}

\begin{proof}
Let $r=\rk(A)$ and let $B_1,\ldots, B_r$ be rank one matrices such that
$$\langle A_0,\ldots, A_q\rangle\subset \langle B_1,\ldots, B_r\rangle,$$
as in Theorem \ref{thm:slices}.
Thus, for any $j=0,\ldots,q$, we may write
$$
A_j= \sum_{i=1}^r \lambda_{ij}B_i
$$
for $\lambda_{ij} \in \C$ for $i=1,\ldots,r$ and $j=0,\dots,q$. Since $A_1,\ldots, A_q$ are linearly independent, after possibly reordering the $B_i$'s, we may assume that the $q\times q$-matrix
$$L=(\lambda_{ij})_{i,j=1,\ldots,q}$$
 is of maximal rank $q$. Thus, the linear system
$$L \cdot X =(-\lambda_{10},-\lambda_{20},\ldots,-\lambda_{q0})^t$$
admits a solution $(\mu_1,\ldots,\mu_q)\in \mathbb C^q$.
It follows that
 $$A_0 + \sum_{i=1}^q \mu_i A_i=\sum_{j=q+1}^{r} s_j B_j$$ for some  $s_{q+1},\ldots,s_r \in \C$.
 Since the rank of $\sum_{j=q+1}^{r} s_j B_j$ is at most $r-q$, our assumption implies that $r-q \ge t$  and the claim follows.
\end{proof}

\section{Points of low rank}\label{s_low}

\begin{definition}
	Given a form $F \in \C[x_0,\ldots,x_b]$ of degree $n \ge 3$ we consider the tensor
of type $(b+1)^{\times (n-1)}$
		defined by the hypermatrix $\mathcal H_F$ of the $(n-1)$-th order derivatives, whose entries are linear forms in $x_0, \ldots, x_b$.
	We say that $F$ is \emph{honest} if for any non-zero element $v \in \mathbb C^{b+1}$, we have  $\mathcal H_F(v) \ne 0$.
    We say that $F$ is  \emph{non-degenerate} if $ \det \mathcal H_F(v) \ne 0$ for some
 non-zero $v \in \mathbb C^{b+1}$
 (note that this definition is consistent with \cite[Page 7929]{CT18} but not with \cite[14.1.A]{GKZ}).
\end{definition}

Note that the locus $p \in \mathbb P^b$ such that  $\mathcal H_F(p) = 0$ is well defined. By \cite[Remark 6.3.5]{Bocci-Chiantini}, the rank $\rk \mathcal H_F(p)$ is also well-defined.

\begin{example}\label{ex:degenerate}
	Consider the  form
	$$
	F(x_0,\ldots,x_4)=\frac{x_0x_1^2}{2} + x_1x_3x_4 + \frac{x_2x_3^2}{2}.
	$$
	Then

	$$
	\mathcal H_F=\begin{pmatrix}
	0   & x_1 & 0   & 0   & 0   \\
	x_1 & x_0 & 0   & x_4 & x_3 \\
	0   & 0   & 0   & x_3 & 0   \\
	0   & x_4 & x_3 & x_2 & x_1 \\
	0   & x_3 & 0   & x_1 & 0
	\end{pmatrix}
	$$
Thus, the form $F$ is honest, but it is degenerate since $\det \mathcal H_F$ is identically zero.
\end{example}

Vaguely speaking, an honest form can be characterised by the fact that it  depends on all the variables. More precisely, we have:
\begin{lemma}\label{lem:H_F(p)=0}
	Let $F \in \C[x_0,\ldots,x_b]$ be a form of degree $n$ and let
    $p:=[1,0,\ldots,0] \in \mathbb P^b$.

