Inducing stability conditions

Macrì, Emanuele; Mehrotra, Sukhendu; Stellari, Paolo

doi:10.1090/S1056-3911-09-00524-4

Authors: Emanuele Macrì, Sukhendu Mehrotra and Paolo Stellari
Journal: J. Algebraic Geom. 18 (2009), 605-649
DOI: https://doi.org/10.1090/S1056-3911-09-00524-4
Published electronically: March 10, 2009
MathSciNet review: 2524593
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Abstract | References | Additional Information

Abstract: We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety, we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived category. As an application, we examine stability conditions on Kummer and Enriques surfaces and we improve the derived version of the Torelli Theorem for the latter surfaces already present in the literature. We also study the relationship between stability conditions on projective spaces and those on their canonical bundles.

References

Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225

A. Bayer, Polynomial Bridgeland stability conditions and the large volume limit, arXiv:0712.1083.

A. A. Beĭlinson, Coherent sheaves on ${\bf P}^{n}$ and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68–69 (Russian). MR 509388

Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR 1299527

A. I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25–44 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 23–42. MR 992977, DOI https://doi.org/10.1070/IM1990v034n01ABEH000583

Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. MR 2373143, DOI https://doi.org/10.4007/annals.2007.166.317

Tom Bridgeland, Stability conditions on $K3$ surfaces, Duke Math. J. 141 (2008), no. 2, 241–291. MR 2376815, DOI https://doi.org/10.1215/S0012-7094-08-14122-5

Tom Bridgeland, t-structures on some local Calabi-Yau varieties, J. Algebra 289 (2005), no. 2, 453–483. MR 2142382, DOI https://doi.org/10.1016/j.jalgebra.2005.03.016

T. Bridgeland, Stability conditions and Kleinian singularities, arXiv:math.AG/ 0508257.

Tom Bridgeland, Stability conditions on a non-compact Calabi-Yau threefold, Comm. Math. Phys. 266 (2006), no. 3, 715–733. MR 2238896, DOI https://doi.org/10.1007/s00220-006-0048-7

T. Bridgeland, Spaces of stability conditions, arXiv:math/0611510.

Tom Bridgeland, Alastair King, and Miles Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535–554. MR 1824990, DOI https://doi.org/10.1090/S0894-0347-01-00368-X

Tom Bridgeland and Antony Maciocia, Complex surfaces with equivalent derived categories, Math. Z. 236 (2001), no. 4, 677–697. MR 1827500, DOI https://doi.org/10.1007/PL00004847

Michael R. Douglas, Dirichlet branes, homological mirror symmetry, and stability, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 395–408. MR 1957548

Werner Geigle and Helmut Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 265–297. MR 915180, DOI https://doi.org/10.1007/BFb0078849

A. L. Gorodentsev and A. N. Rudakov, Exceptional vector bundles on projective spaces, Duke Math. J. 54 (1987), no. 1, 115–130. MR 885779, DOI https://doi.org/10.1215/S0012-7094-87-05409-3

Eiji Horikawa, On the periods of Enriques surfaces. I, Math. Ann. 234 (1978), no. 1, 73–88. MR 491725, DOI https://doi.org/10.1007/BF01409342

Eiji Horikawa, On the periods of Enriques surfaces. II, Math. Ann. 235 (1978), no. 3, 217–246. MR 491726, DOI https://doi.org/10.1007/BF01420123

Shinobu Hosono, Bong H. Lian, Keiji Oguiso, and Shing-Tung Yau, Autoequivalences of derived category of a $K3$ surface and monodromy transformations, J. Algebraic Geom. 13 (2004), no. 3, 513–545. MR 2047679, DOI https://doi.org/10.1090/S1056-3911-04-00364-9

D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR 2244106

Daniel Huybrechts, Emanuele Macrì, and Paolo Stellari, Stability conditions for generic $K3$ categories, Compos. Math. 144 (2008), no. 1, 134–162. MR 2388559, DOI https://doi.org/10.1112/S0010437X07003065

D. Huybrechts, E. Macrì, P. Stellari, Derived equivalences of K3 surfaces and orientation, arXiv:0710.1645.

Daniel Huybrechts and Paolo Stellari, Equivalences of twisted $K3$ surfaces, Math. Ann. 332 (2005), no. 4, 901–936. MR 2179782, DOI https://doi.org/10.1007/s00208-005-0662-2

A. Ishii, K. Ueda, H. Uehara, Stability conditions and $A_n$-singularities, math.AG/ 0609551 .

Akira Ishii and Hokuto Uehara, Autoequivalences of derived categories on the minimal resolutions of $A_n$-singularities on surfaces, J. Differential Geom. 71 (2005), no. 3, 385–435. MR 2198807

Dominic Joyce, Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds, Geom. Topol. 11 (2007), 667–725. MR 2302500, DOI https://doi.org/10.2140/gt.2007.11.667

