Projectivity and birational geometry of Bridgeland moduli spaces

Bayer, Arend^1, 2 ; Macrì, Emanuele³

¹ Department of Mathematics, University of Connecticut U-3009, 196 Auditorium Road, Storrs, Connecticut 06269-3009
² School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, Scotland EH9 3JZ, United Kingdom
³ Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, Ohio 43210-1174

Journal of the American Mathematical Society, Tome 27 (2014) no. 3, pp. 707-752

Résumé

We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wall-crossing for Bridgeland-stability and the minimal model program for the moduli space. We give three applications of our method for classical moduli spaces of sheaves on a K3 surface. 1. We obtain a region in the ample cone in the moduli space of Gieseker-stable sheaves only depending on the lattice of the K3. 2. We determine the nef cone of the Hilbert scheme of $n$ points on a K3 surface of Picard rank one when $n$ is large compared to the genus. 3. We verify the “Hassett-Tschinkel/Huybrechts/Sawon” conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases.

Détail

DOI : 10.1090/S0894-0347-2014-00790-6

Bibliographie

[1] Abramovich, Dan, Polishchuk, Alexander Sheaves of 𝑡-structures and valuative criteria for stable complexes J. Reine Angew. Math. 2006 89 130

[2] Arcara, Daniele, Bertram, Aaron Reider’s theorem and Thaddeus pairs revisited 2011 51 68

[3] Arcara, Daniele, Bertram, Aaron Bridgeland-stable moduli spaces for 𝐾-trivial surfaces J. Eur. Math. Soc. (JEMS) 2013 1 38

[4] Arcara, Daniele, Bertram, Aaron, Coskun, Izzet, Huizenga, Jack The minimal model program for the Hilbert scheme of points on ℙ² and Bridgeland stability Adv. Math. 2013 580 626

[5] Bayer, Arend Polynomial Bridgeland stability conditions and the large volume limit Geom. Topol. 2009 2389 2425

[6] Bayer, Arend, Macrã¬, Emanuele The space of stability conditions on the local projective plane Duke Math. J. 2011 263 322

[7] Bayer, Arend, Macrã¬, Emanuele MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones Lagrangian fibrations Invent. Math. To appear

[8] Bayer, Arend, Macrã¬, Emanuele, Toda, Yukinobu Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities J. Algebraic Geom. 2014 117 163

[9] Beauville, Arnaud Holomorphic symplectic geometry: a problem list 2011 49 63

[10] Bondal, A, Orlov, D Semiorthogonal decomposition for algebraic varieties 1995

[11] Bridgeland, Tom Equivalences of triangulated categories and Fourier-Mukai transforms Bull. London Math. Soc. 1999 25 34

[12] Bridgeland, Tom Flops and derived categories Invent. Math. 2002 613 632

[13] Bridgeland, Tom Stability conditions on triangulated categories Ann. of Math. (2) 2007 317 345

[14] Bridgeland, Tom Stability conditions on 𝐾3 surfaces Duke Math. J. 2008 241 291

[15] Caldararu, Andrei Horia Derived categories of twisted sheaves on Calabi-Yau manifolds 2000 196

[16] CäƒLdäƒRaru, Andrei Nonfine moduli spaces of sheaves on 𝐾3 surfaces Int. Math. Res. Not. 2002 1027 1056

[17] Canonaco, Alberto, Stellari, Paolo Twisted Fourier-Mukai functors Adv. Math. 2007 484 503

[18] Ciliberto, Ciro, Knutsen, Andreas Leopold On 𝑘-gonal loci in Severi varieties on general 𝐾3 surfaces and rational curves on hyperkähler manifolds J. Math. Pure Appl.

[19] Craw, Alastair, Ishii, Akira Flops of 𝐺-Hilb and equivalences of derived categories by variation of GIT quotient Duke Math. J. 2004 259 307

[20] Dolgachev, Igor V., Hu, Yi Variation of geometric invariant theory quotients Inst. Hautes Études Sci. Publ. Math. 1998 5 56

[21] Donaldson, S. K. Polynomial invariants for smooth four-manifolds Topology 1990 257 315

[22] Ellingsrud, Geir, Gã¶Ttsche, Lothar Variation of moduli spaces and Donaldson invariants under change of polarization J. Reine Angew. Math. 1995 1 49