    Then $\mathcal H_F(p) = 0$ if and only if $x_0$ does not appear in the expression of $F$.  In particular, a non-degenerate form is honest.
\end{lemma}

\begin{proof}
	Assume first that $\mathcal H_F(p) = 0$. By  the Euler formula for homogeneous polynomials we have that
	$$
	(n-k)\partial_{i_1 \ldots i_k}F(x_0,\ldots,x_b)= \sum_{\ell=0}^b x_\ell \partial_{\ell, i_1 \ldots i_k}F(x_0, \ldots, x_b)
	$$
	for any $k=0,\ldots, n-1$ and $i_j \in \{0,\ldots, b\}$.
    Proceeding by induction and using the fact that $\mathcal H_F(p) = 0$, we get that $\partial_{i_1 \ldots i_k}F(p)=0$ for any $k=0,\ldots, n-1$ and $i_j \in \{0,\ldots, b\}$. Thus, $x_0$ does not appear in $F(x_0, \ldots, x_n)$.
	The converse is a simple computation.

\medskip

Assume now that $F(x_0, \ldots, x_n)$ is a non-degenerate form. If by contradiction there exists $p\in \mathbb P^b$ such that $\mathcal H_F(p) = 0$ then, up to a  change of coordinates, we may assume $p=[1,0,\ldots,0]$. This means that $F$ does not depends on $x_0$ and so the face of $\mathcal H_F(p)=0$  corresponding to $\partial_0$ is trivial. By \cite[Corollary XIV.1.5(d)]{GKZ}, it follows that $\det \mathcal H_F$ is identically zero, which is a contradiction.
\end{proof}

\begin{pmatrix}
0 & x_1   \\
x_1 & x_0   \\
\end{pmatrix}.
$$
By \cite[Proposition XIV.1.7]{GKZ}, it follows that $F_X$ is degenerate.
\end{example}

\section{Blow-ups}\label{s_blowups}

\begin{lemma}\label{lem:blowup}
Let $X$ be a smooth projective variety of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$.
Let $[E], \beta_1:=f^*\gamma_1, \ldots, \beta_b:=f^*\gamma_b$ be a basis of $H^2(Y,\Q)$ where $E$ is the exceptional divisor and $\gamma_1,\ldots,\gamma_b$ is a basis of $H^2(X,\Q)$.
Then, with respect to this basis, we may write
$$
F_Y(x_0,\ldots,x_n)=ax_0^n +\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \ldots, x_b)
$$
where $a=E^n$ and $R_i \in \mathbb Q[x_1,\ldots,x_b]$ is a form  of degree $i$, for $i=1,\dots,n-k$.

\end{lemma}
\begin{proof}
This follows easily from the projection formula.
\end{proof}

\begin{proposition}\label{prop:blowuprank}
Let $X$ be a smooth K\"ahler manifold of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \le 2$. Let $p=[E] \in H^2(Y,\C)$ be the class of the exceptional divisor.
\begin{enumerate}
\item \label{k=0} If $n\ge 2$ and $k=0$ then $\rk \mathcal H_{F_Y}(p)=1$.
\item \label{k=1}  If $n\ge 3$ and  $k=1$ then $\rk \mathcal H_{F_Y}(p) \ge 2$.
\item \label{k=2} If $n\ge 4$ and $k=2$ then, after a base change, we may write
$$
F_Y(x_0,\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\ldots,x_b) + x_0^{n-2}Q(x_1,\ldots,x_b) + F_X(x_1, \ldots, x_b)
$$
where $a \in \mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\rk \mathcal H_{F_Y}(p) \ge 2q$.

\end{enumerate}
\end{proposition}
\begin{proof}
We use the same notation as in Lemma \ref{lem:blowup}.

