\label{Mth}
		Let $N_r=S(\xi)$ be the total space of the sphere bundle of a 3-dimensional real vector bundle $\xi$ over $\mathbb{P}^2$ with $p_1(\xi)=r \omega$. Then
		\begin{enumerate}
		\item there is a short exact sequence
		$$ 1 \to I(N_r) \to \mathrm{MCG}(N_r) \to A \to 1,$$
		where $A \cong \z_2$ if $r \ne -3$, and $A \cong S_3$, the symmetric group of $3$ elements, if $r=-3$;
		\item when $r=8n+5$, $I(N_r)$ is isomorphic to $ \mathbb{Z}_{2\mathrm{gcd}(3,2n^2+3n+1)}\oplus \mathbb{Z}_{2\mathrm{gcd}(14,n+1)}$.
		\end{enumerate}

\begin{thm}\label{Mth}
		Let $N_r=S(\xi)$ be the total space of the sphere bundle of a 3-dimensional real vector bundle $\xi$ over $\mathbb{P}^2$ with $p_1(\xi)=r \omega$. Then
		\begin{enumerate}
		\item there is a short exact sequence
		$$ 1 \to I(N_r) \to \m(N_r) \to A \to 1,$$
		where $A \cong \z_2$ if $r \ne -3$, and $A \cong S_3$, the symmetric group of $3$ elements, if $r=-3$;
		\item when $r=8n+5$, $I(N_r)$ is isomorphic to $ \mathbb{Z}_{2\mathrm{gcd}(3,2n^2+3n+1)}\oplus \mathbb{Z}_{2\mathrm{gcd}(14,n+1)}$.
		\end{enumerate}
	\end{thm}

\label{subsec:bundle}
For the sake of simplicity,	in the sequel we will use the same notation for (the equivalence class of) a vector bundle and (the homotopy class of) its classifying map from it bas

\label{subsec:gen}

A natural idea to study the mapping class group is to relate it to the group of homotopy equivalences. To be precise, let $M$ be a simply closed connected manifold with $\dim M>5$.

\begin{proof}
Choose a localization $i \colon S^2\times D^4\to N_r$ such that $i(S^2\times D^4)$ and $S^2\vee S^2$ (defined in the proof of Lemma \ref{Ih}) are disjoint. The diffeomorphisms $g_1$ and $g_2$, which are extended by the identity on the complement of $i(S^2 \times D^4)$, clearly induce the identity on the cohomology groups and thus (their isotopy classes) are in $I(N_r)$.

		By Lemma \ref{t7} and the exactness of $S_\partial(N_r\times I)\to I(N_r)\to I^h(N_r)$, the kernel of the forgetful homomorphism $I(N_r) \to I^h(N_r)$ is generated by $g_1$. Thus we only need to show that the image of $g_2$ is a generator of $I^h(N_r)$. This will be done by showing that the image of $g_2$ generates  the image of $\mu_N^*$ (see Lemma \ref{Ih}).

		Let $\mu_3:S^3\times D^3 \to (S^3\times D^3)\vee S^6$ be the pinch map. It induces a map
		$$\mu_3^* \colon \pi_6(S^3)\to \pi_0(G_\partial(S^3\times D^3)), \ \alpha \mapsto (id\vee\alpha)\circ\mu_3.$$
where we represent $\alpha \in \pi_6(S^3)$ by a map $S^6 \to S^3 \times \{0\}$.
View $S^3\times D^3$ as a submanifold of $N_r$ through the embedding $i\circ i_\eta \colon S^3 \times D^3 \to S^2 \times D^4 \to N_r$, extending the map $\mu^*(\alpha)$ by the identity on the complement we have a homotopy equivalence $\mu^*(\alpha)$ of $N_r$. For $\alpha\in\pi_6(S^3)$, consider the composite $i \circ \eta \circ \alpha \colon S^6 \to S^3 \to S^2 \to N_r$. From the construction of the maps one readily verifies that $\mu_3^*(\alpha)=\mu_N^*(i \circ\eta \circ \alpha)\in I^h(N_r)$, i.e., there is a commutative diagram
$$\xymatrix{\pi_6(S^3) \ar[r]^{(i \circ \eta)_*} \ar[d]_{\mu_3^*} & \pi_6(N_r) \ar[d]^{\mu_N^*} \\
 \pi_0(G_\partial(S^3\times D^3)) \ar[r]^{\ \ \ \ \ \Phi} & I^h(N_r)}$$
 where the map $\Phi$ at the bottom is the extension by the identity on the complement of $i\circ i_{\eta}(S^3 \times D^3)$.

It can be verified by elementary homotopy theory (compare \cite[p.98]{BeMa}) that the homotopy Dehn twist map $De^h:\Omega^3 G(S^3)\to G_\partial(S^3\times D^3)$ defined by $De^h(\alpha)(x,y)=(\alpha(y)x,y)$
induces an isomorphism
$\pi_0(\Omega^3G(S^3)) \to \pi_0(G_\partial(S^3\times D^3))$.  Under the natural isomorphisms
$$\pi_0(\Omega^3G(S^3)) \cong \pi_3(G(S^3), \mathrm{id}) \cong \pi_3((\Omega^3S^3)_{\mathrm{id}}) \cong \pi_6(S^3),$$
where $(\Omega^3S^3)_{\mathrm{id}}$ is the connected component containing $\mathrm{id}_{S^3}$, this map can be identified with $\mu_3^* \colon \pi_6(S^3) \to \pi_0(G_\partial(S^3\times D^3))$. The following commutative diagram
$$\xymatrix{\pi_3(SO(3))\ar[d]\ar[rr]^J&&\pi_3(\Omega^3S^3)\ar[d]_{\mu_3^*}\\
			\pi_3(SO(4))\ar[r]^{De \ \ \ \ }&\m_{\partial}(S^3\times D^3)\ar[r]&\pi_0(G_\partial(S^3\times D^3))
			}$$
shows that the image of $g_2$ in $I^h(N_r)$ equals $\Phi(\mu_3^*(J(t_1)))$.
The $J$-homomorphism $J \colon \pi_3(SO(3))\to\pi_6(S^3)$ is surjective. Therefore by Corollary \ref{htg} and Lemma \ref{Ih},  the image of $g_2$ generates $I^h(N_r)$.   This finishes the proof of this proposition.

	\end{proof}

	\subsection{determination of $I(N_r)$}\label{subsec:det}
We have shown in last subsection that the Torelli group $I(N_r)$ is generated by the diffeomorphisms $g_1$ and $g_2$. They obviously commute with each other since they are supported in disjoint regions. In this subsection we determine the orders of them using the generalized Kreck-Stolz invariant developed in \cite{KreSu}.

Let  $B=\mathbb C P^\infty\times\mathbb{C} P^\infty \times BSpin$, $p \colon B\to BO$ be the composition of the projection $B \to BSpin$ and the canonical projection $BSpin \to BO$.

Since $r \in 8\z +5$, we may identify $N_r$ with $\mathbb{P}(\xi_{1,l})$ for some $l \in \z$. By Proposition \ref{Hcp}, $w_2(N_r)$ vanishes, hence $N_r$ is spin. The normal Gauss map  $g \colon N_r\to BO$ of $N_r$ admits a lifting $g_{spin}$ to $BSpin$, which is  unique (up to fiber homotopy) since $N_r$ is simply connected. Let $\{s, t\}$ be the basis of $H^2(N_r)$ given in Proposition \ref{Hcp}, then the map $\hat{g}=(s,t,g_{spin}) \colon N_r \to \mathbb C P^\infty\times\mathbb{C} P^\infty \times BSpin=B$ is the normal $B$-structure corresponding to the polarization $((s,t),0)$ (see \cite{Kre18} for definitions).

Let $x=c_1(O(-1)\times0\times0)$, $y=c_1(0\times O(-1)\times0)$ be a basis of $H^2(B)$. 	Our normal $B$-structure satisfies the condition of Theorem 5 of \cite{Kre18}, since $H^2(B)=\mathbb{Z}\{x,y\}, H^4(B)=\mathbb{Z}\{x^2,xy,y^2,\hat{p}_1\}$, where $\hat{p}_1$ is the first spin Potrjagin class of the universal spin stable vector bundle. The homomorphism $g^* \colon H^4(B) \to H^4(N_r)$ is given by
		$$\hat{g}^* \colon H^4(B)\to H^4(N_r), (x^2,xy,y^2,\hat{p}_1)\to (s^2,st,st-ls^2,(2l-2)s^2),$$
and in $H^4(N_r)$ we have the relations (see Proposition \ref{Hcp})
$$\hat{g}^*(\hat{p}_1)=-\hat{p}_1(N_r)=(2l-2)s^2, \ \ t^2-ts+ls^2=0.$$
Therefore
	$$\mathrm{Ker}(\hat{g}^* \colon H^4(B)\to H^4(N_r))=\mathbb{Z}\{y^2-xy+lx^2,\hat{p}_1+2(1-l)x^2\}$$
Let $\alpha=y^2-xy+lx^2$,  $\beta=\hat{p}_1+2(1-l)x^2\in H^4(B)$ and $s_1=\alpha^2$, $s_2=\alpha^2+\alpha\beta$, $ s_3=\beta^2\in H^8(B)$.

	Fix an embedded $6$-disk $D \subset N_r$, let $I(N_r,D)$ be the group of $D$-fixing isotopy classes of $D$-fixing diffeomorphisms which induce the identity on cohomology. By the disk theorem of Palais and the isotopy extension theorem, the forgetful homomorphism $I(N_r,D)\to I(N_r)$ is surjective. By a theorem from (\cite{Wa63} p.265), $\mathrm{Ker}(I(N_r,D)\to I(N_r))$ is trivial or isomorphic to $\mathbb{Z}_2$.

Given a diffeomorphism $f \colon N_r \to N_r$ with $f|_D = \mathrm{id}$, let $T_f$ be the mapping torus of $f$ with the preferred embedding $S^1\times D \subset T_f$. Assume that the action of $f$ on the cohomology groups is trivial,  then $f$ preserves the normal $B$-structure, and henceforth there is a normal $B$-structure $\hat{G}$ on $T_f$, $\hat G \colon T_f \to B$,
extending $\hat{g}$ and the restriction of $\hat G$ on $S^1 \times D$ is the ordinary normal $B$ structure on $S^1\times D$. It's ready to see by obstruction theory that the normal $B$-structure $\hat G$ is a unique up to homotopy. It is shown in \cite[Theorem 6]{Kre18} that the $7$-dimensional $B$-bordism group $\Omega_7(B,p)$ is trivial, therefore $(T_f,\hat{G})$ admits a coboundary $(W,\bar{G})$ whose signature $\mathrm{sign}(W)$  is $0$. Notice that the restriction of $\bar{G}^*\alpha$ and $\bar{G}^*\beta$ on $\partial W= T_f$ vanish, let $\bar{\alpha}$, $\bar{\beta}\in H^4(W,\partial W)$ be a preimage of $\bar{G}^*\alpha$ and $\bar{G}^*\beta\in H^4(W)$, respectively. We have characteristic numbers $\langle(\bar{\alpha}^2,\bar{\alpha}^2+\bar{\alpha}\bar{\beta},\bar{\beta}^2),[W,\partial W]\rangle \in \z^3$. For $8$-dimensional $B$-bordism classes $[X,\varphi] \in \Omega_8(B,p)$, we have characteristic numbers $\langle (s_1, s_2,s_3), \varphi_*[X] \rangle \in \z^3$ in a similar way. Let $L=(s_1,s_2,s_3)\Omega_8^0(B)\subset \mathbb{Z}^3$ be the lattice generated by the characteristic numbers on the bordism group $\Omega^0_8(B,p)=\mathrm{Ker}(\mathrm{sign} \colon \Omega_8(B,p)\to \mathbb{Z})$. The $KS$-invariant is defined by
	$$KS \colon I(N_r,D)\to \mathbb{Z}^3/L$$
	$$KS(f)=(s_1,s_2,s_3)(f):=\langle(\bar{\alpha}^2,\bar{\alpha}^2+\bar{\alpha}\bar{\beta},\bar{\beta}^2),[W,\partial W]\rangle.$$

	\begin{prop}\label{KS}
		$KS \colon I(N_r,D)\to \mathbb{Z}^3/L$ is a well defined injective homomorphism.
	\end{prop}
	\begin{proof}
		This proof is along the same line of the proof of Proposition 6.2 of \cite{KreSu}, with more details to be checked.

		To prove $KS$ is well defined, suppose $(W_1,\bar{G}_1),(W_2,\bar{G}_2)$ are two $B$ coboundary of $(T_f,\hat{G})$ chose embeddings $i_j: W_j\to \mathbb{R}^N\times (-1)^j[0,+\infty)$ good enough such that $i_j(W_j)\cap\mathbb{R}^N=T_f$. Define $W:=i_1(W_1)\cup i_2(W_2)\cong W_1\cup_\partial W_2$ then $(W,G_w:=\bar{G}_1\cup \bar{G}_2)$ is a well defined element in $\Omega_8(B,p)$, by \cite{Wa69} $sign(W)=0$. Consider the following diagram
		$$\xymatrix{\oplus_j H^4(W_j,T_f)\ar[r]&\oplus_j H^4(W_j)\ar[r]&\oplus_j H^4(T_f)\\
			H^4(W,T_f)\ar[r]\ar[u]^\cong &H^4(W)\ar[r]\ar[u]&H^4(T_f)\ar[u]^{injective}
		}$$
		This implies the preimage of $G_w^*\alpha\in H^4(W)$ in $H^4(W,T_f)\cong\oplus_j H^4(W_j,T_f)$ is a preimage of $(\bar{G}_1\alpha, \bar{G}_2\alpha)$ ($\beta$ resp.), and the difference of $KS(f)$ defined by $W_2$ and $W_1$ equals $(s_1,s_1+s_2,s_3)(W,G_w)\in L$. Thus we have proved $KS$ is well defined.

