Autoequivalences of derived category of a K3 surface and monodromy transformations

Authors:
Shinobu Hosono, Bong H. Lian, Keiji Oguiso and Shing-Tung Yau

Journal:
J. Algebraic Geom. **13** (2004), 513-545

DOI:
https://doi.org/10.1090/S1056-3911-04-00364-9

Published electronically:
January 15, 2004

MathSciNet review:
2047679

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Abstract |
References |
Additional Information

Abstract: We consider autoequivalences of the bounded derived category of coherent sheaves on a K3 surface. We prove that the image of the autoequivalences has index at most two in the group of the Hodge isometries of the Mukai lattice. Motivated by homological mirror symmetry, we also consider the mirror counterpart, i.e., the symplectic version of it. In the case of $\rho (X)=1$, we find an explicit formula which reproduces the number of Fourier-Mukai (FM) partners from the monodromy problem of the mirror K3 family. We present an explicit example in which a monodromy action does not come from an autoequivalence of the mirror side.

[ACHY]ACHY B. Andreas, G. Curio, D. Hernández Ruipérez and S.-T. Yau, *Fourier-Mukai transform and mirror symmetry for D-branes on elliptic Calabi-Yau*, math.AG/0012196.
[Ba1]Ba1 V.V. Batyrev, *Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties*, J. Alg. Geom. 3 (1994) 493–535.
[Ba2]Ba2 V.V. Batyrev, *Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori*, Duke Math. J. 69 (1993) 349–409.
[BO]BO A. Bondal, D. Orlov *Reconstruction of a variety from the derived category and groups of autoequivalences*, Compositio Math. 125 (2001) 327–344.
[BP]BP F. Beukers and C. Peters, *A family of K3-surfaces and $\zeta (3)$*, J. Reine Angew. Math. 351 (1984) 42–54.
[BPV]BPV W. Barth, C. Peters, A. Van de Ven, *Compact complex surfaces*, Springer-Verlag (1984).
[Br]Br T. Bridgeland, *Equivalences of triangulated categories and Fourier-Mukai transforms*, Bull. London Math. Soc. 31 (1999) 25–34.
[BM]BM T. Bridgeland, A. Maciocia, *Complex surfaces with equivalent derived categories*, Math. Z. 236 (2001) 677–697.
[CN]CN J.H. Conway and S.P. Norton, *Monstrous Moonshine*, Bull. London Math. Soc. 11 (1979) 308–339.
[CP]CP F. Campana and T. Peternell, *Algebraicity of the ample cone of projective varieties*, J. Reine Angew. Math. 407 (1990) 160–166.
[Do]Do I.V. Dolgachev, *Mirror symmetry for lattice polarized K3 surfaces*, Algebraic geometry, 4. J. Math. Sci. 81 (1996) 2599–2630.
[Don]Don S.K. Donaldson, *Polynomial invariants for smooth four-manifolds*, Topology 29 (1990) 257–315.
[Fu]Fu K. Fukaya, *Floer homology and mirror symmetry II*, Adv. Studies in Pure Math. 34, “Minimal Surfaces, Geometric Analysis and Symplectic Geometry” (2000) 1–99.
[FO3]FO3 K. Fukaya, Y.G. Oh, H. Ohta and K. Ono, *Lagrangian intersection Floer theory – anomaly and obstruction*, preprint (2000) available at http://www.kusm.kyoto-u.ac.jp/ ̃fukaya.
[GM]GM S.I. Gelfand, Y.I. Manin, *Methods of homological algebra*, Springer-Verlag, Berlin, 1991.
[GW]GW M. Gross and P.M.H. Wilson, *Mirror symmetry via $3$-tori for a class of Calabi-Yau threefolds*, Math. Ann. 309 (1997) 505-531.