We first prove \eqref{k=0}. We have
$$
F_Y(x_0,\ldots,x_n)=ax_0^n  + F_X(x_1, \ldots, x_b)
$$
where $a=E^n \ne 0$. Hence $\rk \mathcal H_{F_Y}(p)=1$.
Thus, \eqref{k=0} follows.
\medskip

We now prove \eqref{k=1}. We have
$$
F_Y(x_0,\ldots,x_n)=ax_0^n+ x_0^{n-1}\cdot \left(\sum_{i=1}^{b}\lambda_ix_i \right )  + F_X(x_1, \ldots, x_b )
$$
where $a=E^n$ and $\lambda_i=Z \cdot \gamma_i$, for $i=1,\dots,b$. After taking a base change that fixes $(1,0,\ldots,0)$, we may assume $\lambda_i=0$ for any $i=2, \ldots ,b$, i.e.

$$
F_Y(x_0,\ldots,x_n)=ax_0^n+ \lambda_1 x_0^{n-1}x_1  + F_X(x_1, \ldots, x_b ),
$$
where $\lambda_1 \ne 0$ because we are blowing up a curve $Z$ in a K\"ahler manifold and so there exists at least one class in $H^2(X,\mathbb Z)$ with non-zero intersection with $Z$. Thus, it is easy to check that the subtensor given by
$$(\partial_0^{n-3}\partial_i\partial_j)_{i,j=0,\dots,b}$$
has rank two,  which implies that $\rk \mathcal H_{F_Y}(p) \ge 2$.

\medskip

We finally prove \eqref{k=2}. Let $j\colon Z \hookrightarrow X$ be the inclusion.
Let  $V \subset H^2(Z, \C)$ be the subspace generated by $j^*\gamma_1, \ldots, j^*\gamma_b$ and let $q$ be the rank of the quadratic form $Q$ obtained restrincting to $V$ the quadratic form of $H^2(Z,\mathbb C)$.
 Note that $q \ge 1$ because a K\"ahler class of $X$ restricts to a K\"ahler class on $Z$.
After a base change that diagonalises $Q$, we can write
$$
F=F_Y(x_0,\ldots,x_b)=\frac{ax_0^n}{n!} + \frac{x_0^{n-1}}{(n-1)!}(c_1x_1+\ldots +c_bx_b) + \frac{x_0^{n-2}}{2(n-2)!}(x_1^2+\ldots+ x_q^2) + F_X(x_1, \ldots, x_b)
$$
for some    $c_1,\dots,c_b \in \C$.

Consider the $(b+1)\times (b+1) \times (b+1)$ subtensor $A$ of $\mathcal H_{F}(p)$ given by the slices $A_0, \ldots, A_b$ where
 $$
A_h= ( \partial^{n-4}_0 \partial_h \partial_i \partial_j F(p))_{i,j=0,\dots,b}
 $$
 for $h=0,\dots,b$.

It is enough to prove that $\rk A \ge 2q$, as it immediately implies that $\rk \mathcal H_{F}(p) \ge 2q$.
We have
$$
A_0 =
\begin{pmatrix}
1 & c_1 & \ldots & c_q & c_{q+1} & \ldots & c_b \\
c_1 & 1  & 0 & \ldots & \ldots & \dots & 0 \\
\vdots & 0 & \ddots  & 0 & \cdots & \cdots & \vdots \\
c_q & 0 & \ldots & 1 & 0 & \ldots & 0  \\
c_{q+1} &\vdots & \vdots  &\vdots & 0 & \ldots & 0 \\
\vdots&\vdots & \vdots  &\vdots & \vdots & \ddots & \vdots \\
c_b & \cdots & \cdots & \cdots & 0 & \cdots & 0 \\
\end{pmatrix}

\label{s_volume}

The goal of this section is to present some open problems on the relationship between birational invariants and topological invariants of a smooth complex projective variety.

\sub

Let $X$ be a compact K\"ahler manifold of dimension $n$. One of the fundamental objects associated with $X$ is the natural form $F_X$ on $H^2(X,\mathbb{Z})$ of degree $n$, induced by the cup product structure. This form serves as a topological invariant of $X$ and plays a crucial role in understanding the intersection theory of higher-dimensional algebraic varieties.