Hiroshige Kajiura, Kyoji Saito, and Atsushi Takahashi, Matrix factorization and representations of quivers. II. Type $ADE$ case, Adv. Math. 211 (2007), no. 1, 327–362. MR 2313537, DOI https://doi.org/10.1016/j.aim.2006.08.005

Jong Hae Keum, Every algebraic Kummer surface is the $K3$-cover of an Enriques surface, Nagoya Math. J. 118 (1990), 99–110. MR 1060704, DOI https://doi.org/10.1017/S0027763000003019

M. Kontsevich, Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.

E. Macrì, Some examples of spaces of stability conditions on derived categories, math.AG/0411613.

E. Macrì, Stability conditions on derived categories, Ph.D. Thesis, SISSA (2006).

Emanuele Macrì, Stability conditions on curves, Math. Res. Lett. 14 (2007), no. 4, 657–672. MR 2335991, DOI https://doi.org/10.4310/MRL.2007.v14.n4.a10

E. Macrì, S. Mehrotra, Remarks on stability conditions on local Calabi–Yau varieties, in preparation.

D. R. Morrison, On $K3$ surfaces with large Picard number, Invent. Math. 75 (1984), no. 1, 105–121. MR 728142, DOI https://doi.org/10.1007/BF01403093

S. Mukai, On the moduli space of bundles on $K3$ surfaces. I, Vector bundles on algebraic varieties (Bombay, 1984) Tata Inst. Fund. Res. Stud. Math., vol. 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 341–413. MR 893604

V.V. Nikulin, Finite automorphism groups of Kähler K3 surfaces, Trans. Moscow Math. Soc. 38 (1980), 71–135.

So Okada, Stability manifold of ${\Bbb P}^1$, J. Algebraic Geom. 15 (2006), no. 3, 487–505. MR 2219846, DOI https://doi.org/10.1090/S1056-3911-06-00432-2

So Okada, On stability manifolds of Calabi-Yau surfaces, Int. Math. Res. Not. , posted on (2006), Art. ID 58743, 16. MR 2276354, DOI https://doi.org/10.1155/IMRN/2006/58743

D. O. Orlov, Equivalences of derived categories and $K3$ surfaces, J. Math. Sci. (New York) 84 (1997), no. 5, 1361–1381. Algebraic geometry, 7. MR 1465519, DOI https://doi.org/10.1007/BF02399195

R. Pandharipande, R. Thomas, Curve counting via stable pairs in the derived category, arXiv:0707.2348.

R. Pandharipande, R. Thomas, Stable pairs and BPS invariants, arXiv:0711.3899.

D. Ploog, Group of autoequivalences of derived categories of smooth projective varieties, Ph.D. Thesis, FU-Berlin (2004).

David Ploog, Equivariant autoequivalences for finite group actions, Adv. Math. 216 (2007), no. 1, 62–74. MR 2353249, DOI https://doi.org/10.1016/j.aim.2007.05.002

A. Polishchuk, Constant families of $t$-structures on derived categories of coherent sheaves, Mosc. Math. J. 7 (2007), no. 1, 109–134, 167 (English, with English and Russian summaries). MR 2324559, DOI https://doi.org/10.17323/1609-4514-2007-7-1-109-134

Paul Seidel and Richard Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1, 37–108. MR 1831820, DOI https://doi.org/10.1215/S0012-7094-01-10812-0

R. P. Thomas, Stability conditions and the braid group, Comm. Anal. Geom. 14 (2006), no. 1, 135–161. MR 2230573

Yukinobu Toda, Stability conditions and crepant small resolutions, Trans. Amer. Math. Soc. 360 (2008), no. 11, 6149–6178. MR 2425708, DOI https://doi.org/10.1090/S0002-9947-08-04509-1

Yukinobu Toda, Stability conditions and Calabi-Yau fibrations, J. Algebraic Geom. 18 (2009), no. 1, 101–133. MR 2448280, DOI https://doi.org/10.1090/S1056-3911-08-00511-0

Y. Toda, Limit stable objects on Calabi–Yau 3-folds, arXiv:0803.2356 .

Y. Toda, Generating functions of stable pair invariants via wall-crossings in derived categories, arXiv:0806.0062.

K. Ueda, Homological Mirror Symmetry and Simple Elliptic Singularities, math.AG/ 0604361.

Additional Information

Emanuele Macrì
Affiliation: Hausdorff Center for Mathematics, Mathematisches Institut, Universität Bonn, Beringstr. 1, 53115 Bonn, Germany
Address at time of publication: Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112-0090
Email: macri@math.uni-bonn.de {\rm and\ }macri@mpim-bonn.mpg.de, macri@math.utah.edu

Sukhendu Mehrotra
Affiliation: Department of Mathematics and Statistics, University of Massachusetts- Amherst, 710 N. Pleasant Street, Amherst, Massachusetts 010003
Email: mehrotra@math.umass.edu

Paolo Stellari
Affiliation: Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy
Email: paolo.stellari@unimi.it

Received by editor(s): April 29, 2007
Received by editor(s) in revised form: April 17, 2008
Published electronically: March 10, 2009