[Hi]Hi E. Hille, *Ordinary differential equations in the complex domain*, Pure & Applied Math, Wiley, 1976.
[Hor]Hor P. Horja, *Hypergeometric functions and mirror symmetry in toric varieties*, math.AG/9912109.
[HLY]HLY S. Hosono, B.H. Lian, and S.-T. Yau, *GKZ-Generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces*, Commun. Math. Phys. 182 (1996) 535–577.
[HLOY]HLOY S. Hosono, B.H. Lian, K. Oguiso and S.-T. Yau, *Fourier-Mukai number of a K3 surface*, math.AG/0202014.
[Ko]Ko M. Kontsevich, *Homological algebra of mirror symmetry*, Proceedings of the International Congress of Mathematicians (Zürich, 1994) Birkhäuser (1995) pp. 120–139.
[LY1]LY1 B. Lian and S.-T. Yau, *Arithmetic properties of mirror map and quantum coupling*, Commun. Math. Phys. 176 (1996) 163–192.
[LY2]LY2 B. Lian and S.-T. Yau, *Mirror maps, modular relations and hypergeometric series I*, hep-th/9507151.
[Mo1]Mo1 D. R. Morrison, *Picard-Fuchs equations and mirror maps for hypersurfaces*, in “Essays on Mirror Manifolds” Ed. S.-T. Yau, International Press, Hong Kong (1992) 241–264.
[Mo2]Mo2 D. R. Morrison, *Geometric aspects of mirror symmetry*, “Mathematics unlimited—2001 and beyond” Springer-Verlag, Berlin (2001) 899–918.
[Mu1]Mu1 S. Mukai, *Symplectic structure of the moduli space of sheaves on an abelian K3 surface*, Invent. Math. 77 (1984) 101–116.
[Mu2]Mu2 S. Mukai, *On the moduli space of bundles on K3 surfaces I*, in: Vector bundles on algebraic varieties, Oxford Univ. Press (1987) 341–413.
[Ni]Ni V. Nikulin, *Integral symmetric bilinear forms and some of their geometric applications*, Math. USSR Izv 14 (1980) 103–167.
[Og1]Og1 K. Oguiso, *Local families of K3 surfaces and applications*, J. Alg. Geom. **12** (2003), 405–433.
[Og2]Og2 K. Oguiso, *K3 surfaces via almost-primes*, Math. Res. Lett. 9 (2002) 47–63.
[Or1]Or1 D. Orlov, *Equivalences of derived categories and K3 surfaces*, Algebraic Geometry, 7. J. Math. Sci. (New York) 84, no. 5 (1997) 1361–1381.
[Or2]Or2 D. Orlov, *On equivalences of derived categories of coherent sheaves on abelian varieties*, math.AG/9712017.
[PS]PS C. Peters and J. Stienstra, *A pencil of K3-surfaces related to Apéry’s recurrence for $\zeta (3)$ and Fermi surfaces for potential zero*, in “Arithmetics of Complex Manifolds”, Lect. Notes in Math., vol. 1399 Springer-Verlag, Berlin (1989) 110–127.
[Sc]Sc F. Scattone, *On the compactification of moduli spaces for algebraic K3 surfaces*, Mem. AMS 70 (1987) No.374.
[ST]ST P. Seidel and R. Thomas, *Braid group actions on derived categories of coherent sheaves*, Duke Math. J. 108, no. 1 (2001) 37–108.
[SYZ]SYZ A. Strominger, S.-T. Yau and E. Zaslow *Mirror symmetry is T-Duality*, Nucl. Phys. B479 (1996) 243–259.
[Sz]Sz B. Szendröi, *Diffeomorphisms and families of Fourier-Mukai transforms in mirror symmetry*, “Applications of Algebraic Geometry to Coding Theory, Physics and Computation”, NATO Science Series Kluwer (2001) 317–337.
[To]To A.N. Todorov, *Application of the Kähler-Einstein-Calabi-Yau metric to moduli of K3 surface*, Invent. Math. 61 (1980) 251–265.
[Ya]Ya S.-T. Yau, *On Calabi’s conjecture and some new results in algebraic geometry*, Proc. Nat. Acad. Sci. U.S.A. 74 (1977) 1798–1799.