For $n=2$, the quadratic form $F_X$ is central to the theory of algebraic surfaces, yielding results in classification theory and the study of their moduli spaces. When $n=3$, the study of the cubic form $F_X$ was initiated in \cite{OV95}, leading to numerous applications in algebraic geometry, differential geometry, and dynamical systems. Recent advancements have further deepened our understanding of $F_X$ and its role in various geometric problems \cite{KW, BCT16, CT18, ST19, ST20, Wilson21, Wilson21b}.

Despite these advances, relatively little is known about the structure and applications of $F_X$ in dimensions $n \geq 4$. The goal of this paper is to initiate a study of $F_X$ in higher dimensions, exploring its fundamental properties and potential applications to birational geometry and topology. In particular, we examine how the rank of the hypermatrix $\mathcal H_{F_X}$ of the $(n-1)$-th derivatives of $F_X$ influence the behavior of divisorial contractions (see Section \ref{s:intersection} for the definition of $\mathcal H_{F_X}$ and some of its basic properties).

\medskip

Our main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:

\label{s:intersection}

Let $M$ be a closed (i.e.\ compact and without boundary) oriented topological manifold of dimension $d=2n$.
Consider the symmetric
$n$-multilinear map defined by the cup prod

Our main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:

This result provides a topological bound on the possible exceptional divisors arising from some of the steps of the Minimal Model Program, establishing a new link between topology and birational geometry. It is a consequence of Theorem \ref{thm:W_F}, where we prove the finiteness of rank one points of $\mathcal H_F$ where $F$ is a non-degenerate $n$-form, a result of independent interest.

The structure of the paper is as follows. In Section \ref{s_preliminaries}, we introduce the necessary background on tensors and forms. Sections \ref{s_low} and \ref{s:intersection} are devoted to the study of the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$ with $n\ge 3$. We show, in particular, that, up to a non-zero scalar, there are at most finitely many classes $e\in H^2(M,\mathbb C)$ such that
$e^n\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.
(cf. Theorem \ref{thm:W_F}). Section \ref{s_blowups} establishes our main theorem by examining the rank of the associated form in the presence of a divisorial contraction. In Section \ref{s_volume}, we present some related open problems.

\subsection{Tensors} Let $a_1,\dots,a_n$ be positive integers. A \emph{tensor} of type $a_1\times \ldots \times a_n$ is a multilinear map
$$T\colon \mathbb C^{a_1}\times \ldots \times \mathbb C^{a_n}\to \mathbb C.$$
For any positive integer $a$, denote $[a]:=\{ 1,2,\ldots, a \}$. Then a tensor $T$ of type $a_1\times \ldots \times a_n$ is determined uniquely by a function
$$\tilde T\colon [a_1]\times \cdots \times [a_n] \to \mathbb{C}.$$
We will refer to $\tilde T$ as the \emph{hypermatrix associated to $T$}.
Given positive integers
$a_j^{'} \le a_j$ for $1 \le j \le n,$ and strictly increasing functions $f_j\colon [a_j^{'}] \to [a_j]$,
we define a \emph{sub-tensor} $T'$ of type $a'_1\times\ldots\times a'_n$ so that if $\tilde T'$ is the hypermatrix associated to $T'$ then
$$\tilde T'=\tilde T(f_1,\dots,f_n)\colon [a'_1]\times \cdots \times [a'_n] \to \mathbb{C}.$$
Note that if $v_i\in \mathbb C^{a_i}$ for $i=1,\dots,n$, then there exists a natural tensor of type $a_1\times \ldots \times a_n$ defined by
$v_1\otimes \ldots \otimes v_n$.

\begin{lemma}\label{lem:blowup}
Let $X$ be a smooth projective variety of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$.
Let $[E], \beta_1:=f^*\gamma_1, \ldots, \beta_b:=f^*\gamma_b$ be a basis of $H^2(Y,\Q)$ where $E$ is the exceptional divisor and $\gamma_1,\ldots,\gamma_b$ is a basis of $H^2(X,\Q)$.
Then, with respect to this basis, we may write
$$
F_Y(x_0,\ldots,x_n)=ax_0^n +\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \ldots, x_b)
$$
where $a=E^n$ and $R_i \in \mathbb Q[x_1,\ldots,x_b]$ is a form of degree $i$, for $i=1,\dots,n-k$.

