相关论文: Lagrangian subvarieties in the Chow ring of some h…

Let $X$ be a hyperk\"ahler variety admitting a Lagrangian fibration. Beauville's "splitting property" conjecture predicts that fibres of the Lagrangian fibration should have a particular behaviour in the Chow ring of $X$. We study this…

Let X be a compact hyperk\"ahler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L. We show that this is indeed the case if X is not…

Let $X$ be a hyperk\"ahler variety. Voisin has conjectured that the classes of Lagrangian constant cycle subvarieties in the Chow ring of $X$ should lie in a subring injecting into cohomology. We study this conjecture for the Fano variety…

Let X be a projective hyperk\"ahler manifold containing a Lagrangian subtorus L. We study intersections of deformations of L, defining a canonical almost holomorphic map called L-reduction, which is not birational if and only if X admits an…

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 $(I, g, \omega)$…

Let $\phi:X\rightarrow B$ be a Lagrangian fibration on a projective irreducible hyper-K\"ahler manifold of dimension $\leq8$. Let $M\in {\rm Pic}\,X$ be a line bundle whose restriction to the general fiber $X_b$ of $\phi$ is topologically…

The Beauville-Voisin conjecture for a hyperk\"ahler manifold X states that the subring of the Chow ring A^*(X) generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of X. We prove a weak…

We study the rank stratification for the differential of a Lagrangian fibration over a smooth basis. We also introduce and study the notion of Lagrangian morphism of vector bundles. As a consequence, we prove some of the vanishing, in the…

This note presents basic restrictions on the topology "general" Lagrangian surfaces of hyper-K\"ahler $4$-folds and a remark on the interaction of a Lagrangian subvariety with a Lagrangian fibration of the associated hyper-K\"ahler variety.

We investigate the Lefschetz standard conjecture for degree $2$ cohomology of hyper-K\"ahler manifolds admitting a covering by Lagrangian subvarieties. In the case of a Lagrangian fibration, we show that the Lefschetz standard conjecture is…

Given a hyperspherical $G$-variety $\mathscr X$ we consider the zero moment level $\Lambda_{\mathscr X}\subset{\mathscr X}$ of the action of a Borel subgroup $B\subset G$. We conjecture that $\Lambda_{\mathscr X}$ is Lagrangian. For the…

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian…

We investigate Beauville's conjecture on the Chow ring of irreducible symplectic varieties. For special irreducible symplectic varieties we relate it to a conjecture on the existence of rational Lagrangian fibrations, which proves…

We study a generalization of a conjecture made by Beauville on the Chow ring of hyper-K\"ahler algebraic varieties. Namely we prove in a number of cases that polynomial cohomological relations involving only CH^1(X) and the Chern classes of…

The Chow rings of hyper-K\"ahler varieties are conjectured to have a particularly rich structure. In this paper, we formulate a conjecture that combines the Beauville-Voisin conjecture regarding the subring generated by divisors and the…

Let $X$ be a hyperk\"ahler variety, and let $G$ be a group of finite order non-symplectic automorphisms of $X$. Beauville's conjectural splitting property predicts that each Chow group of $X$ should split in a finite number of pieces. The…

We show that the image of a dominant meromorphic map from an irreducible compact Calabi-Yau manifold $X$ whose general fiber is of dimension strictly between $0$ and $\dim X$ is rationally connected. Using this result, we construct for any…

Let $X$ be a hyperk\"ahler variety. Beauville has conjectured that a certain subring of the Chow ring of $X$ should inject into cohomology. This note proposes a similar conjecture for the ring of algebraic cycles on $X$ modulo algebraic…

Let $\pi:X\rightarrow\mathbb{P}^n$ be a (holomorphic) Lagrangian fibration that is very general in the moduli space of Lagrangian fibrations. We conjecture that the singular fibres in codimension one must be semistable degenerations of…

We show that if $f:X\to B$ is a Lagrangian fibration from a compact connected K\"ahler hyperk\"ahler manifold $X$ onto a projective normal variety $B$, then $f$ is locally projective. This answers a question raised by L. Kamenova and…