A G ] 2 7 N ov 2 01 7 DERIVED CATEGORIES OF K 3 SURFACES

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 CH0-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 CH0-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.

[1]  A. Marian,et al.  On the group of zero-cycles of holomorphic symplectic varieties , 2017, Épijournal de Géométrie Algébrique.

[2]  D. Huybrechts.  Motives of isogenous K3 surfaces , 2017, Commentarii Mathematici Helvetici.

[3]  Junliang Shen,et al.  $K3$ CATEGORIES, ONE-CYCLES ON CUBIC FOURFOLDS, AND THE BEAUVILLE–VOISIN FILTRATION , 2017, Journal of the Institute of Mathematics of Jussieu.

[4]  Charles Vial.  On the motive of some hyperKaehler varieties , 2014, 1406.1073.

[5]  Arend Bayer,et al.  Derived automorphism groups of K3 surfaces of Picard rank $1$ , 2013, 1310.8266.

[6]  C. Voisin.  Remarks And Questions On Coisotropic Subvarieties and 0-Cycles of Hyper-Kähler Varieties , 2015, 1501.02984.

[7]  T. Latychevskaia,et al.  Atomically resolved structural determination of graphene and its point defects via extrapolation assisted phase retrieval , 2014, 1406.1673.

[8]  Franccois Charles,et al.  Families of rational curves on holomorphic symplectic varieties and applications to zero-cycles , 2014, 1907.10970.

[9]  Ulrike Riess.  On the Chow ring of birational irreducible symplectic varieties , 2013, 1304.4404.

[10]  Richard P. Thomas,et al.  Hodge theory and derived categories of cubic fourfolds , 2012, Duke Mathematical Journal.

[11]  Kieran G. OʼGrady.  Moduli of sheaves and the Chow group of K3 surfaces , 2013 .

[12]  Charles Vial,et al.  The Fourier Transform for Certain HyperKähler Fourfolds , 2013, Memoirs of the American Mathematical Society.

[13]  D. Huybrechts,et al.  Curves and cycles on K3 surfaces , 2013, 1303.4564.

[14]  Arend Bayer,et al.  MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations , 2013, Inventiones mathematicae.

[15]  C. Voisin.  Rational equivalence of 0-cycles on $K3$ surfaces and conjectures of Huybrechts and O'Grady , 2012, 1208.0916.

[16]  K. O’Grady.  Moduli of sheaves and the Chow group of K3 surfaces , 2012, 1205.4119.

[17]  Arend Bayer,et al.  Projectivity and birational geometry of Bridgeland moduli spaces , 2012, 1203.4613.

[18]  A. Kuznetsov.  Derived Categories of Cubic Fourfolds , 2008, 0808.3351.

[19]  Emanuele Macrì,et al.  Fano varieties of cubic fourfolds containing a plane , 2009, 0909.2725.

[20]  D. Huybrechts.  Chow groups of K3 surfaces and spherical objects , 2008, 0809.2606.

[21]  D. Huybrechts,et al.  Stability conditions for generic K3 categories , 2006, Compositio Mathematica.

[22]  C. Voisin.  On the Chow ring of certain algebraic hyper-K\ , 2006, math/0602400.

[23]  A. Beauville.  Algebraic Cycles and Motives: On the Splitting of the Bloch–Beilinson Filtration , 2004, math/0403356.

[24]  C. Maclean.  Chow groups of surfaces with h2,0 ≤ 1 , 2004 .

[25]  T. Bridgeland.  Stability conditions on $K3$ surfaces , 2003, math/0307164.

[26]  T. Bridgeland.  Stability conditions on triangulated categories , 2002, math/0212237.

[27]  A. Beauville.  ON THE CHOW RING OF A K3 SURFACE , 2001, math/0109063.

[28]  K. Yoshioka.  Moduli spaces of stable sheaves on abelian surfaces , 2000, math/0009001.

[29]  M. D. Cataldo,et al.  The Chow Groups and the Motive of the Hilbert Scheme of Points on a Surface , 2000, math/0005249.