[23] Faltings, Gerd Stable 𝐺-bundles and projective connections J. Algebraic Geom. 1993 507 568

[24] Friedman, Robert, Qin, Zhenbo Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces Comm. Anal. Geom. 1995 11 83

[25] Gã¶Ttsche, Lothar, Nakajima, Hiraku, Yoshioka, K\B{O}Ta 𝐾-theoretic Donaldson invariants via instanton counting Pure Appl. Math. Q. 2009 1029 1111

[26] Gross, M., Huybrechts, D., Joyce, D. Calabi-Yau manifolds and related geometries 2003

[27] Hassett, B., Tschinkel, Y. Rational curves on holomorphic symplectic fourfolds Geom. Funct. Anal. 2001 1201 1228

[28] Hassett, Brendan, Tschinkel, Yuri Moving and ample cones of holomorphic symplectic fourfolds Geom. Funct. Anal. 2009 1065 1080

[29] Hassett, Brendan, Tschinkel, Yuri Intersection numbers of extremal rays on holomorphic symplectic varieties Asian J. Math. 2010 303 322

[30] Huybrechts, Daniel Compact hyper-Kähler manifolds: basic results Invent. Math. 1999 63 113

[31] Huybrechts, Daniel, Lehn, Manfred The geometry of moduli spaces of sheaves 2010

[32] Huybrechts, Daniel, Macrã¬, Emanuele, Stellari, Paolo Stability conditions for generic 𝐾3 categories Compos. Math. 2008 134 162

[33] Huybrechts, Daniel, Stellari, Paolo Equivalences of twisted 𝐾3 surfaces Math. Ann. 2005 901 936

[34] Inaba, Michi-Aki Toward a definition of moduli of complexes of coherent sheaves on a projective scheme J. Math. Kyoto Univ. 2002 317 329

[35] Inaba, Michi-Aki Smoothness of the moduli space of complexes of coherent sheaves on an abelian or a projective 𝐾3 surface Adv. Math. 2011 1399 1412

[36] Kaledin, D., Lehn, M., Sorger, Ch. Singular symplectic moduli spaces Invent. Math. 2006 591 614

[37] Kotaro Kawatani Stability of Gieseker stable sheaves on 𝐾3 surfaces in the sense of Bridgeland and some applications 2011

[38] Kimura, Masanori, Yoshioka, K\B{O}Ta Birational maps of moduli spaces of vector bundles on 𝐾3 surfaces Tokyo J. Math. 2011 473 491

[39] King, A. D. Moduli of representations of finite-dimensional algebras Quart. J. Math. Oxford Ser. (2) 1994 515 530

[40] Kollã¡R, Jã¡Nos, Mori, Shigefumi Birational geometry of algebraic varieties 1998

[41] Kontsevich, Maxim, Soibelman, Yan Stability structures, motivic Donaldson-Thomas invariants and cluster transformations 2008

[42] Kuznetsov, A. G. Hyperplane sections and derived categories Izv. Ross. Akad. Nauk Ser. Mat. 2006 23 128

[43] Lazarsfeld, Robert Brill-Noether-Petri without degenerations J. Differential Geom. 1986 299 307

[44] Lazarsfeld, Robert Positivity in algebraic geometry. I 2004

[45] Le Potier, J. Fibré déterminant et courbes de saut sur les surfaces algébriques 1992 213 240

[46] Le Potier, Joseph Dualité étrange, sur les surfaces 2005

[47] Li, Jun Algebraic geometric interpretation of Donaldson’s polynomial invariants J. Differential Geom. 1993 417 466

[48] Li, Jun Picard groups of the moduli spaces of vector bundles over algebraic surfaces 1996 129 146

[49] Lieblich, Max Moduli of complexes on a proper morphism J. Algebraic Geom. 2006 175 206

[50] Lieblich, Max Moduli of twisted sheaves Duke Math. J. 2007 23 118

[51] Lo Jason On some moduli of complexes on 𝐾3 surfaces 2012

[52] Lo Jason, Qin Zhenbo Mini-walls for Bridgeland stability conditions on the derived category of sheaves over surfaces 2011

[53] Maciocia, Antony Computing the walls associated to Bridgeland stability conditions on projective surfaces 2012, To appear in Asian J. Math.