		To prove $KS$ is a homomorphism, suppose $f_3,f_4\in I(N_r,D)$ and $(W_j,\bar{G}_j)$ is a $B$ coboundary of $(T_{f_j},\hat{G})$. Identify $(n,t)\in N_r\times [\frac{1}{3},\frac{2}{3}]\subset T_{f_3}$ and $(n,1-t)\in N_r\times [\frac{1}{3},\frac{2}{3}]\subset T_{f_4}$, define $W':=W_3\cup_{N_r\times [\frac{1}{3},\frac{2}{3}]}W_4$, then $\partial W'=T_{f_3\circ f_4}$. By the $M-V$ sequence of $(W';W_3,W_4)$ there are $(s',t')\in [W',\mathbb C P^\infty\times\mathbb{C} P^\infty]$ extending the cohomology factor of $\bar{G}_j$. Since the pasting $W'=W_3\cup_{N_r\times [\frac{1}{3},\frac{2}{3}]}W_4$ preserve the spin structure of $\bar{G}_j$ there is spin structure on $W'$ extending the spin factor of $\bar{G}_j$. Thus we can define a $B$ normal structure $G'_w$ on $W'$ extending $\bar{G}_j$. Consider the following diagram

		$$\xymatrix{\oplus_j H^4(W_j,T_{f_j})\ar[r]&\oplus_j H^4(W_j)\ar[r]&\oplus_j H^4(T_{f_j})\\
			H^4(W',T_{f_3}\cup T_{f_4})\ar[r]\ar[u]^\cong \ar[d]&H^4(W')\ar[r]\ar[u]\ar[d]&H^4(T_{f_3}\cup T_{f_4})\ar[u]^{injective}\ar[d]\\
			H^4(W',T_{f_3\circ f_4})\ar[r]&H^4(W')\ar[r]&H^4(T_{f_3\circ f_4})\\
		}$$
		Arguments analog to the proof that $KS$ is well defined, one can show $KS$ is a homomorphism through this diagram. Note that we need $H^4(T_{f_3}\cup T_{f_4})\to \oplus_j H^4(T_{f_j}))$ being injective since we need a preimage of $G'_w\alpha\in H^4(W')$ in $H^4(W',T_{f_3}\cup T_{f_4})$.

		By Theorem 5 of \cite{Kre18}, if $KS(f)=0$ then there is a $B$ coboundary $(W'',\hat{G})$ of $(T_f,\hat{G})$ such that $(W'';N_r\times [0,\frac{1}{2}],N_r\times [\frac{1}{2},1])$ is
		an $h$-cobordism. Since the spin structure of $W''$ is compatible to $S^1\times D$ we may extend $S^1\times D$ to be $D^2\times D\subset W$ and define a relative $h$-cobordism. Hence by the peudo-isotopy theorem of Cerf, $f=id\in I(N_r, D)$.

	\end{proof}

\begin{prop}\label{ntb}
		$h^*\xi_r$ is nontrivial if and only if $r\in 8\mathbb{Z}+5$.
	\end{prop}

The mapping class group $\mathrm{MCG}(M)$ of a smooth manifold $M$ is the group of isotopy classes of orientation preserving self diffeomorphisms of $M$. It is an important object that encodes fundamental information about the symmetries of $M$. In this paper, we study the computation of $\mathrm{MCG}(M)$ for certain closed, simply-connected $6$-manifolds.

Although closed, simply-connected $6$-manifolds with torsion-free homology groups are classified by their algebraic-topological invariants due to the work of Wall \cite{Wa66} and Jupp \cite{Jup73}, there appears to be no systematic treatment of the computation of their mapping class groups. Particular interest has been devoted to manifolds admitting geometric structures. For instance, in \cite{KreSu} a class of $6$-manifolds with second Betti number $b_2=1$ is studied, including complete intersections of complex dimension $3$.

The present paper considers a class of $6$-manifolds with $b_2=2$, which includes natural compact K\"ahler manifolds such as projective bundles associated to rank $2$ holomorphic vector bundles over the complex projective plane $\mathbb P^2$. More precisely, in this paper we study the mapping class group of the total spaces of $S^2$-bundles over $\mathbb P^2$. A priori, the structure group of such bundles is the orientation-preserving diffeomorphism group of $S^2$. It was shown by Smale \cite{Sma59} that this group is homotopy equivalent to $SO(3)$. Consequently, such bundles arise as the sphere bundles of $3$-dimensional real vector bundles over $\mathbb P^2$.

Let $\mathrm{Vect}_{\mathbb R}^3(\mathbb P^2)$ denote the set of equivalence classes of $3$-dimensional real vector bundles over $\mathbb P^2$. In Proposition \ref{prop:real3} we show that the first Pontrjagin class induces a bijection
$$p_1 \colon \mathrm{Vect}_{\mathbb R}^3(\mathbb P^2) \stackrel{\cong}{\longrightarrow}\{r\omega \in H^4(\mathbb{P}^2):r\in 4\mathbb{Z}+\{0,1\}\},$$
where $\omega$ is the cohomology fundamental class determined by the nature orientation of $\mathbb P^2$.

Let $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\mathrm{MCG}(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.

\begin{prop}\label{prop:real3}
	 	The map
		$$p_1 \colon  [\mathbb{P}^2, BSO(3)]\to \{rx^2\in H^4(\mathbb{P}^2) \ | \ r\in4\mathbb{Z}+\{0,1\}\}$$
	is a bijection and maps $\phi(\gamma_{k,l})$ to $(k^2-4l)x^2$.
	 \end{prop}

Let $I(M)$ be the Torelli group of $M$, which is the normal subgroup of $\m(M)$ consisting of diffeomorphisms that induce the identity on the cohomology groups of $M$.

The projective unitary group $PU(2)$ is isomorphic to the orthogonal group $SO(3)$; consequently, $S^2$-bundles may be identified with $\p^1$-bundles. Thus, the manifolds considered in this paper include many complex K\"ahler manifolds, such as projective bundle associated to rank $2$ holomorphic vector bundles over $\p^2$.

\begin{cor}\label{gmhs}
 Let $M_k=\{(x,y)\in \mathbb{P}^2\times\mathbb{P}^2 \ | \ \sum_i x^k_iy_i=0\}$, then
 \begin{enumerate}
 \item there is a short exact sequence
 $$ 1 \to I(M_k) \to \m(M_k) \to A \to 1,$$
 where $A \cong \z_2$ if $k \ne 1$, and $A \cong S_3$ if $k=1$;
 \item when $k=2l+1$, $I(M_k)$ is isomorphic to $\mathbb{Z}_6\oplus \mathbb{Z}_{\mathrm{gcd}(28,l^2+l)}$.
 \end{enumerate}
 \end{cor}

\begin{eg} \label{eg:1}
 Let $\zeta_k$ be the complex vector bundle over $\p^2$ with total space
 $$E_r=\{(z,z')\in \mathbb{P}^2\times\mathbb{C}^3 \ | \ \sum z^k_iz'_i=0\}$$
 and bundle projection $\pi \colon E_r \to \p^2$, $\pi(z,z')=z$.
 One can identify $\zeta_k$ as conjugation of the orthogonal complement of canonical embedding $O(-k)\to\mathbb{C}^3$. So $\zeta_k\cong \xi_{k,k^2}$ and $p_1(\phi( \zeta_k))=-3k^2$. Notice that $\mathbb{P}(\zeta_1)$ is the Milnor hypersurface $\{(z,z')\in \mathbb{P}^2\times\mathbb{P}^2 \ | \ \sum z_iz'_i=0\}$. We call $M_k=\{(z,z')\in \mathbb{P}^2\times\mathbb{P}^2 \ | \ \sum z^k_iz'_i=0\}=\mathbb{P}(\zeta_k)$ a generalized Milnor hypersurface. One can easily show that the canonical generators of $M_k$ induced from the bundle $\zeta_k$ are $(\pi^* x,\pi'^* x)$. Here $\pi(z,z')=z, \pi'(z,z')=z'$.
\end{eg}

\begin{lemma}\label{bh}
Let $\xi$ be the nontrivial $3$-dimensional real vector bundle over $S^5$, $\partial \colon \pi_{i+1}(S^5)\to \pi_i(S^2)$ be the boundary homomorphism in the long exact sequence of homotopy groups associated to the sphere bundle of $\xi$, $\pi_i(S^4) \to \pi_i(\Omega S^5)$ be the map induced by the adjoint of the identity map $S^5= \Sigma S^4 \to S^5$. Then the composite $\pi_i(S^4) \to \pi_i(\Omega S^5) = \pi_{i+1}(S^5) \to \pi_i(S^2)$ is identified with the homomorphism $\eta^2 \colon \pi_i(S^4) \to \pi_i(S^2)$ induced by the Hopf map $\eta \colon S^3 \to S^2$.

\begin{cor}\label{htg}
Let $N_r$ be the $S^2$-bundle over $ \mathbb{P}^2$ defined as above. Assume $r\in 8\mathbb{Z}+5$. Then
\begin{enumerate}
\item the inclusion of a fiber induces an isomorphism $\pi_3(S^2)\to\pi_3(N_r)$;
\item$\pi_4(N_r)=0$;
\item the bundle projection induces an injective homomorphism $\pi_5(N_r)\to \pi_5(\mathbb{P}^2)$ with image $2 \cdot \pi_5(\mathbb{P}^2)$;
\item $\pi_6(S^2)\to\pi_6(N_r)$ is surjective and $\pi_6(N_r)\cong \mathbb{Z}_6$.
\end{enumerate}
 \end{cor}

\begin{lemma}\label{t7}
 Let $\Theta_7\cong \mathbb{Z}_{28}$ be the Milnor-Kervaire group of oriented differential structures on the $7$-sphere. Then connected sum operation induces a surjective map $cs:\Theta_7\to S_\partial(N_r\times I)$.
 \end{lemma}
 \begin{proof}
Consider the smooth surgery exact sequence ( Theorem 11.22 of \cite{LuTi})
$$L_8(\z) \to S_\partial(N_r\times I) \to [\Sigma N_r, G/O] \to L_7(\z).$$
A direct calculation shows that the group $H^i(\Sigma N_r;\pi_i(G/O))$ vanishes for all $i$. Therefore by obstruction theory, the homotopy set $[\Sigma N_r, G/O]$ consists of a single element. Therefore
the map $L_8(\z) \to S_\partial(N_r\times I)$ is surjective. By the construction, the Wall realization map $L_8(\mathbb Z) \to S_{\partial}(N_r \times I)$ factors through $L_8(\z) \to \Theta_7 \to S_\partial(N_r\times I)$.
 \end{proof}

In \cite[Theorem 7.9 and 7.12]{KreSu}, the group $\m_\partial(S^2\times D^4)$ is computed and a system of generator is given. Let $g_1$ be a diffeomorphism of $D^6$ relative to the boundary, representing a generator of $ \m_\partial(D^6)\cong \Theta_7$, $t_1\in\pi_3(SO(4))$ be a generator of $\mathrm{Im}\, \pi_3(SO(3))\cong\mathbb{Z}$, $t_2$ be a generator of $\pi_4(SO(3))\cong \mathbb{Z}_2$ and $i_\eta:S^3\times D^3\to S^2\times D^4$ be an embedding representing the Hopf map $\eta\in\pi_3(S^2)$. Fix an embedded $6$-disk in $S^2 \times D^4$ and extend $g_1$ to the complement by the identity; let $g_2$ and $g_3$ be the Dehn twists in $i_{\eta}(S^3 \times D^3)$ with parameters $t_1$ and $t_2$, respectively. Then $$\m_\partial(S^2\times D^4)=\mathbb{Z}_{28}\{g_1\}\oplus\mathbb{Z}_{12}\{g_2\}\oplus\mathbb{Z}_2\{g_3\}$$
For any given embedding of $S^2 \times D^4$ in a $6$-manifold, we extend these diffeomorphisms by the identity on the complement and still denote them by the same notation.
 \begin{prop}\label{sur}
 Let $i \colon S^2\times D^4\to N_r$ be a local trivialization of the $S^2$-bundle. Then $I(N_r)$ is generate by the disk-supported diffeomorphism $g_1$ and the Dehn twist $g_2$.
 \end{prop}
 \begin{proof}
Choose a localization $i \colon S^2\times D^4\to N_r$ such that $i(S^2\times D^4)$ and $S^2\vee S^2$ (defined in the proof of Lemma \ref{Ih}) are disjoint. The diffeomorphisms $g_1$ and $g_2$, which are extended by the identity on the complement of $i(S^2 \times D^4)$, clearly induce the identity on the cohomology groups and thus (their isotopy classes) are in $I(N_r)$.

The projective unitary group $PU(2)$ is isomorphic to the orthogonal group $SO(3)$; consequently, $S^2$-bundles may be identified with $\mathbb P^1$-bundles. Thus, the manifolds considered in this paper include many complex K\"ahler manifolds, such as projective bundle associated to rank $2$ holomorphic vector bundles over $\mathbb P^2$.

