Kyoto Journal of Mathematics

Bundles of generalized theta functions over abelian surfaces

Dragos Oprea

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Abstract

We study the Verlinde bundles of generalized theta functions constructed from moduli spaces of sheaves over abelian surfaces. In degree 0, the splitting type of these bundles is expressed in terms of indecomposable semihomogeneous factors. Furthermore, Fourier–Mukai symmetries of the Verlinde bundles are found consistently with strange duality. Along the way, a transformation formula for the theta bundles is derived, extending a theorem of Drézet–Narasimhan from curves to abelian surfaces.

Article information

Source
Kyoto J. Math., Volume 59, Number 1 (2019), 125-166.

Dates
Received: 13 October 2016
Revised: 28 December 2016
Accepted: 28 December 2016
First available in Project Euclid: 20 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1540001287

Digital Object Identifier
doi:10.1215/21562261-2018-0004

Mathematical Reviews number (MathSciNet)
MR3934625

Zentralblatt MATH identifier
07081624

Subjects
Primary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
Secondary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 14K99: None of the above, but in this section

Keywords
moduli of sheaves Abelian surfaces generalized theta functions

Citation

Oprea, Dragos. Bundles of generalized theta functions over abelian surfaces. Kyoto J. Math. 59 (2019), no. 1, 125--166. doi:10.1215/21562261-2018-0004. https://projecteuclid.org/euclid.kjm/1540001287


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References

  • [1] C. Birkenhake and H. Lange, Complex Abelian Varieties, Grundlehren Math. Wiss. 302, Springer, Berlin, 2000.
  • [2] B. Bolognese, A. Marian, D. Oprea, and K. Yoshioka, On the strange duality conjecture for abelian surfaces, II, J. Algebraic Geom. 26 (2017), 475–511
  • [3] A. Borel and J.-P. Serre, Le théoréme de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97–136.
  • [4] T. Bridgeland and A. Maciocia, Fourier–Mukai transform for quotient varieties, J. Geom. Phys. 122 (2017), 119–127.
  • [5] J. M. Drézet and M. S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), 53–94.
  • [6] G. Ellingsrud, L. Göttsche, and M. Lehn, On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom. 10 (2001), 81–100.
  • [7] B. Hassett and Y. Tschinkel, Moving and ample cones of holomorphic symplectic fourfolds, Geom. Funct. Anal. 19 (2009), 1065–1080.
  • [8] M. Inaba, Smoothness of the moduli space of complexes of coherent sheaves on an abelian or a projective $K3$ surface, Adv. Math. 227 (2011), 1399–1412.
  • [9] J. Le Potier, “Fibré déterminant et courbes de saut sur les surfaces algébriques” in Complex Projective Geometry (Trieste, 1989/Bergen, 1989), London Math. Soc. Lecture Note Ser. 179, Cambridge Univ. Press, Cambridge, 1992, 213–240.
  • [10] J. Le Potier, Dualité étrange sur le plan projectif, lecture, Luminy, France, 1996.
  • [11] J. Li, Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. Differential Geom. 37 (1993), 417–466.
  • [12] A. Marian and D. Oprea, Sheaves on abelian surfaces and strange duality, Math. Ann. 343 (2009), 1–33.
  • [13] S. Mukai, Semi-homogeneous vector bundles on an Abelian variety, J. Math. Kyoto Univ. 18 (1978), 239–272.
  • [14] S. Mukai, Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175.
  • [15] S. Mukai, “Fourier functor and its application to the moduli of bundles on an abelian variety” in Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987, 515–550.
  • [16] M. S. