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K3 and Enriques Surfaces Read chapter Read first chapter

2013 | OriginalPaper | Chapter

Fourier–Mukai Partners and Polarised \(\mathop{\mathrm{K3}}\nolimits\) Surfaces

Authors : K. Hulek, D. Ploog

Published in: Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Publisher: Springer New York

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Abstract

The purpose of this note is twofold. We first review the theory of Fourier–Mukai partners together with the relevant part of Nikulin’s theory of lattice embeddings via discriminants. Then we consider Fourier–Mukai partners of \(\mathop{\mathrm{K3}}\nolimits\) surfaces in the presence of polarisations, in which case we prove a counting formula for the number of partners.

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previous chapter A Structure Theorem for Fibrations on Delsarte Surfaces
next chapter On a Family of K3 Surfaces with Symmetry
Literature
1.
W.L. Baily Jr., A. Borel, Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math. (2) 84, 442–528 (1966)
2.
W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact Complex Surfaces, 2nd edn. (Springer, Berlin, 2004)MATH
3.
C. Bartocci, U. Bruzzo, D. Hernàndez-Ruipérez, Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics (Birkhäuser, Basel, 2009)MATHCrossRef
4.
A.I. Beauville, J.-P. Bourguignon, M. Demazure (eds.), Géométrie des surfaces K3: modules et périodes, in Séminaires Palaiseau. Astérisque, vol. 126. Société Mathématique de France (Paris 1985)
5.
A. Beilinson, J. Bernstein, P. Deligne, Faisceaux perverse, in Astérisque, vol. 100 Société Mathématique de France (Paris 1980)
6.
A.I. Bondal, D.O. Orlov, Semiorthogonal decompositions for algebraic varieties. MPI preprint 15 (1995), also math.AG/9506012
7.
A.I. Bondal, D.O. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences. Compos. Math. 125, 327–344 (2001)MathSciNetMATHCrossRef
8.
9.
10.
T. Bridgeland, Spaces of stability conditions, in Algebraic Geometry, Seattle, 2005. Proceedings of Symposia in Pure Mathematics, vol. 80, Part 1 (American Mathematical Society, Providence, 2009), pp. 1–21
11.
J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices, and Groups, 2nd edn. (Springer, Berlin, 1993)MATHCrossRef
12.
D.A. Cox, Primes of the Form \({x}^{2} + n{y}^{2}\) (Wiley, New York, 1989)
13.
14.
W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 3 (Springer, Berlin, 1998)CrossRef
15.
16.
S.I. Gelfand, Y.I. Manin, Methods of Homological Algebra (Springer, Berlin, 1997)
17.
V. Golyshev, V. Lunts, D.O. Orlov, Mirror symmetry for abelian varieties. J. Algebr. Geom. 10, 433–496 (2001)MathSciNetMATH
18.
19.
D. Happel, in Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. LMS Lecture Notes Series, vol. 119 (Cambridge University Press, Cambridge, 1988)
20.
H. Hartmann, Cusps of the Kähler moduli space and stability conditions on K3 surfaces Math. Ann. doi:10.1007/s00208-011-0719-3 Math. Ann. 354, 1–42 (2012)
21.
R. Hartshorne, in Residues and Duality. Lecture Notes in Mathematics, vol. 20 (Springer, Berlin, 1966)
22.
S. Hosono, B.H. Lian, K. Oguiso, S.-T. Yau, Fourier–Mukai number of a K3 surface, in Algebraic Structures and Moduli Spaces. CRM Proceedings and Lecture Notes, vol. 38 (AMS Providence, 2004), pp. 177–192
23.
D. Huybrechts, Fourier–Mukai Transforms in Algebraic Geometry (Oxford University Press, Oxford, 2006)MATHCrossRef
24.
B.W. Jones, in The Arithmetic Theory of Quadratic Forms. Carus Mathemathical Monographs, vol. 10 (Mathematical Association of America, Buffalo, New York 1950)
25.
26.
S. Ma, Fourier–Mukai partners of a K3 surface and the cusps of its Kahler moduli. Int. J. Math. 20, 727–750 (2009)MATHCrossRef
27.
28.
29.
S. Mukai, Duality between D(X) and \(D(\hat{X})\) with its application to Picard sheaves. Nagoya Math. J. 81, 153–175 (1981)MathSciNetMATH
30.
S. Mukai, On the moduli space of bundles on K3 surfaces I, in Vector Bundles on Algebraic Varieties, vol. 11 (The Tata Institute of Fundamental Research in Mathematics, Bombay, 1984), pp. 341–413
31.
32.
V. Nikulin, Integral symmetric bilinear forms and some of their applications. Math. USSR Izv. 14, 103–167 (1980)MATHCrossRef
33.
34.
D.O. Orlov, Equivalences of derived categories and K3-surfaces. J. Math. Sci. 84, 1361–1381 (1997)MATHCrossRef
35.
D.O. Orlov, Derived categories of coherent sheaves on abelian varieties and equivalences between them. Izv. Math. 66, 569–594 (2002)MathSciNetMATHCrossRef
36.
D.O. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Trudy Steklov Math. Inst. 204, 240–262 (2004)
37.
38.
A. Polishchuk, Symplectic biextensions and generalizations of the Fourier–Mukai transforms. Math. Res. Lett. 3, 813–828 (1996)MathSciNetMATH
39.
A. Rudakov et al., Helices and Vector Bundles. London Mathematical Society Lecture Note Series, vol. 148 (1990)
40.
41.
C.L. Siegel, Equivalence of quadratic forms. Am. J. Math. 63, 658–680 (1941)CrossRef
42.
P. Stellari, Some remarks about the FM-partners of K3 surfaces with small Picard numbers 1 and 2. Geom. Dedicata 108, 1–13 (2004)MathSciNetMATHCrossRef
43.
44.
45.
J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Ph.D. thesis, 1967; Asterisque, vol. 239, 1996
46.
G.L. Watson, Integral Quadratic Forms (Cambridge University Press, Cambridge, 1960)MATH
47.
C. Weibel, An Introduction to Homological Algebra (Cambrige University Press, Cambrige, 1994)MATHCrossRef
48.
Metadata
Title
Fourier–Mukai Partners and Polarised Surfaces
Authors
K. Hulek
D. Ploog
Copyright Year
2013
Publisher
Springer New York
Book
Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Print ISBN: 978-1-4614-6402-0

Electronic ISBN: 978-1-4614-6403-7

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-6403-7_11

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