[ACHY]ACHY B. Andreas, G. Curio, D. Hernández Ruipérez and S.-T. Yau, *Fourier-Mukai transform and mirror symmetry for D-branes on elliptic Calabi-Yau*, math.AG/0012196.
[Ba1]Ba1 V.V. Batyrev, *Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties*, J. Alg. Geom. 3 (1994) 493–535.
[Ba2]Ba2 V.V. Batyrev, *Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori*, Duke Math. J. 69 (1993) 349–409.
[BO]BO A. Bondal, D. Orlov *Reconstruction of a variety from the derived category and groups of autoequivalences*, Compositio Math. 125 (2001) 327–344.
[BP]BP F. Beukers and C. Peters, *A family of K3-surfaces and $\zeta (3)$*, J. Reine Angew. Math. 351 (1984) 42–54.
[BPV]BPV W. Barth, C. Peters, A. Van de Ven, *Compact complex surfaces*, Springer-Verlag (1984).
[Br]Br T. Bridgeland, *Equivalences of triangulated categories and Fourier-Mukai transforms*, Bull. London Math. Soc. 31 (1999) 25–34.
[BM]BM T. Bridgeland, A. Maciocia, *Complex surfaces with equivalent derived categories*, Math. Z. 236 (2001) 677–697.
[CN]CN J.H. Conway and S.P. Norton, *Monstrous Moonshine*, Bull. London Math. Soc. 11 (1979) 308–339.
[CP]CP F. Campana and T. Peternell, *Algebraicity of the ample cone of projective varieties*, J. Reine Angew. Math. 407 (1990) 160–166.
[Do]Do I.V. Dolgachev, *Mirror symmetry for lattice polarized K3 surfaces*, Algebraic geometry, 4. J. Math. Sci. 81 (1996) 2599–2630.
[Don]Don S.K. Donaldson, *Polynomial invariants for smooth four-manifolds*, Topology 29 (1990) 257–315.
[Fu]Fu K. Fukaya, *Floer homology and mirror symmetry II*, Adv. Studies in Pure Math. 34, “Minimal Surfaces, Geometric Analysis and Symplectic Geometry” (2000) 1–99.
[FO3]FO3 K. Fukaya, Y.G. Oh, H. Ohta and K. Ono, *Lagrangian intersection Floer theory – anomaly and obstruction*, preprint (2000) available at http://www.kusm.kyoto-u.ac.jp/ ̃fukaya.
[GM]GM S.I. Gelfand, Y.I. Manin, *Methods of homological algebra*, Springer-Verlag, Berlin, 1991.
[GW]GW M. Gross and P.M.H. Wilson, *Mirror symmetry via $3$-tori for a class of Calabi-Yau threefolds*, Math. Ann. 309 (1997) 505-531.
[Hi]Hi E. Hille, *Ordinary differential equations in the complex domain*, Pure & Applied Math, Wiley, 1976.
[Hor]Hor P. Horja, *Hypergeometric functions and mirror symmetry in toric varieties*, math.AG/9912109.
[HLY]HLY S. Hosono, B.H. Lian, and S.-T. Yau, *GKZ-Generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces*, Commun. Math. Phys. 182 (1996) 535–577.
[HLOY]HLOY S. Hosono, B.H. Lian, K. Oguiso and S.-T. Yau, *Fourier-Mukai number of a K3 surface*, math.AG/0202014.
[Ko]Ko M. Kontsevich, *Homological algebra of mirror symmetry*, Proceedings of the International Congress of Mathematicians (Zürich, 1994) Birkhäuser (1995) pp. 120–139.
[LY1]LY1 B. Lian and S.-T. Yau, *Arithmetic properties of mirror map and quantum coupling*, Commun. Math. Phys. 176 (1996) 163–192.
[LY2]LY2 B. Lian and S.-T. Yau, *Mirror maps, modular relations and hypergeometric series I*, hep-th/9507151.
