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Let $p\colon X\to B$ be a Lagrangian fibration on an irreducible holomorphic symplectic manifold over $\mathbb C$. Assume that the base $B$ is smooth, hence $\mathbb P^n$. Define the relative tangent sheaf $T_{X/B}$ as the kernel of the map $T_X \to p^*T_B$. The holomorphic symplectic form induces an injective morphism $\Omega_B \to p_*T_{X/B}$. This morphism need not be an iso in general; yet, it is in many cases.

Observation. Assume that a general singular fiber of $p$ has a smooth point. Then the map $\Omega_B \to p_* T_{X/B}$ is an isomorphism. In particular, the sheaf $p_*T_{X/B}$ is locally free.

The proof will appear in the revised version of my preprint [AR21], let me know if it is already written somewhere! Here is a sketch. The map $p^*\Omega_B \to T_{X/B}$ is an isomorphism on the set $X'$ of points of $X$ where $p$ is smooth. We can conclude that the map $\Omega_B \to p_*T_{X/B}$ is an iso on the image $B'$ of $X'$. By the assumption of the theorem, $B'$ is the complement to a subset of codimension at least two. Both sheaves are reflexive, hence the map in question is an iso over $B$.

The fact that $p_*T_{X/B}$ is locally free gives evidence that the following should be true.

Question 1. Consider a Lagrangian fibration satisfying the condition of the observation above. Is there a smooth commutative group scheme $P\to B$ acting on $X$ such that each connected component of $X'_b$ is a torsor over $P_b$? Here $X'_b$ and $P_b$ are the fibers over $b\in B$ of the morphisms $X'\to B$ and $P\to B$.

The scheme $P$ should be obtained as follows. Let $B^0\subset B$ be the image of smooth fibers, $X^0:=p^{-1}(B^0)$ and $Aut^0_{X^0/B^0}$ the connected component of unity of the relative automorphism scheme. Then $P$ should be the closure of $Aut^0_{X^0/B^0}$ in $Aut_{X/B}$.

Dima Arinkin and Roman Fedorov gave an affirmative answer to the question when all fibers are integral [AF16]. Another piece of evidence is the following theorem of Yasunari Nagai [Na05]. He constructed a smooth commutative group scheme $P^\vee\to B$ which is a partial compactification of the group scheme $Pic^0(X^0/B^0)$. The scheme $P^\vee$ should be thought of as dual to $P$. Nagai assumes that $B' = B$; yet, Thorsten Beckmann and Daniel Huybrechts remark in [BH21, Remark 5.4] that the assumption in the observation above is enough (if I understood their remark correctly).

Bibliography

[AR21] Anna Abasheva, Vasily Rogov. Shafarevich-Tate groups of holomorphic Lagrangian fibrations. arXiv:2112.10921.

[AF16] Dima Arinkin, Roman Fedorov. Partial Fourier–Mukai transform for integrable systems with applications to Hitchin fibration. Duke Math. J. 165 (2016), 2991–3042.

[BH21] Thorsten Beckmann, Daniel Huybrechts. Dual Lagrangian fibrations. pdf

[Na05] Yasunari Nagai. Dual fibration of a projective Lagrangian fibration. PhD thesis. pdf

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