\begin{proposition}\label{prop:blowuprank}
Let $X$ be a smooth K\"ahler manifold of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \le 2$. Let $p=[E] \in H^2(Y,\C)$ be the class of the exceptional divisor.
\begin{enumerate}
\item \label{k=0} If $n\ge 2$ and $k=0$ then $\rk \mathcal H_{F_Y}(p)=1$.
\item \label{k=1} If $n\ge 3$ and $k=1$ then $\rk \mathcal H_{F_Y}(p) \ge 2$.
\item \label{k=2} If $n\ge 4$ and $k=2$ then, after a base change, we may write
$$
F_Y(x_0,\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\ldots,x_b) + x_0^{n-2}Q(x_1,\ldots,x_b) + F_X(x_1, \ldots, x_b)
$$
where $a \in \mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\rk \mathcal H_{F_Y}(p) \ge 2q$.

\begin{example}
Let $X$ be a complex projective manifold of dimension $n$ and $b_2(X)=1$. Let $f\colon Y \to X$ be the blow-up along a smooth curve $Z$ of $X$. Let $p=[E] \in H^2(Y,\C)$ be the class of the exceptional divisor. Then we can write
$$
F_Y(x_0,\ldots,x_n)=\frac{a}{n!}x_0^n+ \frac{x_0^{n-1}x_1}{(n-1)!} + x_1^n,
$$
for some $a \in \mathbb Q$. If $n=3$, then
$$
\mathcal H_{F_Y}(p)=
\begin{pmatrix}
a & 1 \\
1 & 0 \\
\end{pmatrix}
$$
has rank 2. If $n=4$, then
$$
\mathcal H_{F_Y}(p)=
\left[ \begin{pmatrix}
a & 1 \\
1 & 0 \\
\end{pmatrix},
\begin{pmatrix}
1 & 0 \\
0 & 0 \\
\end{pmatrix}
\right]
$$
has rank at least 3 by Lemma \ref{lem:trick}. On the other hand, 3 is the maximal rank for a $2\times 2 \times 2$ tensor (see, for instance, \cite[Section 3]{Kruskal}). Thus, $\mathcal H_{F_Y}(p)$ has rank $3$.
\end{example}

\begin{question}\label{q:volume}
Let $M$ be a closed topological manifold of dimension $2n$. Is there a constant $C$, depending only on $M$, such that for any smooth complex projective variety $X$ whose underlying topological space is $M$, we have
\[ \vol(X) \le C? \]
\end{question}

The proof of Theorem~\ref{thm_main} is obtained by reducing it to a finiteness statement about certain cohomology classes detected by the intersection form. Concretely, the paper studies “the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$,” and shows that “up to a non-zero scalar, there are at most finitely many classes $e\in H^2(M,\mathbb C)$ such that $e^n\neq 0$ and the rank of the Hessian of $F$ at $e$ is one” (cf. Theorem~\ref{thm:W_F}), which itself comes from “the finiteness of rank one points of $\mathcal H_F$ where $F$ is a non-degenerate $n$-form.” Then Section~\ref{s_blowups} “establishes our main theorem by examining the rank of the associated form in the presence of a divisorial contraction,” relating the exceptional divisor class $[E]$ of a contraction $f\colon X\to Y$ to one of those finitely many rank-one Hessian classes, yielding the finite list $e_1,\ldots,e_q$.

