30 problems

Let a point of the caustic be fixed, and consider the moduli space of gradient lines and the moduli space of pseudoholomorphic discs associated with the Lagrangian fibration near t…

Let $X$ be an irreducible holomorphic symplectic manifold of dimension $2g$, and let $q_X$ denote its Beauville–Bogomolov form. A line bundle $L$ on $X$ is called non-trivial if it…

Lagrangian-fibration conjecture. The morphism $f$ admits a multiplicative decomposition theorem

A primitive symplectic variety is a normal compact Kähler variety with symplectic singularities whose smooth locus carries a symplectic form and whose reflexive powers of the canon…

Lagrangian-fibration base conjecture. The base $B$ is isomorphic to a quotient of

Hyperkähler SYZ conjecture. 1. $X$ has a Lagrangian fibration if and only if it admits a non-trivial isotropic nef line bundle $L$. 2. $X$ admits a rational Lagrangian fibration if…

Let $\pi:\overline{J}_C\to B$ be a Lagrangian compactified Jacobian fibration as in the motivic Beauville decomposition conjecture, with motivic summands $h_{2i}(\overline{J}_C/B)$…

Let $X^{2n}$ be a hyperkähler manifold, and let $f:X\to B$ be a holomorphic Lagrangian fibration, where $B$ is its base. Base projective-space conjecture. If $X$ is a hyperkähler m…

Let $Y$ be a hyper-Kähler variety, let $q$ be its Beauville–Bogomolov quadratic form on $H^2(Y)$, and let $l$ be a divisor class on $Y$. The class $l$ is potentially Lagrangian if…

Let $\pi\colon M\to B$ be the Lagrangian fibration under consideration, and let … be the associated complexes of coherent $\mathscr{O}_B$-modules, where $i$ records the cohomologic…

Let $X$ be a projective irreducible holomorphic symplectic (IHS) manifold and let $D$ be an integral divisor on $X$ that is isotropic with respect to the Beauville–Bogomolov–Fujiki…

Let $Y$ be a symplectic Fano manifold with smooth anticanonical divisor $D$, such that $Y\setminus D$ supports a Lagrangian torus fibration collapsing an $S^1$ along $D$. Let $L$ b…

Hyper-Kähler SYZ deformation conjecture. Any hyper-Kähler manifold can be deformed into a hyper-Kähler manifold with a Lagrangian fibration.

Let \mathcal{X}\xymatrix@1@=15pt{\ar[r]&}\Delta be a projective minimal divisorial log terminal type III degeneration of hyperkähler manifolds, let $\mathcal{X}_t$ be a smooth fi…

Let $X$ be a hyperkähler variety of dimension $4$ admitting a Lagrangian fibration, and let $A$ be a general fibre. Write $A^i(X):=CH^i(X)_{\mathbb{Q}}$ for the Chow groups with ra…

Common cycle conjecture. The class $e'\cap e”$ is primitive in both $H_{1}(e',\mathbb{Z})$ and $H_{1}(e”,\mathbb{Z})$, and is monodromy invariant with respect to both $\pi'$ and…

Let $M$ be a compact hyperkähler manifold admitting a holomorphic Lagrangian torus fibration $\ell:M\to B$, and let $\beta$ be the pullback of an ample divisor on $B$. Let $M_1$ be…

Let $X\rightarrow\mathbb{P}^g$ be a Debarre system and let $\widehat{X}$ be its dual fibration, with fibres related by dual exact sequences of Jacobians and abelian varieties. The…

Let $S'$ be a K3 surface that is a branched double cover of a degree-one del Pezzo surface $T'$, and let $D'$ be a genus-two curve on $T'$ covered by a genus-five curve $C'$ on…

Let $\mathcal{C}\rightarrow\mathbb{P}^n$ be a family of reduced and irreducible curves of arithmetic genus $n$. Suppose that the relative compactified Jacobian … is an irreducible h…

Rational Lagrangian fibration conjecture. One has

Let $X$ be a compact hyperkähler manifold, and let $L$ be a nef line bundle. Let $q$ denote the Beauville–Bogomolov–Fujiki form; $L$ is parabolic when $q(L,L)=0$. SYZ conjecture. A…

Let $X$ be a hyperkähler manifold, and let $L$ be a nontrivial nef line bundle on $X$. Denote by $q(L)$ its Beauville–Bogomolov square. Lagrangian conjecture. If … then some power…

Matsushita's conjecture. The fibration has either maximal variation or minimal variation.

Finiteness conjecture for compactified torsors. Up to deformation, there are finitely many compactified torsors $\pi:X\rightarrow\mathbb{P}^n$ associated to the fixed Lagrangian fi…