Filtrations on the derived category of twisted K3 surfaces (2402.13793v3)
Abstract: We introduce and study the Shen-Yin-Zhao filtration on derived categories of twisted K3 surfaces. A main contribution is the construction of a twisted Beauville-Voisin class $\mathfrak{o}{\mathscr{X}} \in \operatorname{CH}_0(X)$ that extends fundamental results of O'Grady and Shen-Yin-Zhao \cite{OG13, SYZ20} to twisted settings. This class enables: 1. A derived equivalence-invariant filtration $\mathbf{S}\bullet(\mathrm{D}{(1)}(\mathscr{X}))$ preserved under Fourier-Mukai transforms, 2. A birational invariant filtration $\mathbf{S}{\mathrm{SYZ}}_\bullet \operatorname{CH}0$ on Bridgeland moduli spaces. We prove $\mathbf{S}{\mathrm{SYZ}}\bullet \operatorname{CH}0$ coincides with Voisin's filtration $\mathbf{S}{\mathrm{BV}}\bullet\operatorname{CH}_0$ (Theorem 1.4), providing a canonical candidate for the conjectural Beauville-Voisin filtration. Applications include Bloch's conjecture for (anti)-symplectic automorphisms and existence of algebraically coisotropic subvarieties.
- Nicolas Addington. On two rationality conjectures for cubic fourfolds. Math. Res. Lett., 23(1):1–13, 2016.
- MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations. Invent. Math., 198(3):505–590, 2014.
- On the Chow ring of a K3𝐾3K3italic_K 3 surface. J. Algebraic Geom., 13(3):417–426, 2004.
- Twistor spaces for supersingular K3𝐾3K3italic_K 3 surfaces. arXiv: Algebraic Geometry, 2018.
- Tom Bridgeland. Stability conditions on K3𝐾3K3italic_K 3 surfaces. Duke Math. J., 141(2):241–291, 2008.
- Constantin Bănică. Smooth reflexive sheaves. In Proceedings of the Colloquium on Complex Analysis and the Sixth Romanian-Finnish Seminar, volume 36, pages 571–593, 1991.
- Formality conjecture for K3 surfaces. Compos. Math., 155(5):902–911, 2019.
- D. Huybrechts. Curves and cycles on K3 surfaces. Algebr. Geom., 1(1):69–106, 2014. With an appendix by C. Voisin.
- Daniel Huybrechts. Chow groups of K3 surfaces and spherical objects. J. Eur. Math. Soc. (JEMS), 12(6):1533–1551, 2010.
- Daniel Huybrechts. The K3 category of a cubic fourfold. Compos. Math., 153(3):586–620, 2017.
- The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig, 1997.
- Derived categories of projectivizations and flops. Adv. Math., 396:Paper No. 108169, 44, 2022.
- Birational motives I: Pure birational motives. Ann. K-Theory, 1(4):379–440, 2016.
- Bloch’s conjecture for (anti-)autoequivalences on K3𝐾3K3italic_K 3 surfaces, 2023.
- Beauville–Voisin Filtrations on Zero-Cycles of Moduli Space of Stable Sheaves on K3 Surfaces. International Mathematics Research Notices, 06 2022. rnac161.
- Max Lieblich. Moduli of twisted sheaves. Duke Math. J., 138(1):23–118, 2007.
- Finiteness of K3 surfaces and the Tate conjecture. Ann. Sci. Éc. Norm. Supér. (4), 47(2):285–308, 2014.
- Hsueh-Yung Lin. Lagrangian Constant Cycle Subvarieties in Lagrangian Fibrations. International Mathematics Research Notices, 2020(1):14–24, 02 2018.
- On the group of zero-cycles of holomorphic symplectic varieties. Épijournal Géom. Algébrique, 4:Art. 3, 5, 2020.
- Kieran G. O’Grady. Moduli of sheaves and the Chow group of K3𝐾3K3italic_K 3 surfaces. J. Math. Pures Appl. (9), 100(5):701–718, 2013.
- Derived categories of K3𝐾3K3italic_K 3 surfaces, O’Grady’s filtration, and zero-cycles on holomorphic symplectic varieties. Compos. Math., 156(1):179–197, 2020.
- Charles Vial. On the birational motive of hyper-Kähler varieties. J. Math. Pures Appl. (9), 163:577–624, 2022.
- Angelo Vistoli. Intersection theory on algebraic stacks and on their moduli spaces. Inventiones mathematicae, 97:613–670, 1989.
- Claire Voisin. Hodge Theory and Complex Algebraic Geometry II: Volume 2, volume 77. Cambridge University Press, 2003.
- Claire Voisin. On the Chow ring of certain algebraic hyper-Kähler manifolds. Pure Appl. Math. Q., 4(3, Special Issue: In honor of Fedor Bogomolov. Part 2):613–649, 2008.
- Claire Voisin. Rational equivalence of 0-cycles on K3𝐾3K3italic_K 3 surfaces and conjectures of Huybrechts and O’Grady. In Recent advances in algebraic geometry, volume 417 of London Math. Soc. Lecture Note Ser., pages 422–436. Cambridge Univ. Press, Cambridge, 2015.
- Kōta Yoshioka. Moduli spaces of stable sheaves on abelian surfaces. Math. Ann., 321(4):817–884, 2001.
- Kōta Yoshioka. Moduli spaces of twisted sheaves on a projective variety. In Moduli spaces and arithmetic geometry, volume 45 of Adv. Stud. Pure Math., pages 1–30. Math. Soc. Japan, Tokyo, 2006.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.