arXiv subjects

Xiaolei Zhao

Publications and source records attributed to Xiaolei Zhao.

26 records · Page 2Linked to original sources

arXiv2020

Elliptic quintics on cubic fourfolds, O'Grady 10, and Lagrangian fibrations

For a smooth cubic fourfold Y, we study the moduli space M of semistable objects of Mukai vector $2λ_1+2λ_2$ in the Kuznetsov component of Y. We show that with a certain choice of stability conditions, M admits a symplectic resolution $\tilde M$, which is a smooth projective hyperkähler manifold, deformation equivalent to the 10-dimensional examples constructed by O'Grady. As applications, we show that a birational model of $\tilde M$ provides a hyperkähler compactification of the twisted family of intermediate Jacobians associated to Y. This generalizes the previous result of Voisin arXiv:1611.06679 in the very general case. We also prove that $\tilde M$ is the MRC quotient of the main component of the Hilbert scheme of elliptic quintic curves in Y, confirming a conjecture of Castravet.

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arXiv2020

On the group of zero-cycles of holomorphic symplectic varieties

For a moduli space of Bridgeland-stable objects on a K3 surface, we show that the Chow class of a point is determined by the Chern class of the corresponding object on the surface. This establishes a conjecture of Junliang Shen, Qizheng Yin, and the second author.

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arXiv2019

Period integrals and Hodge modules

We define a map $\mathcal{P}_M$ attached to any polarized Hodge module $M$ such that the restriction of $\mathcal{P}_M$ to a locus on which $M$ is a variation of Hodge structures induces the usual period integral pairing for this variation of Hodge structures. In the case that $M$ is the minimal extension of a simple polarized variation of Hodge structures $V$, we show that the homotopy image of $\mathcal{P}_M$ is the minimal extension of the graph morphism of the usual period integral map for $V$.

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arXiv2018

Nef cones of Hilbert schemes of points on surfaces

Let X be a smooth projective surface of irregularity 0. The Hilbert scheme of n points on X parameterizes zero-dimensional subschemes of X of length n. In this paper, we discuss general methods for studying the cone of ample divisors on the Hilbert scheme. We then use these techniques to compute the cone of ample divisors on the Hilbert scheme for several surfaces where the cone was previously unknown. Our examples include families of surfaces of general type and del Pezzo surfaces of degree 1. The methods rely on Bridgeland stability and the Positivity Lemma of Bayer and Macri.

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arXiv2017

Derived categories of K3 surfaces, O'Grady's filtration, and zero-cycles on holomorphic symplectic varieties

Moduli spaces of stable objects in the derived category of a $K3$ surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on holomorphic symplectic varieties which arise as moduli spaces. First, we show that the second Chern class of any object in the derived category lies in a suitable piece of O'Grady's filtration on the $\mathrm{CH}_0$-group of the $K3$ surface. This solves a conjecture of O'Grady and improves on previous results of Huybrechts, O'Grady, and Voisin. Then we propose a candidate of the Beauville-Voisin filtration on the $\mathrm{CH}_0$-group of the moduli space of stable objects. We discuss its connection with Voisin's recent proposal via constant cycle subvarieties. In particular, we deduce the existence of algebraic coisotropic subvarieties in the moduli space. Further, for a generic cubic fourfold containing a plane, we establish a connection between zero-cycles on the Fano variety of lines and on the associated $K3$ surface.

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arXiv2017

Bridgeland Stability Conditions on Fano Threefolds

We show the existence of Bridgeland stability conditions on all Fano threefolds, by proving a modified version of a conjecture by Bayer, Toda, and the second author. The key technical ingredient is a strong Bogomolov inequality, proved recently by Chunyi Li. Additionally, we prove the original conjecture for some toric threefolds by using the toric Frobenius morphism.

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arXiv2016

Birational models of moduli spaces of coherent sheaves on the projective plane

We study the birational geometry of moduli spaces of semistable sheaves on the projective plane via Bridgeland stability conditions. We show that the entire MMP of their moduli spaces can be run via wall-crossing. Via a description of the walls, we give a numerical description of their movable cones, along with its chamber decomposition corresponding to minimal models. As an application, we show that for primitive vectors, all birational models corresponding to open chambers in the movable cone are smooth and irreducible.

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arXiv2013

The MMP for deformations of Hilbert schemes of points on the projective plane

We study the birational geometry of deformations of Hilbert schemes of points on the projective plane P2. We complete the picture in the paper of Daniele Arcara, Aaron Bertram, Izzet Coskun and Jack Huizenga by giving an explicit correspondence between the stable base locus walls on the Neron-Severi space and the actual walls on the Bridgeland stability space. We show that the birational geometry of a deformed Hilb P2 is different from that of Hilb P2.

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