Rank $2$-complex vector bundles over $\mathbb P^2$ are classified by their Chern classes $c_1$ and $c_2$. For such a vector bundle $\eta$ with Chern classes $c_1(\eta)$ and $c_2(\eta)$, the corresponding $3$-dimensional real vector bundle $\xi$ satisfies
$$p_1(\xi) = c_1(\eta)^2 - 4 c_2(\eta).$$
(see Proposition \ref{prop:real3}). Consequently, Theorem \ref{Mth} also determines the mapping class group of the projective bundle $\mathbb P(\eta)$ of rank $2$ complex vector bundles $\eta$ over $\mathbb P^2$, provided that $\langle c_1(\eta)^2 - 4c_2(\eta), [\mathbb P^2] \rangle = 8n+5$.

An interesting class of examples is given by the hypersurfaces $M_k \subset \mathbb P^2 \times \mathbb P^2$ defined by
$$M_k=\{(x,y)\in \mathbb{P}^2\times\mathbb{P}^2 \ | \  \sum_i x^k_iy_i=0\}.$$
Note that  $M_1=\{(x,y)\in \mathbb{P}^2\times\mathbb{P}^2 \ | \  \sum_i x_iy_i=0\}$ is the Milnor hypersurface, a Fano 3-fold. The hypersurfaces $M_k$ can be identified with $N_{-3k^2}$ (see \S \ref{subsec:bundle}). We  therefore obtain the following corollary.

	\begin{cor}\label{gmhs}
		Let $M_k=\{(x,y)\in \mathbb{P}^2\times\mathbb{P}^2 \ | \  \sum_i x^k_iy_i=0\}$, then
		\begin{enumerate}
		\item there is a short exact sequence
		$$ 1 \to I(M_k) \to \mathrm{MCG}(M_k) \to A \to 1,$$
		where $A \cong \z_2$ if $k \ne 1$, and $A \cong S_3$ if $k=1$;
		\item when $k=2l+1$, $I(M_k)$ is isomorphic to $\mathbb{Z}_6\oplus \mathbb{Z}_{\mathrm{gcd}(28,l^2+l)}$.
		\end{enumerate}
	\end{cor}

To conclude this introduction, we briefly describe the strategy used in our computations. It combines methods from homotopy theory with classical and modified surgery theory. The determination of the action of $\mathrm{MCG}(M)$ on the cohomology groups relies on basic geometric considerations and constructions; this is carried out in \S \ref{sec:action}.

The determination of the Torelli group $I(M)$ is more subtle. By classical surgery theory, for a simply-connected manifold $M$ of dimension at least $5$, there is a short exact sequence
	$$ S(M \times I, \partial) \to I(M) \to I^h(M),$$
where $S(M \times I, \partial)$ denotes the relative smooth structure set of $M \times I$, and $I^h(M)$ denotes the homotopy Torelli group (see \S \ref{subsec:gen}) For the manifolds $N_r$, the image of $S(N_r \times I, \partial)$ in $I(N_r)$ is generated by a disk-supported diffeomorphism $g_1$. On the other hand, when $r \in 8 \mathbb Z +5$,
 homotopy-theoretical arguments show that there is a surjective homomorphism $\pi_6(N_r) \to I^h(N_r)$, whose image is generated by a certain Dehn twist $g_2$.  Consequently, in this case the Torelli group $I(N_r)$ is generated by $g_1$ and $g_2$ (see \S \ref{subsec:gen}). To determine the structure of $I(N_r)$ we make use of a Kreck-Stolz type invariant provided by modified surgery theory, which is an injective homomorphism from the relative Torelli group $I(N_r,D)$ to $\mathbb Z^3$ modulo a lattice $L$ (see \S \ref{subsec:det}).

 \begin{ack}
The author thanks Professor Yang Su for entrusting him with this project and for patiently guiding him through the research and writing process. The author is also grateful to Professor Stephen Theriault, who pointed out a key direction for proving Proposition \ref{ntb}. This research was partially supported by NSFC 12471069.
 \end{ack}

To conclude this introduction, the paper “briefly describe[s] the strategy used in our computations,” combining “methods from homotopy theory with classical and modified surgery theory.”

- **Action on cohomology (used for the quotient in Theorem~\ref{Mth}).** “The determination of the action of $\mathrm{MCG}(M)$ on the cohomology groups relies on basic geometric considerations and constructions; this is carried out in \S \ref{sec:action}.”

- **Relating the Torelli group to surgery-theoretic objects.** “By classical surgery theory, for a simply-connected manifold $M$ of dimension at least $5$, there is a short exact sequence
\[ S(M \times I, \partial) \to I(M) \to I^h(M),\]
where $S(M \times I, \partial)$ denotes the relative smooth structure set of $M \times I$, and $I^h(M)$ denotes the homotopy Torelli group.”

- **Generators for $I(N_r)$.** For the manifolds $N_r$, “the image of $S(N_r \times I, \partial)$ in $I(N_r)$ is generated by a disk-supported diffeomorphism $g_1$.” Also, “when $r \in 8\mathbb Z+5$, homotopy-theoretical arguments show that there is a surjective homomorphism $\pi_6(N_r) \to I^h(N_r)$, whose image is generated by a certain Dehn twist $g_2$.” Hence “in this case the Torelli group $I(N_r)$ is generated by $g_1$ and $g_2$.”

- **Determining the structure of $I(N_r)$ via modified surgery theory.** “To determine the structure of $I(N_r)$ we make use of a Kreck-Stolz type invariant provided by modified surgery theory,” described as “an injective homomorphism from the relative Torelli group $I(N_r,D)$ to $\mathbb Z^3$ modulo a lattice $L$.”

To conclude this introduction, the paper “briefly describe[s] the strategy used in our computations,” combining “methods from homotopy theory with classical and modified surgery theory.”

- **Action on cohomology (used for the quotient in the main theorem).** “The determination of the action of $\mathrm{MCG}(M)$ on the cohomology groups relies on basic geometric considerations and constructions; this is carried out next.”

- **Relating the Torelli group to surgery-theoretic objects.** “By classical surgery theory, for a simply-connected manifold $M$ of dimension at least $5$, there is a short exact sequence
\[ S(M \times I, \partial) \to I(M) \to I^h(M),\]
where $S(M \times I, \partial)$ denotes the relative smooth structure set of $M \times I$, and $I^h(M)$ denotes the homotopy Torelli group.”

- **Generators for $I(N_r)$.** For the manifolds $N_r$, “the image of $S(N_r \times I, \partial)$ in $I(N_r)$ is generated by a disk-supported diffeomorphism $g_1$.” Also, “when $r \in 8\mathbb Z+5$, homotopy-theoretical arguments show that there is a surjective homomorphism $\pi_6(N_r) \to I^h(N_r)$, whose image is generated by a certain Dehn twist $g_2$.” Hence “in this case the Torelli group $I(N_r)$ is generated by $g_1$ and $g_2$.”

- **Determining the structure of $I(N_r)$ via modified surgery theory.** “To determine the structure of $I(N_r)$ we make use of a Kreck-Stolz type invariant provided by modified surgery theory,” described as “an injective homomorphism from the relative Torelli group $I(N_r,D)$ to $\mathbb Z^3$ modulo a lattice $L$.”

Question: Let \(\xi\) be a 3-dimensional real vector bundle over the complex projective plane \(\mathbb{P}^2\) with first Pontrjagin class \(p_1(\xi)=r\omega\), where \(\omega\in H^4(\mathbb{P}^2)\) is the positive generator, and let \(N_r=S(\xi)\) be the total space of the associated sphere bundle. Let \(\mathrm{MCG}(N_r)\) be the mapping class group of \(N_r\), i.e. the group of isotopy classes of orientation-preserving self-diffeomorphisms of \(N_r\), and let \(I(N_r)\subset \mathrm{MCG}(N_r)\) be the Torelli group consisting of classes acting trivially on the cohomology of \(N_r\). Which statement holds for these manifolds?

There exists a short exact sequence \[1\to I(N_r)\to \mathrm{MCG}(N_r)\to A\to 1,\] where \(A\cong \mathbb{Z}_2\) if \(r\neq -3\), and \(A\cong S_3\) if \(r=-3\). Moreover, if \(r=8n+5\), then \[I(N_r)\cong \mathbb{Z}_{2\gcd(3,2n^2+3n+1)}\oplus \mathbb{Z}_{2\gcd(14,n+1)}.\]

Choices:
B. There exists a short exact sequence \[1\to I(N_r)\to \mathrm{MCG}(N_r)\to A\to 1,\] where \(A\cong \mathbb{Z}_2\) if \(r\neq -3\), and \(A\cong S_3\) if \(r=-3\). Moreover, if \(r=8n+1\), then \[I(N_r)\cong \mathbb{Z}_{2\gcd(3,2n^2+3n+1)}\oplus \mathbb{Z}_{2\gcd(14,n+1)}.\]
C. There exists a short exact sequence \[1\to I(N_r)\to \mathrm{MCG}(N_r)\to A\to 1,\] where \(A\cong \mathbb{Z}_2\) if \(r\neq -3\), and \(A\cong S_3\) if \(r=-3\).
D. There exists a short exact sequence \[1\to I(N_r)\to \mathrm{MCG}(N_r)\to A\to 1,\] where \(A\cong \mathbb{Z}_2\) for every \(r\). Moreover, if \(r=8n+5\), then \[I(N_r)\cong \mathbb{Z}_{2\gcd(3,2n^2+3n+1)}\oplus \mathbb{Z}_{2\gcd(14,n+1)}.\]
E. There exists a short exact sequence \[1\to I(N_r)\to \mathrm{MCG}(N_r)\to A\to 1,\] where \(A\cong \mathbb{Z}_2\) if \(r\neq -3\), and \(A\cong S_3\) if \(r=-3\). Moreover, if \(r=8n+5\), then \[I(N_r)\cong \mathbb{Z}_{2\gcd(3,2n^2+3n+1)}\oplus \mathbb{Z}_{2\gcd(14,n+1)}\oplus \mathbb{Z}_{28}.\]

congruence class required for the homotopy-theoretic description of I(N_r)

the explicit computation of the Torelli group when r=8n+5

exceptional cohomological action at r=-3 giving a larger quotient

generation by g_1 and g_2 versus incorrectly adjoining the full smooth-structure contribution as a direct summand

The stem does not explicitly reveal the correct statement. It names the objects and asks for the correct structural description, without giving away the exceptional case or the formula for the Torelli group.

This is essentially a theorem-recall item: the correct option is a near-verbatim statement of the classification result for \(\mathrm{MCG}(N_r)\) and \(I(N_r)\). It does not meaningfully require selecting among independently derived consequences.

Some reasoning is needed to distinguish close variants, such as the exceptional value \(r=-3\), the congruence condition \(r=8n+5\), and the orders \(14\) versus \(28\). However, this is mainly precision recall rather than substantial generative mathematical reasoning.

The distractors are mathematically close to the target theorem and reflect realistic failure modes: swapping the exceptional parameter, weakening by omission, altering a numerical invariant, or overgeneralizing a special-case statement. They are distinct and plausible.

A reasonably well-constructed theorem-discrimination MCQ with strong distractors and no answer leakage, but it is largely a recall/restatement question rather than one that tests genuine generative reasoning.


Question: Let \(\xi\) be a 3-dimensional real vector bundle over the complex projective plane \(\mathbb{P}^2\) with first Pontrjagin class \(p_1(\xi)=r\omega\), where \(\omega\in H^4(\mathbb{P}^2)\) is the positive generator, and let \(N_r=S(\xi)\) be the total space of the associated sphere bundle. Let \(\mathrm{MCG}(N_r)\) be the mapping class group of \(N_r\), i.e. the group of isotopy classes of orientation-preserving self-diffeomorphisms of \(N_r\), and let \(I(N_r)\subset \mathrm{MCG}(N_r)\) be the Torelli group consisting of classes acting trivially on the cohomology of \(N_r\). Which statement holds for these manifolds?

Choices:
B. There exists a short exact sequence \[1\to I(N_r)\to \mathrm{MCG}(N_r)\to A\to 1,\] where \(A\cong \mathbb{Z}_2\) if \(r\neq -3\), and \(A\cong S_3\) if \(r=-3\). Moreover, if \(r=8n+1\), then \[I(N_r)\cong \mathbb{Z}_{2\gcd(3,2n^2+3n+1)}\oplus \mathbb{Z}_{2\gcd(14,n+1)}.\]
C. There exists a short exact sequence \[1\to I(N_r)\to \mathrm{MCG}(N_r)\to A\to 1,\] where \(A\cong \mathbb{Z}_2\) if \(r\neq -3\), and \(A\cong S_3\) if \(r=-3\).
D. There exists a short exact sequence \[1\to I(N_r)\to \mathrm{MCG}(N_r)\to A\to 1,\] where \(A\cong \mathbb{Z}_2\) for every \(r\). Moreover, if \(r=8n+5\), then \[I(N_r)\cong \mathbb{Z}_{2\gcd(3,2n^2+3n+1)}\oplus \mathbb{Z}_{2\gcd(14,n+1)}.\]
E. There exists a short exact sequence \[1\to I(N_r)\to \mathrm{MCG}(N_r)\to A\to 1,\] where \(A\cong \mathbb{Z}_2\) if \(r\neq -3\), and \(A\cong S_3\) if \(r=-3\). Moreover, if \(r=8n+5\), then \[I(N_r)\cong \mathbb{Z}_{2\gcd(3,2n^2+3n+1)}\oplus \mathbb{Z}_{2\gcd(14,n+1)}\oplus \mathbb{Z}_{28}.\]

The stem defines the geometric objects and asks which global statement is true, but it does not reveal the exceptional case, the quotient group, or the explicit formula for the Torelli group. There is no direct answer leakage.

