相关论文: A hyper-K\"ahler compactification of the Intermedi…

The starting point of this note is our recent paper with Laza and Sacc\`a on the construction of deformations of O'Grady's $10$-dimensional manifolds as compactifications of intermediate Jacobian fibrations associated to cubic fourfolds.…

We show that the intermediate Jacobian fibration associated to any smooth cubic fourfold $X$ admits a hyper-K\"ahler compactification $J(X)$ with a regular Lagrangian fibration $J \to \mathbb P^5$. This builds upon arXiv:1602.05534, where…

We give a description of the intermediate Jacobian fibration attached to a general complex cubic fourfold $X$ containing a plane as a Lagrangian subfibration of a moduli space of torsion sheaves on the K3 surface associated to $X$ up to a…

We provide a modular construction of the Laza--Sacc\`a--Voisin compactification of the intermediate Jacobian fibration of a cubic fourfold. Additionally, we construct infinitely many $20$-dimensional families of polarized hyper-K\"ahler…

Given a cubic 4-fold $Y$, we provide an easy Hodge-theoretic proof of the following result of Iliev--Manivel: the relative intermediate Jacobian of the universal family of cubic 5-folds $Z$ extending $Y$ is a Lagrangian fibration.

We study the sections, Tate--Shafarevich twists, and the period for an OG10 hyperk\"ahler Lagrangian associated to a cubic fourfold. To do so, we introduce the analytic relative Jacobian sheaf for a Lagrangian fibration of a hyperk\"ahler…

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known that a given compact manifold admits only finitely many holomorphic symplectic structures, up to…

Given a Lagrangian fibration $\pi : X \to \mathbb{P}^n$ of a compact hyper-K\"ahler manifold of $\text{K3}^{[n]}$, $\text{Kum}_n$, $\text{OG10}$ or $\text{OG6}$-type, we construct a natural compactification of its dual torus fibration.…

In this paper we analyse the birational geometry of O'Grady ten dimensional manifolds, giving a characterisation of Kaehler classes and lagrangian fibrations. Moreover, we study symplectic compactifications of intermediate jacobian…

Let $|H|$ be a linear system on a smooth surface $S$. We study the cohomology classes of sections of the universal Jacobian over lines in $|H|$. When $S$ is a K3 surface, the universal compactified Jacobian is a hyperk\"ahler manifold, and…

Every fibration of a projective hyper-K\"ahler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We study those cases where the…

A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of…

We suggest a general framework for compactifing quasi-projective Lagrangian fibrations of geometric origin by holomorphic symplectic varieties. This framework includes a compactification criterion, which we then apply to various fibrations…

This article initiates the study of isotrivial Lagrangian fibrations of compact hyper-K\"ahler manifolds. We present four foundational results that extend well-known facts about isotrivial elliptic fibrations of K3 surfaces. First, we prove…

We establish a compact analog of the P = W conjecture. For a holomorphic symplectic variety with a Lagrangian fibration, we show that the perverse numbers associated with the fibration match perfectly with the Hodge numbers of the total…

For a smooth cubic fourfold Y, we study the moduli space M of semistable objects of Mukai vector $2\lambda_1+2\lambda_2$ in the Kuznetsov component of Y. We show that with a certain choice of stability conditions, M admits a symplectic…

In this paper we define the notion of a hyperk\"ahler manifold (potentially) of Jacobian type. If we view hyperk\"ahler manifolds as "abelian varieties", then those of Jacobian type should be viewed as "Jacobian varieties". Under a minor…

Lagrangian submanifolds in strict nearly K\"ahler 6-manifolds are related to special Lagrangian submanifolds in Calabi-Yau 6-manifolds and coassociative cones in $G_2$-manifolds. We prove that the mean curvature of a Lagrangian submanifold…

In this work we construct an analytically completely integrable Hamiltonian system which is canonically associated to any family of Calabi-Yau threefolds. The base of this system is a moduli space of gauged Calabi-Yaus in the family, and…

Let $M$ be a holomorphic symplectic K\"ahler manifold equipped with a Lagrangian fibration $\pi$ with compact fibers. The base of this manifold is equipped with a special K\"ahler structure, that is, a K\"ahler structure $(I, g, \omega)$…