[54] Maciocia, Antony, Meachan, Ciaran Rank 1 Bridgeland stable moduli spaces on a principally polarized abelian surface Int. Math. Res. Not. IMRN 2013 2054 2077

[55] Marian, Alina, Oprea, Dragos Generic strange duality for 𝐾3 surfaces Duke Math. J. 2013 1463 1501

[56] Markman, Eyal On the monodromy of moduli spaces of sheaves on 𝐾3 surfaces II 2003

[57] Markman, Eyal On the monodromy of moduli spaces of sheaves on 𝐾3 surfaces J. Algebraic Geom. 2008 29 99

[58] Markman, Eyal A survey of Torelli and monodromy results for holomorphic-symplectic varieties 2011 257 322

[59] Markman, Eyal Private communication 2012

[60] Markushevich, Dimitri Rational Lagrangian fibrations on punctual Hilbert schemes of 𝐾3 surfaces Manuscripta Math. 2006 131 150

[61] Matsuki, Kenji, Wentworth, Richard Mumford-Thaddeus principle on the moduli space of vector bundles on an algebraic surface Internat. J. Math. 1997 97 148

[62] Matsushita, Daisuke On fibre space structures of a projective irreducible symplectic manifold Topology 1999 79 83

[63] Matsushita, Daisuke On fibre space structures of a projective irreducible symplectic manifold Topology 1999 79 83

[64] Meachan Ciaran Moduli of Bridgeland-stable objects 2012

[65] Minamide, Hiroki, Yanagida, Shintarou, Yoshioka, KåTa Fourier-mukai transforms and the wall-crossing behavior for Bridgeland’s stability conditions 2011

[66] Minamide, Hiroki, Yanagida, Shintarou, Yoshioka, KåTa Some moduli spaces of Bridgeland’s stability conditions 2011

[67] Mukai, S. On the moduli space of bundles on 𝐾3 surfaces. I 1987 341 413

[68] O’Grady, Kieran G. Moduli of vector-bundles on surfaces 1997 101 126

[69] O’Grady, Kieran G. Desingularized moduli spaces of sheaves on a 𝐾3 J. Reine Angew. Math. 1999 49 117

[70] Orlov, D. O. Equivalences of derived categories and 𝐾3 surfaces J. Math. Sci. (New York) 1997 1361 1381

[71] Perego, Arvid, Rapagnetta, Antonio Factoriality properties of moduli spaces of sheaves on abelian and 𝐾3 surfaces 2011

[72] Polishchuk, A. Constant families of 𝑡-structures on derived categories of coherent sheaves Mosc. Math. J. 2007

[73] Sawon, Justin Abelian fibred holomorphic symplectic manifolds Turkish J. Math. 2003 197 230

[74] Sawon, Justin Lagrangian fibrations on Hilbert schemes of points on 𝐾3 surfaces J. Algebraic Geom. 2007 477 497

[75] Seidel, Paul, Thomas, Richard Braid group actions on derived categories of coherent sheaves Duke Math. J. 2001 37 108

[76] Thaddeus, Michael Geometric invariant theory and flips J. Amer. Math. Soc. 1996 691 723

[77] Toda, Yukinobu Moduli stacks and invariants of semistable objects on 𝐾3 surfaces Adv. Math. 2008 2736 2781

[78] Toda, Yukinobu Limit stable objects on Calabi-Yau 3-folds Duke Math. J. 2009 157 208

[79] Toda, Yukinobu Introduction and open problems of Donaldson-Thomas theory 2011

[80] Verbitsky, Misha HyperKähler SYZ conjecture and semipositive line bundles Geom. Funct. Anal. 2010 1481 1493

[81] Yanagida, Shintarou, Yoshioka, K\B{O}Ta Bridgeland’s stabilities on abelian surfaces Math. Z. 2014 571 610

[82] Yoshioka, K\B{O}Ta Brill-Noether problem for sheaves on 𝐾3 surfaces Sūrikaisekikenkyūsho K\B{o}kyūroku 2001 109 124

[83] Yoshioka, K\B{O}Ta Moduli spaces of stable sheaves on abelian surfaces Math. Ann. 2001 817 884

[84] Yoshioka, K\B{O}Ta Moduli spaces of twisted sheaves on a projective variety 2006 1 30

[85] Yoshioka, K\B{O}Ta Stability and the Fourier-Mukai transform. II Compos. Math. 2009 112 142