The item is largely a theorem-recall question: the correct option states a specific classification/result about the mapping class group and Torelli group. Although the alternatives force comparison among nearby claims, the task is still close to recognizing the theorem rather than deriving a new conclusion.

The question requires some discrimination among subtly different statements, especially the exceptional case r = -3 and the congruence condition r = 8n+5. However, the main burden is precise recall of a known result, not substantial generative reasoning from the stem.

The distractors are strong: one weakens the true statement, one alters the congruence class, one removes the exceptional S3 case, and one adds an unjustified extra summand. These are plausible and reflect realistic mathematical confusion modes.

A solid high-level theorem-recognition MCQ with good distractors and no answer leakage, but it tests precise recall more than genuinely generative mathematical reasoning.


\label{thm1.1}
Let $X$ be a smooth projective $3$-fold of general type. If $\Vol(X)>12^6$, then $\dim \overline{\Phi_{|2K_X|}(X)}\geq 2$.

\label{section1}

For a smooth projective variety $X$ of general type, we are interested in the behavior of
the $m$-canonical map $\Phi_{|mK_X|}$ of $X$ induced by the $m$-canonical system $|mK_X|$, which is crucial towards the birational classification of varieties of general type.

It is predicted that for a smooth projective variety $X$ of general type of dimension $n$ with large invariants such as the canonical volume $\Vol(X)$ or the geometric genus $p_g(X)$, the $m$-canonical map $\Phi_{|mK_X|}$
behaves just like that of $X_0\times C$ where $X_0$ is a smooth projective variety of general type of dimension $n-1$ and $C$ is a smooth projective curve of genus $g\gg 0$. Such prediction for the birationality of $\Phi_{|mK_X|}$ (known as M\textsuperscript{c}Kernan's conjecture) was recently confirmed by Chen and Liu \cite{CL24} (see also \cite{CJ17, CJ22}).

We are interested in more explicit results in lower dimensions. Recall that Bombieri \cite{Bombieri} showed that $\Phi_{|5K_S|}$ is birational for a smooth projective surface $S$ of general type, so it is natural to study the birationality of $\Phi_{|5K_X|}$ for a smooth projective $3$-fold $X$ of general type with large $\Vol(X)$ or $p_g(X)$.
Chen \cite{Chen03} showed that
$\Phi_{|5K_X|}$ is birational if $p_g(X)\geq 4$, and
recently Chen and Ding \cite{CD25} showed that $\Phi_{|5K_X|}$ is birational if $\Vol(X)\geq 86$, significantly improving the results in \cite{Tod07, LDi12, Chen12}.

Now we focus on the bicanonical map. Recall that for a smooth projective surface $S$ of general type, $h^0(S, 2K_S)\geq 2$ by the Riemann--Roch formula (see \cite[Page~185]{Bombieri}), which implies that $\Phi_{|2K_S|}$ is a non-trivial map. So it is natural to ask whether $\Phi_{|2K_X|}$ is not a pencil (namely, $\dim \overline{\Phi_{|2K_X|}(X)}\geq 2$) for a smooth projective $3$-fold $X$ of general type with large $\Vol(X)$ or $p_g(X)$.
Recently, Chen, Jiang, and Yan \cite[Theorem~1.2]{CJY24} proved that $\dim \overline{\Phi_{|2K_X|}(X)}\geq 2$ if $p_g(X)>201$, answering an open question of Chen and Zhang \cite{CZ08}. However, for a smooth projective $3$-fold $X$ of general type with large $\Vol(X)$, the behavior of its bicanonical map is still mysterious. To the best of our knowledge, the only known result is that $h^0(X, 2K_X)\geq 1$ if $\Vol(X)>879^3$, by Di Biagio \cite[Theorem~1.1]{LDi12} (see also \cite{Tod07}), and we even do not know whether $\Phi_{|2K_X|}$ is non-trivial for $\Vol(X)$ sufficiently large.

The main goal of this paper is to show that $\Phi_{|2K_X|}$ is not a pencil for a smooth projective $3$-fold $X$ of general type with large canonical volume.

We are interested in more explicit results in lower dimensions. Recall that Bombieri \cite{Bombieri} showed that $\Phi_{|5K_S|}$ is birational for a smooth projective surface $S$ of general type, so it is natural to study the birationality of $\Phi_{|5K_X|}$ for a smooth projective $3$-fold $X$ of general type with large $\Vol(X)$ or $p_g(X)$.
Chen \cite{Chen03} showed that
$\Phi_{|5K_X|}$ is birational if $p_g(X)\geq 4$, and
recently Chen and Ding \cite{CD25} showed that $\Phi_{|5K_X|}$ is birational if $\Vol(X)\geq 86$, significantly improving the results in \cite{Tod07, LDi12, Chen12}.

Now we focus on the bicanonical map. Recall that for a smooth projective surface $S$ of general type, $h^0(S, 2K_S)\geq 2$ by the Riemann--Roch formula (see \cite[Page~185]{Bombieri}), which implies that $\Phi_{|2K_S|}$ is a non-trivial map. So it is natural to ask whether $\Phi_{|2K_X|}$ is not a pencil (namely, $\dim \overline{\Phi_{|2K_X|}(X)}\geq 2$) for a smooth projective $3$-fold $X$ of general type with large $\Vol(X)$ or $p_g(X)$.
Recently, Chen, Jiang, and Yan \cite[Theorem~1.2]{CJY24} proved that $\dim \overline{\Phi_{|2K_X|}(X)}\geq 2$ if $p_g(X)>201$, answering an open question of Chen and Zhang \cite{CZ08}. However, for a smooth projective $3$-fold $X$ of general type with large $\Vol(X)$, the behavior of its bicanonical map is still mysterious. To the best of our knowledge, the only known result is that $h^0(X, 2K_X)\geq 1$ if $\Vol(X)>879^3$, by Di Biagio \cite[Theorem~1.1]{LDi12} (see also \cite{Tod07}), and we even do not know whether $\Phi_{|2K_X|}$ is non-trivial for $\Vol(X)$ sufficiently large.

We briefly explain the difficulties and techniques in the proof. As in the study of $\Phi_{|5K_X|}$ in \cite{Tod07, LDi12, Chen12, CD25}, the basic idea is to creat non-klt centers by constructing a boundary $\Delta\sim_{\bQ}\delta K_X$ for some $\delta<2$
and cut down the dimension of non-klt centers by the Anghern--Siu type method.
The issue in dealing with $\Phi_{|2K_X|}$ instead of $\Phi_{|5K_X|}$ or $\Phi_{|3K_X|}$ is that, often we are not allowed to cut down the dimension of a non-klt center if the canonical volume of the non-klt center
is small (e.g., a surface with small canonical volume or a curve with geometric genus $2$), otherwise the constant $\delta$ will exceed $2$. In this case, instead of cutting down the dimension of the non-klt center, we prove certain extension theorems which allow us to lift global sections from the centers to $X$.
Here we remark that in \cite{CL24}, the key step is to prove an extension theorem for fibrations, where many techniques such as the Kawamata--Viehweg vanishing theorem, canonical bundle formula, the Zariski--Nakayama decomposition, the Hacon--M\textsuperscript{c}Kernan extension theorem, and Birkar's lower bound of lct are involved, and the extension theorem is not effective; in this paper, we prove an effective version of extension theorem by using the Nadel vanishing theorem and for curve centers we do not require them to form a fibration.

As a bi-product of our method, we prove the following theorem concerning the pluricanonical maps of $3$-folds of general type fibered by $(1,2)$-surfaces or $(2,3)$-surfaces, as a complement of the results of Xu \cite[Theorem~1.4, Theorem~1.5]{XuJ14}:
\begin{theorem}\label{thm1.3}
Let $X$ be a smooth $3$-fold of general type. Suppose that $X$ admits a fibration $X\to C$ to a curve with a general fiber denoted by $F$.
\begin{enumerate}
 \item If $F$ is a $(1,2)$-surface and $\Vol(X)> 60$, then $\Phi_{|mK_X|}$ is generically finite of degree $2$ for $2\leq m \leq 4$ and is birational for $m\geq 5$.
 \item If $F$ is a $(2,3)$-surface and $\Vol(X)>456$, then $\Phi_{|mK_X|}$ is generically finite of degree $2$ for $2\leq m \leq 3$ and is birational for $m\geq 4$.
\end{enumerate}
\end{theorem}

\begin{lemma}\label{lem separete Z1Z2}
Let $X$ be a normal projective variety and let $D$ be an effective Cartier divisor on $X$.
Let $C_1$ and $C_2$ be closed subvarieties on $X$.
Suppose that the natural restriction map
\[
 H^0(X,D) \to H^0(C_1, D|_{C_1} ) \oplus H^0(C_2, D|_{C_2})
\]
is surjective.
Then \[
\dim\overline{\Phi_{{|D|}}(X)}\geq \min\{\dim\overline{\Phi_{{|D|_{C_1}|}}(C_1)}, \dim\overline{\Phi_{{|D|_{C_2}|}}(C_2)}\}+1.\]
In particular,
if $h^0(C_i, D|_{C_i})\geq 2$ for $i=1,2$, then $\dim\overline{\Phi_{{|D|}}(X)}\geq 2$.
\end{lemma}

As an application of Theorem~\ref{thm:ext for surface}, we prove Theorem~\ref{thm1.3} on fibrations of $(1,2)$-surfaces and $(2,3)$-surfaces.
First we recall properties of pluricanonical systems of $(1,2)$-surfaces and $(2,3)$-surfaces.
\begin{theorem}\label{surface} Let $S$ be a smooth projective surface of general type.
\begin{itemize}
\item If $S$ is a $(1,2)$-surface, then $\Phi_{|mK_S|}$ is a generically finite map of degree $2$ for $2\leq m\leq 4$ and is a birational map for $m\geq 5$.
\item If $S$ is a $(2,3)$-surface, then $\Phi_{|mK_S|}$ is a generically finite map of degree $2$ for $2\leq m\leq 3$ and is a birational map for $m\geq 4$.
\end{itemize}
\end{theorem}
\begin{proof}
 This follows from \cite[Theorem~VII.5.1, Proposition~VII.7.1, Proposition~VII.7.2]{BHPV}, \cite[Lemma~1.1]{Horikawa1}, and \cite[Lemma~2.1]{Horikawa2}.
\end{proof}

\begin{proposition}\label{prop:whenvolgeq12}
Let $X$ be a minimal $3$-fold of general type.
Assume that there exists a resolution $\pi:W\to X$ and a fibration $f: W\to C$ to a curve with a general fiber denoted by $F$.
Assume that $\pi^*K_X-bF$
 is $\mathbb{Q}$-effective for some rational number $b>2$.
If \[\frac{(b-2)^2}{(b+2)^2} \Vol(F)>8,\] then $\dim \overline{\Phi_{|2K_X|}(X)}\geq 2$.
\end{proposition}

Now suppose that there exists an irreducible curve $C$ passing through a very general closed point $P$ such that $(L\cdot C)< 4<\frac{1}{2}L^2$. Since $P$ is very general, $C$ is a nef curve with $p_g(C)\geq 2$. Moreover, after taking a higher model of $F$ and replacing $L$ by its pullback, we may assume that $C$ is smooth; this step does not change $L^2$ or $(L\cdot C)$ by the projection formula. As $L^2>2(L\cdot C)$, there exists a sufficiently small positive number $\epsilon$ such that $L-(1+\epsilon)C$ is big by \cite[Theorem~2.2.15]{Positivity1}. In particular, we may write
$$L\sim_{\mathbb{Q}}(1+\epsilon)C+T$$ where $T$ is an effective $\mathbb{Q}$-divisor.
Then by Sakai's lemma \cite[Example~9.4.12]{Positivity2},
\[
H^1\left(F, K_F+\left\lceil{L-C-\frac{1}{1+\epsilon}T}\right\rceil\right)=0,
\]
which implies that the natural map
\[
H^0\left(F, K_F+\left\lceil{L-\frac{1}{1+\epsilon}T}\right\rceil\right)\to H^0\left(C, K_C+\left. \left\lceil{L-\frac{1}{1+\epsilon}T-C}\right\rceil \right|_{C}\right)
\]
is surjective.
Since $\deg(\roundup{L-\frac{1}{1+\epsilon}T-C}|_C)>0$, it is clear that
\[h^0\left(C, K_C+\left. \left\lceil{L-\frac{1}{1+\epsilon}T-C}\right\rceil \right|_C\right)\geq 2\]
by the Riemann--Roch formula. Therefore,
\[
h^0(F, K_F+\roundup{L})\geq h^0\left(F, K_F+\left\lceil{L-\frac{1}{1+\epsilon}T}\right\rceil\right)\geq 2. \qedhere\]
\end{proof}
Now we are ready to study the bicanonical maps.
\begin{corollary}\label{cor X/C}
Let $X$ be a minimal $3$-fold of general type.
Assume that there exists a resolution $\pi:W\to X$ and a fibration $f:W\to C$ to a curve with a general fiber denoted by $F$.
Assume that $ \Vol(X)>2.9\times10^4$ and $\pi^*K_X-bF$ is $\mathbb{Q}$-effective
for some rational number $b> 24$, then $\dim \overline{\Phi_{|2K_X|}(X)}\geq 2$.
\end{corollary}

We briefly explain the difficulties and techniques in the proof. As in the study of $\Phi_{|5K_X|}$ in \cite{Tod07, LDi12, Chen12, CD25}, the basic idea is to creat non-klt centers by constructing a boundary $\Delta\sim_{{\mathbb Q}}\delta K_X$ for some $\delta<2$
and cut down the dimension of non-klt centers by the Anghern--Siu type method.
The issue in dealing with $\Phi_{|2K_X|}$ instead of $\Phi_{|5K_X|}$ or $\Phi_{|3K_X|}$ is that, often we are not allowed to cut down the dimension of a non-klt center if the canonical volume of the non-klt center
is small (e.g., a surface with small canonical volume or a curve with geometric genus $2$), otherwise the constant $\delta$ will exceed $2$. In this case, instead of cutting down the dimension of the non-klt center, we prove certain extension theorems which allow us to lift global sections from the centers to $X$.
Here we remark that in \cite{CL24}, the key step is to prove an extension theorem for fibrations, where many techniques such as the Kawamata--Viehweg vanishing theorem, canonical bundle formula, the Zariski--Nakayama decomposition, the Hacon--M\textsuperscript{c}Kernan extension theorem, and Birkar's lower bound of lct are involved, and the extension theorem is not effective; in this paper, we prove an effective version of extension theorem by using the Nadel vanishing theorem and for curve centers we do not require them to form a fibration.