[30]  B. Hassett.  Special Cubic Fourfolds , 2000, Compositio Mathematica.

[31]  D. Huybrechts.  Compact hyperkähler manifolds: basic results , 1997, alg-geom/9705025.

[32]  D. Huybrechts.  Birational symplectic manifolds and their deformations , 1996, alg-geom/9601015.

[33]  K. O’Grady.  The weight-two hodge structure of moduli spaces of sheaves on A K3 surface , 1995, alg-geom/9510001.

[34]  S. Mukai.  Symplectic structure of the moduli space of sheaves on an abelian or K3 surface , 1984 .

[35]  A. Beauville,et al.  Variétés Kähleriennes dont la première classe de Chern est nulle , 1983 .

[36]  A. A. Rojtman.  The Torsion of the Group of 0-Cycles Modulo Rational Equivalence , 1980 .

[37]  D. Huybrechts.  Motives of isogenous K3 surfaces , 2017, Commentarii Mathematici Helvetici.

[38]  Arend Bayer,et al.  Derived automorphism groups of K3 surfaces of Picard rank $1$ , 2013, 1310.8266.

[39]  C. Voisin.  Remarks And Questions On Coisotropic Subvarieties and 0-Cycles of Hyper-Kähler Varieties , 2015, 1501.02984.

[40]  Charles Vial,et al.  The Fourier Transform for Certain HyperKähler Fourfolds , 2013, Memoirs of the American Mathematical Society.

[41]  Ulrike Riess.  On the Chow ring of birational irreducible symplectic varieties , 2013, 1304.4404.

[42]  Richard P. Thomas,et al.  Hodge theory and derived categories of cubic fourfolds , 2012, Duke Mathematical Journal.

[43]  D. Huybrechts,et al.  Curves and cycles on K3 surfaces , 2013, 1303.4564.

[44]  Arend Bayer,et al.  MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations , 2013, Inventiones mathematicae.

[45]  C. Voisin.  Rational equivalence of 0-cycles on $K3$ surfaces and conjectures of Huybrechts and O'Grady , 2012, 1208.0916.

[46]  K. O’Grady.  Moduli of sheaves and the Chow group of K3 surfaces , 2012, 1205.4119.

[47]  Arend Bayer,et al.  Projectivity and birational geometry of Bridgeland moduli spaces , 2012, 1203.4613.

[48]  A. Kuznetsov.  Derived Categories of Cubic Fourfolds , 2008, 0808.3351.

[49]  Y. Tschinkel,et al.  Cohomological and Geometric Approaches to Rationality Problems , 2010 .

[50]  Emanuele Macrì,et al.  Fano varieties of cubic fourfolds containing a plane , 2009, 0909.2725.

[51]  D. Huybrechts.  Chow groups of K3 surfaces and spherical objects , 2008, 0809.2606.

[52]  C. Voisin.  Intrinsic pseudovolume forms and K -correspondences , 2008 .

[53]  C. Voisin.  On the Chow ring of certain algebraic hyper-K\ , 2006, math/0602400.

[54]  A. Beauville.  Algebraic Cycles and Motives: On the Splitting of the Bloch–Beilinson Filtration , 2004, math/0403356.

[55]  C. Maclean.  Chow groups of surfaces with h2,0 ≤ 1 , 2004 .

[56]  T. Bridgeland.  Stability conditions on $K3$ surfaces , 2003, math/0307164.

[57]  T. Bridgeland.  Stability conditions on triangulated categories , 2002, math/0212237.

[58]  A. Beauville.  ON THE CHOW RING OF A K3 SURFACE , 2001, math/0109063.

[59]  B. Hassett.  Special Cubic Fourfolds , 2000, Compositio Mathematica.

[60]  D. Huybrechts,et al.  The geometry of moduli spaces of sheaves , 1997 .

[61]  A. Beauville,et al.  Variétés Kähleriennes dont la première classe de Chern est nulle , 1983 .

[62]  A. A. Rojtman.  The Torsion of the Group of 0-Cycles Modulo Rational Equivalence , 1980 .