[Mo1]Mo1 D. R. Morrison, *Picard-Fuchs equations and mirror maps for hypersurfaces*, in “Essays on Mirror Manifolds” Ed. S.-T. Yau, International Press, Hong Kong (1992) 241–264.
[Mo2]Mo2 D. R. Morrison, *Geometric aspects of mirror symmetry*, “Mathematics unlimited—2001 and beyond” Springer-Verlag, Berlin (2001) 899–918.
[Mu1]Mu1 S. Mukai, *Symplectic structure of the moduli space of sheaves on an abelian K3 surface*, Invent. Math. 77 (1984) 101–116.
[Mu2]Mu2 S. Mukai, *On the moduli space of bundles on K3 surfaces I*, in: Vector bundles on algebraic varieties, Oxford Univ. Press (1987) 341–413.
[Ni]Ni V. Nikulin, *Integral symmetric bilinear forms and some of their geometric applications*, Math. USSR Izv 14 (1980) 103–167.
[Og1]Og1 K. Oguiso, *Local families of K3 surfaces and applications*, J. Alg. Geom. **12** (2003), 405–433.
[Og2]Og2 K. Oguiso, *K3 surfaces via almost-primes*, Math. Res. Lett. 9 (2002) 47–63.
[Or1]Or1 D. Orlov, *Equivalences of derived categories and K3 surfaces*, Algebraic Geometry, 7. J. Math. Sci. (New York) 84, no. 5 (1997) 1361–1381.
[Or2]Or2 D. Orlov, *On equivalences of derived categories of coherent sheaves on abelian varieties*, math.AG/9712017.
[PS]PS C. Peters and J. Stienstra, *A pencil of K3-surfaces related to Apéry’s recurrence for $\zeta (3)$ and Fermi surfaces for potential zero*, in “Arithmetics of Complex Manifolds”, Lect. Notes in Math., vol. 1399 Springer-Verlag, Berlin (1989) 110–127.
[Sc]Sc F. Scattone, *On the compactification of moduli spaces for algebraic K3 surfaces*, Mem. AMS 70 (1987) No.374.
[ST]ST P. Seidel and R. Thomas, *Braid group actions on derived categories of coherent sheaves*, Duke Math. J. 108, no. 1 (2001) 37–108.
[SYZ]SYZ A. Strominger, S.-T. Yau and E. Zaslow *Mirror symmetry is T-Duality*, Nucl. Phys. B479 (1996) 243–259.
[Sz]Sz B. Szendröi, *Diffeomorphisms and families of Fourier-Mukai transforms in mirror symmetry*, “Applications of Algebraic Geometry to Coding Theory, Physics and Computation”, NATO Science Series Kluwer (2001) 317–337.
[To]To A.N. Todorov, *Application of the Kähler-Einstein-Calabi-Yau metric to moduli of K3 surface*, Invent. Math. 61 (1980) 251–265.
[Ya]Ya S.-T. Yau, *On Calabi’s conjecture and some new results in algebraic geometry*, Proc. Nat. Acad. Sci. U.S.A. 74 (1977) 1798–1799.

Additional Information

**Shinobu Hosono**

Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, Komaba Meguro-ku, Tokyo 153-8914, Japan

Email:
hosono@ms.u-tokyo.ac.jp

**Bong H. Lian**

Affiliation:
Department of Mathematics, Brandeis University, Waltham, Massachusetts 02154

Email:
lian@brandeis.edu

**Keiji Oguiso**

Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, Komaba Meguro-ku, Tokyo 153-8914, Japan

Email:
oguiso@ms.u-tokyo.ac.jp

**Shing-Tung Yau**

Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

MR Author ID:
185480

ORCID:
0000-0003-3394-2187

Email:
yau@math.harvard.edu

Received by editor(s):
February 1, 2002

Received by editor(s) in revised form:
September 18, 2002

Published electronically:
January 15, 2004