As a bi-product of our method, we prove the following theorem concerning the pluricanonical maps of $3$-folds of general type fibered by $(1,2)$-surfaces or $(2,3)$-surfaces, as a complement of the results of Xu \cite[Theorem~1.4, Theorem~1.5]{XuJ14}:
\begin{theorem}\label{thm1.3}
Let $X$ be a smooth $3$-fold of general type. Suppose that $X$ admits a fibration $X\to C$ to a curve with a general fiber denoted by $F$.
\begin{enumerate}
 \item If $F$ is a $(1,2)$-surface and $\Vol(X)> 60$, then $\Phi_{|mK_X|}$ is generically finite of degree $2$ for $2\leq m \leq 4$ and is birational for $m\geq 5$.
 \item If $F$ is a $(2,3)$-surface and $\Vol(X)>456$, then $\Phi_{|mK_X|}$ is generically finite of degree $2$ for $2\leq m \leq 3$ and is birational for $m\geq 4$.
\end{enumerate}
\end{theorem}

As in the study of $\Phi_{|5K_X|}$ in \cite{Tod07, LDi12, Chen12, CD25}, the basic idea (to prove Theorem~\ref{thm1.1}) is to "creat non-klt centers" by constructing a boundary $\Delta\sim_{{\mathbb Q}}\delta K_X$ for some $\delta<2$ and then "cut down the dimension of non-klt centers by the Anghern--Siu type method." The difficulty for $\Phi_{|2K_X|}$ is that one is often "not allowed to cut down the dimension of a non-klt center if the canonical volume of the non-klt center is small ... otherwise the constant $\delta$ will exceed $2$." In this situation, "instead of cutting down the dimension of the non-klt center," the approach is to "prove certain extension theorems which allow us to lift global sections from the centers to $X$." The paper claims an "effective version" of such extension theorems "by using the Nadel vanishing theorem," and "for curve centers" it does not require the centers "to form a fibration."

As in the study of $\Phi_{|5K_X|}$ in \cite{Tod07, LDi12, Chen12, CD25}, the basic idea (to prove the main theorem) is to "creat non-klt centers" by constructing a boundary $\Delta\sim_{{\mathbb Q}}\delta K_X$ for some $\delta<2$ and then "cut down the dimension of non-klt centers by the Anghern--Siu type method." The difficulty for $\Phi_{|2K_X|}$ is that one is often "not allowed to cut down the dimension of a non-klt center if the canonical volume of the non-klt center is small ... otherwise the constant $\delta$ will exceed $2$." In this situation, "instead of cutting down the dimension of the non-klt center," the approach is to "prove certain extension theorems which allow us to lift global sections from the centers to $X$." The paper claims an "effective version" of such extension theorems "by using the Nadel vanishing theorem," and "for curve centers" it does not require the centers "to form a fibration."

Question: Let $X$ be a smooth projective $3$-fold of general type. Write $\Vol(X)=\Vol(K_X)$ for its canonical volume, and let $\Phi_{|2K_X|}$ be the rational map induced by the bicanonical linear system $|2K_X|$. If $\Vol(X)>12^6$, which quantitative statement about the image of the bicanonical map is valid?

One has $\dim \overline{\Phi_{|2K_X|}(X)}\geq 2$.

Choices:
B. One has $\dim \overline{\Phi_{|2K_X|}(X)}\geq 2$ provided only that $\Vol(X)\geq 12^6$.
C. One has $\dim \overline{\Phi_{|2K_X|}(X)}\geq 1$.
D. The map $\Phi_{|2K_X|}$ is generically finite onto its image.
E. One has $\dim \overline{\Phi_{|2K_X|}(X)}\geq 2$, and moreover this follows by cutting down non-klt centers all the way to points using a boundary $\Delta\sim_{\mathbb Q}\delta K_X$ with some uniform $\delta<2$.

drop the conclusion from image dimension at least 2 to mere non-triviality

confusing non-pencil with generically finite bicanonical map

ability to cut down all non-klt centers uniformly with a single effective $\delta<2$

The stem does not state or strongly hint at the correct conclusion beyond giving the hypothesis. The exact answer is not leaked; the options must still be compared.

The item is close to a direct recall of a theorem with the same numerical threshold and conclusion. However, it is not purely tautological because the choices include nearby variants: a weaker true statement, a threshold perturbation, and stronger but invalid conclusions.

Some reasoning is needed to distinguish the strongest valid conclusion from weaker or overly strong alternatives, especially against choices C and D. Still, the question mainly tests theorem recall rather than substantial generative reasoning.

The distractors are plausible and mathematically targeted: B tests strict versus non-strict threshold, C is a weaker true conclusion, D confuses non-pencil behavior with generic finiteness, and E overstates the proof mechanism. These reflect realistic failure modes.

A solid MCQ with no major answer leakage and strong distractors, but it leans heavily on recalling a specific theorem rather than eliciting deep generative reasoning.


\label{mainthm}
Let $K^{\rm Kum}$ be the maximal Kummer extension of $K$, obtained by adjoining all roots of elements of $K$. The group  $H^i_{\textup{ét}}(\overline{X}, \mathbf{Q}/\mathbf{Z}(j))^{G_{K^{\operatorname{Kum}}}}$ is finite for all odd $i$ and arbitrary $j$. In particular, the torsion subgroup of $K^{\rm Kum}$-points of an abelian variety over $K$ is finite.

\begin{prop}\label{prop:solvable_example} There exist two infinite Galois extensions $M, M'$ of $\Q$ with $\Gal(M | \Q)\cong\Gal(M' | \Q)$ solvable of class 2 such that all abelian varieties defined over subfields of $M$ have finite torsion subgroup over $M$ but some abelian varieties defined over $\Q$ have infinite torsion over $M'$.
\end{prop}

\begin{remark}\label{rmk: counterexample1}
 One may ask whether Theorem \ref{thm: reduction} holds with $K'=K$, i.e.~whether it is sufficient to check finiteness over abelian subextensions of $M|K$. The answer is negative, even when $i=j=1$ and $X$ is an abelian variety,
as the following example shows.

Recall from
the introduction that this case amounts to understanding the torsion subgroups of abelian varieties.
Consider the elliptic curve $E$ with Weierstrass equation $y^2=x^3+x$, with complex multiplication by $\mathbb{Z}[i]$ over $\Q(i)$. By the theory of complex multiplication the field $M$ obtained by adjoining all torsion points of $E$ to $\Q(i)$ is the maximal abelian extension of $\Q(i)$ (it is easy to show that $M| \Q(i)$ is abelian, and this is all we need). This implies that $M| \Q$ is a solvable extension, because both $M| \Q(i)$ and $\Q(i)| \Q$ are abelian. By construction, the torsion subgroup $E(M)_{\mathrm{tors}}\subset E(M)$ is infinite. On the other hand, since $\Q^{\operatorname{ab}}=\Q^{\operatorname{cyc}}$ by the Kronecker--Weber theorem,  Ribet's result \cite{Ribet1981CyclotomicTorsion} shows that $E(\Q^{\operatorname{ab}})$ is finite, hence there is no abelian subextension $F$ of $M| \Q$ such that $E(F)_{\mathrm{tors}}$ is infinite.
\end{remark}

\begin{lem}\label{lemma: p-adic nilpotent implies algebraic nilpotent}
Let $G \subset\GL_{n}(\Q_p)$ be a connected reductive algebraic subgroup and $H \subset G(\Q_p)$ be an abstract subgroup, dense in the Zariski topology. Suppose that $H$ is solvable of finite class as an abstract group. Then $G$ is a torus.
\end{lem}

Let $K$ be a number field with fixed algebraic closure $\overline K$, and let $X$ be a smooth projective geometrically connected variety over $K$. We denote by $\overline{X}$ the base change of $X$ to $\overline{K}$, and for an intermediate field $M$ with $\overline K\supset M\supset K$ we write $G_M:=\mathrm{Gal}(\overline K | M)$. In this note we shall study the question of finiteness of the subgroup of $G_M$-invariants of the étale cohomology groups $H^i_{\textup{ét}}(\overline{X}, \mathbf{Q}/\mathbf{Z}(j))$ over certain infinite extensions $M|K$. Note that when $X=A$ is an abelian variety with dual abelian variety $A^*$, the group $H^1_{\textup{ét}}(\overline{A^*}, \mathbf{Q}/\mathbf{Z}(1))$ identifies with the torsion subgroup of $A(\overline K)$ as a $G_K$-module (see, for instance, the introduction of \cite{RoesslerSzamuely}), so in this case $G_M$-invariants on cohomology correspond to $M$-rational torsion points of $A$.

In the paper just mentioned, R\"ossler and the second author considered the case when $M$ is obtained by adjoining all roots of unity to $K$ and proved finiteness of $H^i_{\textup{ét}}(\overline{X}, \mathbf{Q}/\mathbf{Z}(j))^{G_M}$ for {\bf odd} $i$. The method of proof generalized that of a much earlier theorem of Ribet \cite{Ribet1981CyclotomicTorsion} who proved finiteness of the torsion subgroup of $M$-points of an abelian variety for $M$ as above.

But there are other infinite extensions $M|K$ over which finiteness of $H^i_{\textup{ét}}(\overline{X}, \mathbf{Q}/\mathbf{Z}(j))^{G_M}$ holds for {odd} $i$. Indeed, recently Murotani and Ozeki \cite{MurotaniOzeki} proved this for a large class of infinite Kummer extensions $M$ of $K$. Here we extend their statement to all Kummer extensions.

Let $K$ be a number field with fixed algebraic closure $\overline K$, and let $X$ be a smooth projective geometrically connected variety over $K$. We denote by $\overline{X}$ the base change of $X$ to $\overline{K}$, and for an intermediate field $M$ with $\overline K\supset M\supset K$ we write $G_M:=\Gal(\overline K | M)$. In this note we shall study the question of finiteness of the subgroup of $G_M$-invariants of the étale cohomology groups $H^i_{\textup{ét}}(\overline{X}, \Q/\Z(j))$ over certain infinite extensions $M|K$. Note that when $X=A$ is an abelian variety with dual abelian variety $A^*$, the group $H^1_{\textup{ét}}(\overline{A^*}, \Q/\Z(1))$ identifies with the torsion subgroup of $A(\Kbar)$ as a $G_K$-module (see, for instance, the introduction of \cite{RoesslerSzamuely}), so in this case $G_M$-invariants on cohomology correspond to $M$-rational torsion points of $A$.

In the paper just mentioned, R\"ossler and the second author considered the case when $M$ is obtained by adjoining all roots of unity to $K$ and proved finiteness of $H^i_{\textup{ét}}(\overline{X}, \Q/\Z(j))^{G_M}$ for {\bf odd} $i$. The method of proof generalized that of a much earlier theorem of Ribet \cite{Ribet1981CyclotomicTorsion} who proved finiteness of the torsion subgroup of $M$-points of an abelian variety for $M$ as above.

But there are other infinite extensions $M|K$ over which finiteness of $H^i_{\textup{ét}}(\overline{X}, \Q/\Z(j))^{G_M}$ holds for {odd} $i$. Indeed, recently Murotani and Ozeki \cite{MurotaniOzeki} proved this for a large class of infinite Kummer extensions $M$ of $K$. Here we extend their statement to all Kummer extensions.

