93 problems

Let $M$ be a hyperkähler manifold of dimension $2n$ and let $f:M\longrightarrow B$ be a Lagrangian fibration, where $B$ is its normal projective base of dimension $n$. In the known…

Lagrangian conjecture. If

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…

Let $X$ be a hyperkähler manifold of complex dimension $2n$, and let $f:X\to B$ be a Lagrangian fibration, where $B$ is a normal analytic variety. Matsushita's conjecture. The base…

Let $X\to B$ and $X^\phi\to B$ be projective Lagrangian fibrations related by a Tate-Shafarevich twist, where a Tate-Shafarevich twist is obtained by modifying the transition funct…

Let $S$ be a K3 surface, let $H$ be a primitive ample divisor on $S$, and let $m\geq 2$. The linear system $|mH|$ parametrizes curves whose variation in the moduli space of curves…

Rational Lagrangian fibration conjecture. One has

Let $F$ be an irreducible holomorphic symplectic fourfold, and let $\lambda$ be a primitive divisor class of Beauville square zero on the boundary of the closure of the ample cone.…

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…

A special Lagrangian fibration is a fibration whose fibres are special Lagrangian submanifolds. Joyce's conjecture. Special Lagrangian fibrations should in general be piecewise smo…

Let $Y\subseteq ({\mathbb{C}}^\ast)^{n-1}$ be a smooth algebraic hypersurface, and define … Let…

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…

Let two Lagrangian torus bundles over a torus be equipped with their fibrations. A symplectomorphism is fiber-preserving when it maps fibers of one bundle to fibers of the other. M…

Singular-locus resemblance conjecture. The singular locus of the generalized special Lagrangian torus fibration should resemble $\tilde{\Gamma}$.

Singular-locus resemblance conjecture. The singular locus of the generalized special Lagrangian torus fibration should resemble $\tilde{\Gamma}$.

Precise SYZ mirror conjecture. For any Calabi–Yau $3$-fold $X$ with Calabi–Yau metric $\omega_g$ and holomorphic volume form $\Omega$, there exists a special Lagrangian fibration

SYZ Conjecture. The line bundle $L$ is semiample.

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

Let $X$ be an irreducible holomorphic symplectic manifold. A lagrangian fibration is a surjective morphism with connected fibers from $X$ to a positive-dimensional base. An isotrop…

Let $X$ vary in the moduli space of hyperkähler varieties. A hyperkähler variety is said to admit a Lagrangian fibration if it admits a fibration whose fibers are Lagrangian with r…

Weak hyperkähler SYZ conjecture. If $q(M)=0$ and $h^0(X,\mathcal{O}_X(M))\geq n+1$, then $M$ is semiample; that is, some positive multiple of $M$ is globally generated.

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…

Let $M$ be a hyper-Kähler manifold of dimension $2n$, and let $f\colon M \rightarrow X$ be a projective Lagrangian fibration. The projective Lagrangian fibration base conjecture. T…

SYZ conjecture. Every such $X$ admits a locally trivial deformation to a primitive symplectic variety with a Lagrangian fibration.

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