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

Part of: Surfaces and higher-dimensional varieties (Co)homology theory Cycles and subschemes

Published online by Cambridge University Press:  26 November 2019

Junliang Shen ,
Qizheng Yin  and
Xiaolei Zhao
Junliang Shen
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, Simons Building, 77 Massachusetts Avenue, Cambridge, MA 02139, USA email jlshen@mit.edu
Qizheng Yin
Affiliation:
Peking University, Beijing International Center for Mathematical Research, Jingchunyuan Courtyard #78, 5 Yiheyuan Road, Haidian District, Beijing 100871, China email qizheng@math.pku.edu.cn
Xiaolei Zhao
Affiliation:
University of California, Santa Barbara, Department of Mathematics, South Hall, Santa Barbara, CA 93106, USA email xlzhao@ucsb.edu
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Abstract

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 $\text{CH}_{0}$-group of the $K3$ surface. This solves a conjecture of O’Grady and improves on previous results of Huybrechts, O’Grady, and Voisin. Second, we propose a candidate for the Beauville–Voisin filtration on the $\text{CH}_{0}$-group of the moduli space of stable objects. We discuss its connection with Voisin’s recent proposal via constant cycle subvarieties, and prove a conjecture of hers on the existence of special algebraically coisotropic subvarieties for the moduli space.

Keywords

K3 surfacesderived categoriesChow groupsmoduli spaces of sheaveshyper-Kähler varieties

MSC classification

Primary: 14J28: $K3$ surfaces and Enriques surfaces 14F05: Sheaves, derived categories of sheaves and related constructions 14C15: (Equivariant) Chow groups and rings; motives
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 1 , January 2020 , pp. 179 - 197
Copyright
© The Authors 2019 

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