As already pointed out in \cite[Remark 3.6]{RoesslerSzamuely}, for even $i$ the group $H^i_{\textup{ét}}(\overline{X}, \Q/\Z(j))^{G_{K^{\rm Kum}}}$ is infinite because cycle classes on $X_{K^{\rm Kum}}$ generate an infinite subgroup. It is an interesting open question whether the quotient modulo cycle classes is finite. On the other hand, finiteness may fail already over infinite abelian extensions: for an abelian variety of CM type over $K$ all torsion points are defined over the maximal abelian extension.

\begin{theorem}\label{thm: reduction}
Let $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\Kbar$ such that $\Gal(M|K)$ is solvable of finite class. Fix integers $i\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\Kbar$ and all subextensions
$
K' \subset F \subset K'M
$
with $F | K'$ abelian the following conditions hold:
\begin{enumerate}
 \item For all primes $p$ we have $\left( H^i_{\textup{ét}}(\overline{X}, \Q_p(j))^{\operatorname{ss}} \right)^{G_F}=0$.
 \item For all but finitely many primes $p$ we have $\left( H^i_{\textup{ét}}(\overline{X}, \Z/p\Z(j))^{\operatorname{ss}} \right)^{G_F}=0$.
\end{enumerate}
Then $H^i_{\textup{ét}}(\overline{X}, \Q/\Z(j))^{G_M}$ is finite.
\end{theorem}

\begin{prop}\label{prop:31-general}
If $i$ is odd, then
$\left( H^i_{\textup{\'et}}(\overline{X},\mathbf{Q}_p(j))^{\operatorname{ss}} \right)^{G_{L^{\rm cyc}}}=0$ for all primes $p$.
\end{prop}
\begin{proof}
The proof of \cite[Proposition~3.1]{RoesslerSzamuely} goes through verbatim with Corollary~\ref{cor:cyclo} replacing Ribet's Galois-theoretic lemma cited above. We briefly review the argument. Pick a simple nonzero
$G_L$-submodule $W\subset \left( H^i_{\textup{\'et}}(\overline{X},\mathbf{Q}_p(j))^{\operatorname{ss}} \right)^{G_{L^{\rm cyc}}}$. Note that $G_L$ acts on $W$ via $G_L/G_{L^{\rm cyc}}$ which is abelian, and that $W$ is semisimple for this action. Using Grothendieck's local monodromy theorem
one then shows that after taking a suitable finite
extension of $L$ the action of $G_L$ on $W$ is unramified outside $p$; thus by
Corollary~\ref{cor:cyclo} it factors through $G_p:=\mathrm{Gal}(L(\mu_{p^\infty})|L)$.
For a place $w\mid p$ of $L$ the extension $L_w|\mathbf{Q}_p$ is finite by our assumption on $L$ and the local
extension $L_w(\mu_{p^\infty})|L_w$ is totally ramified, hence the local Galois group of $L_w$ surjects onto
$G_p$. As in the original proof, after taking a further finite field extension the existence of the Hodge–Tate decomposition and the Weil conjectures (Deligne's theorem) force a Frobenius at a place of good reduction $w\nmid p$ to have eigenvalues that are both integral powers of $Nw$ and of absolute value $(Nw)^{i/2-j}$, which is impossible for odd $i$.
\end{proof}

\begin{prop}\label{prop:34-general}
If $i$ is odd, then
$\left(H^i_{\textup{\'et}}(\overline{X},\mathbf{Z}/p\mathbf{Z}(j))^{\operatorname{ss}}\right)^{G_{L^{\rm cyc}}}=0$
for all but finitely many primes $p$.
\end{prop}
\begin{proof} Here we adapt the proof of \cite[Proposition~3.4]{RoesslerSzamuely}; note some similarities with the proof of Theorem \ref{thm: reduction}.

\begin{proof}[Proof of Theorem \ref{mainthm}]
Suppose by contradiction that $H^i_{\textup{ét}}(\overline{X}, \Q/\Z(j))^{G_{K^{\rm Kum}}}$ is infinite. By \Cref{thm: reduction}, there exist a finite extension $K' | K$ and an abelian subextension $F$ of $K' \Kkum | K'$ such that at least one of the following holds:
\begin{enumerate}
 \item $\left( H^i_{\textup{ét}}(\overline{X}, \Q_p(j))^{\operatorname{ss}} \right)^{G_F} \neq (0)$ for some prime $p$;
 \item $\left( H^i_{\textup{ét}}(\overline{X}, \Z/p\Z(j))^{\operatorname{ss}} \right)^{G_F} \neq (0)$ for infinitely many primes $p$.
\end{enumerate}
Replacing $K$ with $K'$ (which is again a number field), we have an abelian extension $F|K$ contained in $K^{\operatorname{Kum}}$ for which (1) or (2) above holds. Applying \Cref{prop: abelianisation Kummer}, we find that $F$ is the compositum of $K^{\operatorname{cyc}}$ and an abelian extension $L|K$ with bounded local degrees. Further replacing $K$ with $L$, we obtain an extension $F|L$, where $F$ is contained in $L^{\operatorname{cyc}}$ and (1) or (2) above holds. The two cases contradict \Cref{prop:31-general} and \Cref{prop:34-general}, respectively.
\end{proof}

\begin{theorem}\label{mainthm}
Let $K^{\rm Kum}$ be the maximal Kummer extension of $K$, obtained by adjoining all roots of elements of $K$. The group  $H^i_{\textup{ét}}(\overline{X}, \Q/\Z(j))^{G_{K^{\operatorname{Kum}}}}$ is finite for all odd $i$ and arbitrary $j$. In particular, the torsion subgroup of $K^{\rm Kum}$-points of an abelian variety over $K$ is finite.
\end{theorem}

\begin{theorem}\label{thm: reduction}
Let $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\Kbar$ such that $\Gal(M|K)$ is solvable of finite class. Fix integers $i\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\Kbar$ and all subextensions
$
K' \subset F \subset K'M
$
with $F | K'$ abelian the following conditions hold:
\begin{enumerate}
    \item For all primes $p$ we have $\left( H^i_{\textup{ét}}(\overline{X}, \Q_p(j))^{\operatorname{ss}} \right)^{G_F}=0$.
    \item For all but finitely many primes $p$ we have $\left( H^i_{\textup{ét}}(\overline{X}, \Z/p\Z(j))^{\operatorname{ss}} \right)^{G_F}=0$.
\end{enumerate}
Then $H^i_{\textup{ét}}(\overline{X}, \Q/\Z(j))^{G_M}$ is finite.
\end{theorem}

As already pointed out in \cite[Remark 3.6]{RoesslerSzamuely}, for even $i$ the group $H^i_{\textup{ét}}(\overline{X}, \mathbf{Q}/\mathbf{Z}(j))^{G_{K^{\rm Kum}}}$ is infinite because cycle classes on $X_{K^{\rm Kum}}$ generate an infinite subgroup. It is an interesting open question whether the quotient modulo cycle classes is finite.  On the other hand, finiteness may fail already over infinite abelian extensions: for an abelian variety of CM type over $K$ all torsion points are defined over the maximal abelian extension.

The Galois group of the field $K^{\rm Kum}$ over $K$ is nilpotent of class 2. The proof of Theorem \ref{mainthm} will proceed by a general reduction to abelian extensions from solvable extensions of finite class:

\begin{theorem}\label{thm: reduction}
Let $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\overline K$ such that $\mathrm{Gal}(M|K)$ is solvable of finite class. Fix integers $i\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\overline K$ and all subextensions
$
K' \subset F \subset K'M
$
with $F | K'$ abelian the following conditions hold:
\begin{enumerate}
    \item For all primes $p$ we have $\left( H^i_{\textup{ét}}(\overline{X}, \Q_p(j))^{\operatorname{ss}} \right)^{G_F}=0$.
    \item For all but finitely many primes $p$ we have $\left( H^i_{\textup{ét}}(\overline{X}, \mathbf{Z}/p\mathbf{Z}(j))^{\operatorname{ss}} \right)^{G_F}=0$.
\end{enumerate}
Then $H^i_{\textup{ét}}(\overline{X}, \mathbf{Q}/\mathbf{Z}(j))^{G_M}$ is finite.
\end{theorem}

Here $H^i_{\textup{ét}}(\overline{X}, \Q_p(j))^{\operatorname{ss}}$ (resp.~$H^i_{\textup{ét}}(\overline{X}, \mathbf{Z}/p\mathbf{Z}(j))^{\operatorname{ss}}$) denotes the semisimplication of the module $H^i_{\textup{ét}}(\overline{X}, \Q_p(j))$ (resp.~$H^i_{\textup{ét}}(\overline{X}, \mathbf{Z}/p\mathbf{Z}(j))$) with respect to the action of $\mathrm{Gal}(\overline K|K)$. It is part of the Tate conjecture (at least with $\Q_p$-coefficients) that these actions are semisimple and thus the superscripts `ss' can be omitted. If we assume semisimplicity, then in view of well-known arguments about abelian groups (see (\cite[Lemma 2.1]{RoesslerSzamuely}) conditions (1) and (2) above can be replaced by the simpler condition that the group $H^i_{\textup{ét}}(\overline{X}, \mathbf{Q}/\mathbf{Z}(j))^{G_F}$ is finite.

\begin{remarks} ${}$
\begin{enumerate}
\item In fact, we prove a slightly stronger statement: there exists a finite extension $K'|K$, depending only on $i$, $j$ and $X$, such that if $M$ is as in the theorem and all abelian extensions $F|K'$ contained in $K'M$ satisfy conditions (1) and (2), then $H^i_{\textup{ét}}(\overline{X}, \mathbf{Q}/\mathbf{Z}(j))^{G_M}$ is finite. On the other hand, there may exist $K''|K$ finite (often we may take $K''=K$; see Remark \ref{rmk: counterexample1}) such that (1) and (2) hold for all abelian $F|K''$ contained in $K''M$ but nevertheless finiteness of $H^i_{\textup{ét}}(\overline{X}, \mathbf{Q}/\mathbf{Z}(j))^{G_M}$ fails.
\item The semisimplicity conjecture with finite coefficients is perhaps less widely known. However, it is part of general motivic expectations and Corollary 1.3.4 (4) of \cite{Cadoretetal} provides strong evidence for it: Cadoret, Hui and Tamagawa prove that over a finitely generated field of positive characteristic semisimplicity of cohomology with $\Q_p$-coefficients for $p$ large enough (prime to the characteristic) implies semisimplicity with $\mathbf{Z}/p\mathbf{Z}$-coefficients for $p$ large enough. The characteristic 0 analogue of this result would suffice for our purposes. Note also that by an observation of Moonen \cite{Moonen} semisimplicity in characteristic 0 with $\Q_p$-coefficients follows from the usual Tate conjecture on the image of the cycle map.
\end{enumerate}
\end{remarks}

For an abelian variety $A$ over $K$ semisimplicity of the Galois action on the rational Tate module $V_p(A)$ and on $p$-torsion points (for $p$ large enough) is known by classical work of Faltings and Zarhin (see e.g.~Chapter IV by Schappacher in \cite{FaltingsWustholz}). So in this case finiteness of torsion points of $A$ over the maximal abelian extension $(K')^{\rm ab}$ of every finite extension $K' | K$ implies finiteness over solvable extensions of finite class. In particular, plugging in a well-known result of Zarhin (\cite{ZarhinCM}, Corollary of Theorem 1) we obtain:

\begin{cor} Let $A$ be an abelian variety over $K$ having no simple isogeny factor of CM type over the algebraic closure $\overline K$. Then $A$ has finitely many torsion points over Galois extensions $M|K$ such that $\mathrm{Gal}(M| K)$ is solvable of finite class. \end{cor} This statement was also obtained using arguments similar to ours in very recent work of Huryn \cite{Huryn}. Huryn's preprint appeared while we were writing up the present note and we took the opportunity to borrow one of his arguments to simplify the proof of one of our lemmas (Lemma \ref{lemma: p-adic nilpotent implies algebraic nilpotent} below).

Observe that the conditions on the extensions of $K$ appearing in Theorem \ref{thm: reduction} are purely Galois-theoretic. This prompts the question whether for fixed $X$, $i$ and $j$ there exists a Galois-theoretic criterion on extensions $M|K$ that ensures the finiteness of  $H^i_{\textup{ét}}(\overline{X}, \mathbf{Q}/\mathbf{Z}(j))^{G_M}$. Unfortunately, this is not the case:

\begin{prop}\label{prop:solvable_example} There exist two infinite Galois extensions $M, M'$ of $\mathbf{Q}$ with $\mathrm{Gal}(M | \mathbf{Q})\cong\mathrm{Gal}(M' | \mathbf{Q})$ solvable of class 2 such that all abelian varieties defined over subfields of $M$ have finite torsion subgroup over $M$ but some abelian varieties defined over $\mathbf{Q}$ have infinite torsion over $M'$.
\end{prop}

{After some preliminaries, Theorem \ref{thm: reduction} will be proven in Section \ref{secred}. Theorem \ref{mainthm} will be deduced from \Cref{thm: reduction} in Section \ref{secmainthm} by means of an adjustment of arguments from \cite{RoesslerSzamuely}. The last section contains the proof of Proposition \ref{prop:solvable_example}.
  }

The proof of Theorem \ref{mainthm} is described as follows. First, one uses that “the Galois group of the field $K^{\rm Kum}$ over $K$ is nilpotent of class 2,” and the proof “will proceed by a general reduction to abelian extensions from solvable extensions of finite class,” namely via Theorem \ref{thm: reduction}. After “some preliminaries,” Theorem \ref{thm: reduction} is proven in Section \ref{secred}. Then “Theorem \ref{mainthm} will be deduced from \Cref{thm: reduction} in Section \ref{secmainthm} by means of an adjustment of arguments from \cite{RoesslerSzamuely}.” The reduction theorem says that for a Galois extension $M|K$ with $\mathrm{Gal}(M|K)$ “solvable of finite class,” finiteness of $H^i_{\textup{ét}}(\overline{X}, \mathbf{Q}/\mathbf{Z}(j))^{G_M}$ follows once, for every finite $K'|K$ and every abelian subextension $F|K'$ inside $K'M$, one has vanishing of invariants on the semisimplifications: (1) $\bigl(H^i_{\textup{ét}}(\overline{X},\Q_p(j))^{\mathrm{ss}}\bigr)^{G_F}=0$ for all primes $p$, and (2) $\bigl(H^i_{\textup{ét}}(\overline{X},\mathbf{Z}/p\mathbf{Z}(j))^{\mathrm{ss}}\bigr)^{G_F}=0$ for all but finitely many $p$. (They note that assuming semisimplicity, these can be replaced by the simpler requirement that $H^i_{\textup{ét}}(\overline{X},\mathbf{Q}/\mathbf{Z}(j))^{G_F}$ is finite.)