The proof of Theorem~\ref{thm_main} is obtained by reducing it to a finiteness statement about certain cohomology classes detected by the intersection form. Concretely, the paper studies “the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$,” and shows that “up to a non-zero scalar, there are at most finitely many classes $e\in H^2(M,\mathbb C)$ such that $e^n\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.” We first record the precise finiteness statement:

\begin{theorem}\label{thm:W_F}
Let $F \in \C[x_0, x_1, \ldots,x_b]$ be a form of degree $n \ge 3$.
\begin{enumerate}
\item If $F$ is honest, then $W_F \cap \{F \ne 0\}$ is a finite set, and more precisely,
$$
|W_F  \cap \{F \ne 0\} | \le b+1.
$$

\item If $F$ is non-degenerate, then $W_F$ is a finite set.
\end{enumerate}
\end{theorem}

This itself comes from “the finiteness of rank one points of $\mathcal H_F$ where $F$ is a non-degenerate $n$-form.” Next we study blow-ups and use them to connect exceptional divisor classes to these finitely many rank-one Hessian classes. In particular, we use the following structural description of the intersection form under blow-up:

\begin{lemma}\label{lem:blowup}
Let $X$ be a smooth projective variety of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$.
Let $[E], \beta_1:=f^*\gamma_1, \ldots, \beta_b:=f^*\gamma_b$ be a basis of $H^2(Y,\Q)$ where $E$ is the exceptional divisor and $\gamma_1,\ldots,\gamma_b$ is a basis of $H^2(X,\Q)$.
Then, with respect to this basis, we may write
$$
F_Y(x_0,\ldots,x_n)=ax_0^n +\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \ldots, x_b)
$$
where $a=E^n$ and $R_i \in \mathbb Q[x_1,\ldots,x_b]$ is a form  of degree $i$, for $i=1,\dots,n-k$.

\end{lemma}

We then control the rank of the Hessian at the exceptional divisor class $p=[E]$ in low-dimensional centers of blow-up via the following result:

\begin{proposition}\label{prop:blowuprank}
Let $X$ be a smooth K\"ahler manifold of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \le 2$. Let $p=[E] \in H^2(Y,\C)$ be the class of the exceptional divisor.
\begin{enumerate}
\item \label{k=0} If $n\ge 2$ and $k=0$ then $\rk \mathcal H_{F_Y}(p)=1$.
\item \label{k=1}  If $n\ge 3$ and  $k=1$ then $\rk \mathcal H_{F_Y}(p) \ge 2$.
\item \label{k=2} If $n\ge 4$ and $k=2$ then, after a base change, we may write
$$
F_Y(x_0,\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\ldots,x_b) + x_0^{n-2}Q(x_1,\ldots,x_b) + F_X(x_1, \ldots, x_b)
$$
where $a \in \mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\rk \mathcal H_{F_Y}(p) \ge 2q$.

\end{enumerate}
\end{proposition}

In establishing the main theorem, we apply these blow-up computations to examine “the rank of the associated form in the presence of a divisorial contraction,” relating the exceptional divisor class $[E]$ of a contraction $f\colon X\to Y$ to one of the finitely many rank-one Hessian classes provided by Theorem~\ref{thm:W_F}, yielding the finite list $e_1,\ldots,e_q$.

Question: Let $M$ be a closed (compact and without boundary) topological manifold of real dimension $2n$, and set $b=\dim_{\mathbb C} H^2(M,\mathbb C)$. For any smooth complex projective variety $X$ of complex dimension $n$ whose underlying topological space is $M$, identify $H^2(X,\mathbb C)$ with $H^2(M,\mathbb C)$. If $f\colon X\to Y$ is a divisorial contraction to a point, let $E$ denote its exceptional divisor and let $[E]\in H^2(M,\mathbb C)$ be its cohomology class under this identification. Which existence statement holds?

There exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ with $q\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\colon X\to Y$ to a point with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]=e_i$.

Choices:
B. There exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ with $q\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\colon X\to Y$ with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]=e_i$.
C. There exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ for some finite $q$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\colon X\to Y$ to a point with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]=e_i$.
D. For every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, there exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ with $q\le b+1$ such that for every divisorial contraction $f\colon X\to Y$ to a point with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]=e_i$.
E. There exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ with $q\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\colon X\to Y$ to a point with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]$ is proportional to $e_i$ by a nonzero scalar.