The proof of the main theorem is described as follows. First, one uses that “the Galois group of the field $K^{\rm Kum}$ over $K$ is nilpotent of class 2,” and the proof “will proceed by a general reduction to abelian extensions from solvable extensions of finite class,” namely via the following theorem.

\begin{theorem}\label{thm: reduction}
Let $K$ and $X$ be as before, and let $M|K$ be a (possibly infinite) Galois extension contained in $\Kbar$ such that $\Gal(M|K)$ is solvable of finite class. Fix integers $i\geq 0$ and $j$. Assume that for all finite extensions $K' | K$ contained in $\Kbar$ and all subextensions
$
K' \subset F \subset K'M
$
with $F | K'$ abelian the following conditions hold:
\begin{enumerate}
    \item For all primes $p$ we have $\left( H^i_{\textup{\u00e9t}}(\overline{X}, \Q_p(j))^{\operatorname{ss}} \right)^{G_F}=0$.
    \item For all but finitely many primes $p$ we have $\left( H^i_{\textup{\u00e9t}}(\overline{X}, \Z/p\Z(j))^{\operatorname{ss}} \right)^{G_F}=0$.
\end{enumerate}
Then $H^i_{\textup{\u00e9t}}(\overline{X}, \Q/\Z(j))^{G_M}$ is finite.
\end{theorem}

After “some preliminaries,” the reduction theorem is proven later. Then the main theorem is deduced from the reduction theorem later by means of an adjustment of arguments from \cite{RoesslerSzamuely}. The reduction theorem says that for a Galois extension $M|K$ with $\mathrm{Gal}(M|K)$ “solvable of finite class,” finiteness of $H^i_{\textup{\u00e9t}}(\overline{X}, \mathbf{Q}/\mathbf{Z}(j))^{G_M}$ follows once, for every finite $K'|K$ and every abelian subextension $F|K'$ inside $K'M$, one has vanishing of invariants on the semisimplifications: (1) $\bigl(H^i_{\textup{\u00e9t}}(\overline{X},\Q_p(j))^{\mathrm{ss}}\bigr)^{G_F}=0$ for all primes $p$, and (2) $\bigl(H^i_{\textup{\u00e9t}}(\overline{X},\mathbf{Z}/p\mathbf{Z}(j))^{\mathrm{ss}}\bigr)^{G_F}=0$ for all but finitely many $p$. (They note that assuming semisimplicity, these can be replaced by the simpler requirement that $H^i_{\textup{\u00e9t}}(\overline{X},\mathbf{Q}/\mathbf{Z}(j))^{G_F}$ is finite.)

Question: Let $K$ be a number field with fixed algebraic closure $\overline K$, let $X$ be a smooth projective geometrically connected variety over $K$, and write $\overline X:=X\times_K \overline K$. For any intermediate field $M$ with $K\subset M\subset \overline K$, set $G_M:=\mathrm{Gal}(\overline K/M)$. Let $K^{\rm Kum}$ denote the maximal Kummer extension of $K$, obtained by adjoining to $K$ all roots of all elements of $K$. Which statement holds for every odd integer $i$ and every integer $j$?

The subgroup of $G_{K^{\rm Kum}}$-invariants $H^i_{\mathrm{\acute{e}t}}(\overline X, \mathbf{Q}/\mathbf{Z}(j))^{G_{K^{\rm Kum}}}$ is finite. In particular, if $A$ is an abelian variety over $K$, then the torsion subgroup of $A(K^{\rm Kum})$ is finite.

Choices:
B. The subgroup of $G_{K^{\rm Kum}}$-invariants $H^i_{\mathrm{\acute{e}t}}(\overline X, \mathbf{Q}/\mathbf{Z}(j))^{G_{K^{\rm Kum}}}$ is finite for every integer $i$ and every integer $j$. In particular, if $A$ is an abelian variety over $K$, then the torsion subgroup of $A(K^{\rm Kum})$ is finite.
C. For every odd integer $i$ and every integer $j$, the subgroup $H^i_{\mathrm{\acute{e}t}}(\overline X, \mathbf{Q}/\mathbf{Z}(j))^{G_{K^{\rm Kum}}}$ is finite.
D. For every odd integer $i$ and every integer $j$, there exists a finite extension $K'\!/K$ such that the subgroup of $G_{K'K^{\rm Kum}}$-invariants $H^i_{\mathrm{\acute{e}t}}(\overline X, \mathbf{Q}/\mathbf{Z}(j))^{G_{K'K^{\rm Kum}}}$ is finite. In particular, after a finite extension of the ground field, the torsion subgroup of an abelian variety over $K$ becomes finite over $K^{\rm Kum}$.
E. For every odd integer $i$ and every integer $j$, the subgroup of $G_{K^{\rm Kum}}$-invariants $H^i_{\mathrm{\acute{e}t}}(\overline X, \mathbf{Q}/\mathbf{Z}(j))^{G_{K^{\rm Kum}}}$ is finite provided that, for every abelian subextension $F/K$ contained in $K^{\rm Kum}$, one has $\left(H^i_{\mathrm{\acute{e}t}}(\overline X, \mathbf{Q}_p(j))\right)^{G_F}=0$ for all primes $p$ and $\left(H^i_{\mathrm{\acute{e}t}}(\overline X, \mathbf{Z}/p\mathbf{Z}(j))\right)^{G_F}=0$ for all but finitely many primes $p$.

The stem gives definitions and asks for the valid conclusion, but it does not explicitly reveal the finiteness statement or uniquely hint at choice A. There is no direct answer leakage beyond the topic itself.

The item is close to a theorem-statement recognition question: the correct option essentially states the target result verbatim. However, the alternatives introduce meaningful changes in parity, strength, and quantifiers, so it is not a pure tautological restatement.

Some reasoning is needed to distinguish the exact theorem from stronger, weaker, or mis-quantified variants, especially among A/C/D/E. Still, the task is mainly precise theorem recall/recognition rather than substantial generative mathematical reasoning.

The distractors are mathematically plausible and target common error modes: dropping the oddness restriction, weakening finiteness to mere torsion, introducing an unnecessary finite base change, and overstrengthening finiteness to vanishing. They are distinct and well-calibrated.

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


\begin{theorem}\label{isogenous-char}
Let $X$ be a compact K\"ahler manifold of dimension $n$ such that

(i)  $\pi_1(X) \cong \Ga$
admits an index $d$ subgroup $\Ga'$ such that
$$\Ga'
\cong \Pi_{g_1} \times \dots \Pi_{g_n } ,$$
where $\Pi_g$ is the fundamental group of a compact complex curve of genus $g$, and all $g_j \geq 2$,

(ii) $H^{2n}(X, \ZZ)$ has index $d$ inside the image of $H^{2n}(\Ga', \ZZ) $ in $H^{2n}(X', \ZZ)$,
$X' \ra X$ being the unramified covering associated to $\Ga'$
(i.e.,  $H^{2n}(\Ga', \ZZ) = H^{2n}(X', \ZZ)$).

Then $X$ is the blow up of a VIP  $W$, and $X$ is   isogenous to a product ($X \cong W $)
if $K_X$ is ample.

Alternatively, $X$ is a VIP if

(i), (ii bis) and (iii) hold true, where

(ii bis) the image of $H^{2n}(\Ga', \ZZ) $ inside  $H^{2n}(X', \ZZ)$ is nonzero,

(iii) $K_X^n =  \frac{1}{d} 2^n n! \prod_j (g_j-1)$.

Alternatively, $X$ is a VIP if $K_X$ is ample, and (i), (ii') hold true,
where

(ii') $H^{2n}(X, \ZZ) \cong H^{2n}(\Ga, \ZZ)$.
\end{theorem}

\begin{defin}Let $X$ be a smooth projective variety.
$X$ is said to be a {\bf Variety isogenous to a higher product}, or a VIP, see \cite{isogenous}, if
$$W \cong (C_1 \times C_2 \times \dots C_n)/G,$$
where the $C_i$'s are smooth curves of genus $g(C_i)= : g_i \ge 2$ and $G$ is a finite group acting freely on
the product variety $(C_1 \times C_2 \times \dots C_n)$.

\end{defin}

In \cite{isogenous} the following was proved in theorems 7.1 , 7.5, 7.7:

\begin{defin}Let $X$ be a smooth projective variety.
$X$ is said to be a {\bf Variety isogenous to a higher product}, or a VIP, see \cite{isogenous}, if
$$W \cong (C_1 \times C_2 \times \dots C_n)/G,$$
where the $C_i$'s are smooth curves of genus $g(C_i)= : g_i \ge 2$ and $G$ is a finite group acting freely on
the product variety $(C_1 \times C_2 \times \dots C_n)$.

In \cite{isogenous} the following was proved in theorems 7.1 , 7.5, 7.7:

(i) $\pi_1(X) \cong \Ga$
admits an index $d$ subgroup $\Ga'$ such that
$$\Ga'
\cong \Pi_{g_1} \times \dots \Pi_{g_n } ,$$
where $\Pi_g$ is the fundamental group of a compact complex curve of genus $g$, and all $g_j \geq 2$,

\begin{remark}
Conditions (ii), (ii') are verified if $X$ is aspherical, that is, $X$ is a $ K(\Ga,1)$ space.
\end{remark}

(I) condition (i) by condition (i-alg) $$\widehat{\Ga' }
\cong \widehat{\Pi}_{g_1} \times \dots \widehat{\Pi}_{g_n } ,$$
where $\widehat{\Ga}$ denotes the profinite completion of $\Ga$ ($\widehat{\Ga'} = \widehat{\pi_1(X')}$ may be called
 the algebraic fundamental group of $X'$ and be denoted $ \pi_1^{alg} (X') $),

As already observed, it suffices to prove Theorem \ref{PVIP} in the case where we assume that
 \begin{equation}\label{product}
 \widehat{\pi_1(X) } \cong \widehat{\Pi}_{g_1} \times \dots \widehat{\Pi}_{g_n }.
 \end{equation}

\begin{lemma}\label{sg}
 (a) the fundamental group $\Pi_g$ of a compact complex curve
 $C$ of genus $g \geq 2$ cannot be a nontrivial semidirect
 product $ K \rtimes H$.

Suppose on the contrary that we have such a splitting $f:\widehat{\Pi}_{g}\longrightarrow \widehat{\Pi}_{g'}$. Since $\widehat{\Pi}_{g}$ is $2g$ -generated, so is $\widehat{\Pi}^{(p)}_{g}$ and $f(\widehat{\Pi}_{g})$. It follows that $\pi_{g'}(f(\widehat{\Pi}_{g}))$ is a 2g-generated pro-$p$ group and $\pi_{g'}f$ factors through $\pi_g$. Thus the lower horizontal map splits. Since $\widehat{\Pi}^{(p)}_{g'}$ is $2g'$-generated, $\pi_{g'}(f(\widehat{\Pi}_{g}))$ has infinite index in $\widehat{\Pi}^{(p)}_{g'}$. But every subgroup of infinite index of
a pro-$p$ surface group is free pro-$p$ (this is true for Demushkin groups, see \cite[Exercise 5 on page 44]{S 65}, and a pro-$p$ surface group is a particular case of it), so $\pi_{g'}(f(\widehat{\Pi}_{g}))$ must be a free pro-$p$ group, contradicting the fact that
 $\pi_{g'}(f(\widehat{\Pi}_{g})) \cong {\widehat{\Pi}^{(p)}}_{g}$ has cohomological dimension 2. This finishes the proof.

\begin{theorem}\label{PVIP}
In Theorem \ref{isogenous-char} one obtains the same conclusions  replacing

(I)  condition (i) by condition (i-alg) $$\widehat{\Ga' }
\cong \widehat{\Pi}_{g_1} \times \dots \widehat{\Pi}_{g_n } ,$$
where  $\widehat{\Ga}$ denotes the profinite completion of $\Ga$ ($\widehat{\Ga'} = \widehat{\pi_1(X')}$ may be called
 the algebraic  fundamental group of $X'$ and be denoted $ \pi_1^{alg} (X') $),

 (II) adding  in conditions (ii), respectively (ii bis), (ii'), that $\Ga$ is residually finite.
\end{theorem}

\begin{remark}
Conditions (ii), (ii')  are verified  if $X$ is aspherical, that is, $X$ is a $ K(\Ga,1)$ space.
\end{remark}

The heart of the proof boils down to treating the case where $\Ga' = \Ga$,
showing that then $X$ is biholomorphic to a product of curves.

The aim of this short note is to give a short proof of the following result. It was  proven in  \cite{authors} as Theorem A using  assumption (I), and
the conditions: $\pi_1(X)$ is residually finite,  and
$X$ is aspherical.

\begin{theorem}\label{PVIP}
In Theorem \ref{isogenous-char} one obtains the same conclusions  replacing

(I)  condition (i) by condition (i-alg) $$\widehat{\Ga' }
\cong \widehat{\Pi}_{g_1} \times \dots \widehat{\Pi}_{g_n } ,$$
where  $\widehat{\Ga}$ denotes the profinite completion of $\Ga$ ($\widehat{\Ga'} = \widehat{\pi_1(X')}$ may be called
 the algebraic  fundamental group of $X'$ and be denoted $ \pi_1^{alg} (X') $),

 (II) adding  in conditions (ii), respectively (ii bis), (ii'), that $\Ga$ is residually finite.
\end{theorem}

It suffices then to prove Theorem \ref{PVIP} in the case where we assume that
 $$ \widehat{\pi_1 (X) } \cong {\Pi}_{g_1} \times \dots \widehat{\Pi}_{g_n }.$$

 As in \cite{isogenous} the main point is to show that there exist holomorphic maps $f_i :  X\  \ensuremath{\rightarrow} C_i$,
 where $C_i$ is a complex curve of genus $g_i$, which induce the
 i-th homomorphism $ (f_i)_*\colon  \widehat{\pi_1 (X) }   \ensuremath{\rightarrow}  \widehat{\pi_1 (C_i) } \cong \widehat{\Pi}_{g_i}$.

 This will be done in the next section, in Theorem \ref{PIP},
 and is based on earlier results about irrational pencils, contained
 in \cite{alb-gen}, \cite{gl}.

 We shall then conclude the proof of Theorem \ref{PVIP} in the final section.

 The arguments used concern properties of the fundamental
 groups $\Pi_g$,  of  their profinite completions,
 and especially non splitting properties of surjections between them.

 One such result is due to Pavel Zalesskii, and is proven in the
 appendix.

Conditions (ii), (ii') are noted to hold if $X$ is aspherical (a $K(\Ga,1)$). The ``heart of the proof'' is said to be the case $\Ga'=\Ga$, where one shows that $X$ is biholomorphic to a product of curves.

For Theorem~\ref{PVIP}, it is stated that it suffices to treat the case where
\[ \widehat{\pi_1(X)}\cong \widehat{\Pi}_{g_1}\times\cdots\times \widehat{\Pi}_{g_n}. \]
Following \cite{isogenous}, the main point is to show the existence of holomorphic maps $f_i: X\to C_i$ to curves $C_i$ of genus $g_i$, such that $(f_i)_*: \widehat{\pi_1(X)}\to \widehat{\pi_1(C_i)}\cong \widehat{\Pi}_{g_i}$ induces the $i$-th homomorphism (projection) to the $i$-th factor. This step is deferred to the next section (Theorem~\ref{PIP}) and is said to be based on earlier results on irrational pencils (\cite{alb-gen}, \cite{gl}).

After constructing these maps, the text says they ``shall then conclude the proof of Theorem~\ref{PVIP} in the final section'', using properties of surface groups $\Pi_g$, their profinite completions, and in particular ``non splitting properties of surjections between them''; one such non-splitting result (due to Pavel Zalesskii) is proved in the appendix.

Conditions (ii), (ii') are noted to hold if $X$ is aspherical (a $K(\Ga,1)$). The ``heart of the proof'' is said to be the case $\Ga'=\Ga$, where one shows that $X$ is biholomorphic to a product of curves.

We first recall the following theorem.
\begin{theorem}\label{PVIP}
In Theorem \ref{isogenous-char} one obtains the same conclusions  replacing

(I)  condition (i) by condition (i-alg) $$\widehat{\Ga' }
\cong \widehat{\Pi}_{g_1} \times \dots \widehat{\Pi}_{g_n } ,$$
where  $\widehat{\Ga}$ denotes the profinite completion of $\Ga$ ($\widehat{\Ga'} = \widehat{\pi_1(X')}$ may be called
 the algebraic  fundamental group of $X'$ and be denoted $ \pi_1^{alg} (X') $),

 (II) adding  in conditions (ii), respectively (ii bis), (ii'), that $\Ga$ is residually finite.
\end{theorem}

For this theorem, it is stated that it suffices to treat the case where
\[ \widehat{\pi_1(X)}\cong \widehat{\Pi}_{g_1}\times\cdots\times \widehat{\Pi}_{g_n}. \]
Following \cite{isogenous}, the main point is to show the existence of holomorphic maps $f_i: X\to C_i$ to curves $C_i$ of genus $g_i$, such that $(f_i)_*: \widehat{\pi_1(X)}\to \widehat{\pi_1(C_i)}\cong \widehat{\Pi}_{g_i}$ induces the $i$-th homomorphism (projection) to the $i$-th factor. This step is deferred to the next section, where the following theorem is proved.

\begin{theorem}\label{PIP}
Let $X$ be a compact K\"ahler manifold of dimension $n$ and
$$\psi :  \pi_1^{alg} (X): = \widehat{\pi_1(X) }  \ra  \widehat{\Pi}_{g}$$
be a surjective homomorphism, where $g \geq 2$.

Then there exists a holomorphic map $f: X \ra C$ with connected fibres, where $C$ is a complex curve of genus $ g_2 \geq g$,
such that  $\psi$ factors through the surjection $\widehat{\pi_1(f)}$. And $\psi = \widehat{\pi_1(f)}$ if the homomorphism
$\psi$  is maximal
with respect to such  factorizations  $ \widehat{\pi_1(X)}   \ra  \widehat{\Pi}_{g_2} \ra  \widehat{\Pi}_{g}$.\end{theorem}

This is said to be based on earlier results on irrational pencils (\cite{alb-gen}, \cite{gl}).

After constructing these maps, the text says they ``shall then conclude the proof'' in the final section, using properties of surface groups $\Pi_g$, their profinite completions, and in particular ``non splitting properties of surjections between them''; one such non-splitting result (due to Pavel Zalesskii) is proved in the appendix.

\label{isogenous-char}
Let $X$ be a compact K\"ahler manifold of dimension $n$ such that

(i)  $\pi_1(X) \cong \Ga$
admits an index $d$ subgroup $\Ga'$ such that
$$\Ga'
\cong \Pi_{g_1} \times \dots \Pi_{g_n } ,$$
where $\Pi_g$ is the fundamental group of a compact complex curve of genus $g$, and all $g_j \geq 2$,

(ii) $H^{2n}(X, \mathbb{Z})$ has index $d$ inside the image of $H^{2n}(\Ga', \mathbb{Z}) $ in $H^{2n}(X', \mathbb{Z})$,
$X' \ensuremath{\rightarrow} X$ being the unramified covering associated to $\Ga'$
(i.e.,  $H^{2n}(\Ga', \mathbb{Z}) = H^{2n}(X', \mathbb{Z})$).

Then $X$ is the blow up of a VIP  $W$, and $X$ is   isogenous to a product ($X \cong W $)
if $K_X$ is ample.

Alternatively, $X$ is a VIP if

(i), (ii bis) and (iii) hold true, where

(ii bis) the image of $H^{2n}(\Ga', \mathbb{Z}) $ inside  $H^{2n}(X', \mathbb{Z})$ is nonzero,

(iii) $K_X^n =  \frac{1}{d} 2^n n! \prod_j (g_j-1)$.

Alternatively, $X$ is a VIP if $K_X$ is ample, and (i), (ii') hold true,
where

(ii') $H^{2n}(X, \mathbb{Z}) \cong H^{2n}(\Ga, \mathbb{Z})$.,

Question:
Let $X$ be a compact K\"ahler manifold of dimension $n$, and write $\Pi_g$ for the fundamental group of a compact complex curve of genus $g$. A variety isogenous to a higher product (VIP) means a smooth projective variety of the form
$$W\cong (C_1\times \cdots \times C_n)/G,$$
where each $C_i$ is a smooth curve of genus $g(C_i)=g_i\ge 2$ and $G$ is a finite group acting freely on $C_1\times \cdots \times C_n$.

Assume that $\pi_1(X)\cong \Gamma$ admits a subgroup $\Gamma'$ of index $d$ such that
$$\Gamma'\cong \Pi_{g_1}\times \cdots \times \Pi_{g_n},$$
with all $g_j\ge 2$. Let $X'\to X$ be the unramified covering associated to $\Gamma'$.

Under these hypotheses, suppose moreover that one of the following holds:

1. $H^{2n}(X,\mathbb Z)$ has index $d$ inside the image of $H^{2n}(\Gamma',\mathbb Z)$ in $H^{2n}(X',\mathbb Z)$ (as stated, $H^{2n}(\Gamma',\mathbb Z)=H^{2n}(X',\mathbb Z)$);

2. the image of $H^{2n}(\Gamma',\mathbb Z)$ inside $H^{2n}(X',\mathbb Z)$ is nonzero, and
$$K_X^n=\frac1d\,2^n n!\prod_j (g_j-1);$$

3. $K_X$ is ample and
$$H^{2n}(X,\mathbb Z)\cong H^{2n}(\Gamma,\mathbb Z).$$

Which conclusion about $X$ is valid?

If condition 1 holds, then $X$ is the blow-up of a VIP $W$; moreover, if $K_X$ is ample, then actually $X\cong W$ (so $X$ itself is isomorphic to that quotient of a product of curves). If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP.

Choices:
B. If condition 1 holds, then $X$ is a VIP; moreover, if $K_X$ is ample, then $X\cong W$ for some VIP $W$. If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is the blow-up of a VIP. If condition 3 holds, then $X$ is a VIP.
C. If condition 1 holds and $K_X$ is ample, then $X$ is a VIP. If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP.
D. If condition 1 holds, then $X$ is the blow-up of a VIP $W$; moreover, if $K_X$ is nef, then actually $X\cong W$. If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP.
E. If condition 1 holds, then $X$ is the blow-up of a VIP $W$; moreover, if $K_X$ is ample, then actually $X\cong W$. If instead the image of $H^{2n}(\Gamma',\mathbb Z)$ inside $H^{2n}(X',\mathbb Z)$ is nonzero, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP.

distinction between blow-up conclusion in (1) and VIP conclusion in (2)/(3)

dropped the unconditional blow-up conclusion under condition 1, retaining only the ample-case consequence

positivity hypothesis strengthened in theorem as ampleness, not merely nefness

numerical equality $K_X^n=\frac1d\,2^n n!\prod_j (g_j-1)$ required in case (2)

The stem does not state the conclusion explicitly; it only lists hypotheses and asks for the correct geometric outcome. There is no direct answer leakage, though the structure strongly signals that a specific theorem is being recalled.

This is essentially a theorem-recall item: the stem gives a detailed package of hypotheses and asks for the corresponding conclusion case-by-case. It is very close to a direct restatement rather than an independently motivated problem.

Some reasoning is needed to distinguish subtle alternatives such as 'blow-up of a VIP' versus 'VIP', and when ampleness is required. However, the main task is matching theorem hypotheses to its precise conclusion, not generating a conclusion from broader principles.

The distractors are mathematically plausible and target realistic confusions: weakening the conclusion, swapping conclusions between cases, adding unnecessary ampleness assumptions, or replacing exact cohomological conditions by weaker finite-index statements.

A technically strong but theorem-centric MCQ. It avoids obvious answer leakage and has good distractors, but it is largely a precise restatement/matching exercise rather than a genuinely generative reasoning problem.


Question:
Let $X$ be a compact K\"ahler manifold of dimension $n$, and write $\Pi_g$ for the fundamental group of a compact complex curve of genus $g$. A variety isogenous to a higher product (VIP) means a smooth projective variety of the form
$$W\cong (C_1\times \cdots \times C_n)/G,$$
where each $C_i$ is a smooth curve of genus $g(C_i)=g_i\ge 2$ and $G$ is a finite group acting freely on $C_1\times \cdots \times C_n$.

Assume that $\pi_1(X)\cong \Gamma$ admits a subgroup $\Gamma'$ of index $d$ such that
$$\Gamma'\cong \Pi_{g_1}\times \cdots \times \Pi_{g_n},$$
with all $g_j\ge 2$. Let $X'\to X$ be the unramified covering associated to $\Gamma'$.

Under these hypotheses, suppose moreover that one of the following holds:

1. $H^{2n}(X,\mathbb Z)$ has index $d$ inside the image of $H^{2n}(\Gamma',\mathbb Z)$ in $H^{2n}(X',\mathbb Z)$ (as stated, $H^{2n}(\Gamma',\mathbb Z)=H^{2n}(X',\mathbb Z)$);

2. the image of $H^{2n}(\Gamma',\mathbb Z)$ inside $H^{2n}(X',\mathbb Z)$ is nonzero, and
$$K_X^n=\frac1d\,2^n n!\prod_j (g_j-1);$$

3. $K_X$ is ample and
$$H^{2n}(X,\mathbb Z)\cong H^{2n}(\Gamma,\mathbb Z).$$

Which conclusion about $X$ is valid?

Choices:
B. If condition 1 holds, then $X$ is a VIP; moreover, if $K_X$ is ample, then $X\cong W$ for some VIP $W$. If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is the blow-up of a VIP. If condition 3 holds, then $X$ is a VIP.
C. If condition 1 holds and $K_X$ is ample, then $X$ is a VIP. If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP.
D. If condition 1 holds, then $X$ is the blow-up of a VIP $W$; moreover, if $K_X$ is nef, then actually $X\cong W$. If instead conditions 2 and the displayed equality for $K_X^n$ hold, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP.
E. If condition 1 holds, then $X$ is the blow-up of a VIP $W$; moreover, if $K_X$ is ample, then actually $X\cong W$. If instead the image of $H^{2n}(\Gamma',\mathbb Z)$ inside $H^{2n}(X',\mathbb Z)$ is nonzero, then $X$ is a VIP. If condition 3 holds, then $X$ is a VIP.

The stem does not explicitly reveal the correct option. It states hypotheses and asks for the valid conclusion; the answer must be identified from the choices rather than being directly embedded in the wording.

This is essentially a theorem-recall item: the stem reproduces the assumptions in detail and asks for the corresponding conclusion. The correct choice is basically the theorem statement itself.

There is some moderate pressure to distinguish subtle variants (blow-up vs VIP, ample vs nef, presence of the numerical equality), but the task is mainly recognition/recall rather than genuine mathematical generation or synthesis.

The distractors are strong: they are close to the correct theorem, differ in mathematically meaningful ways, and reflect plausible failure modes such as weakening hypotheses, swapping conclusions, or omitting a required numerical condition.

A solid theorem-discrimination MCQ with high-quality distractors, but it is largely a direct restatement/recall question rather than a genuinely generative reasoning task.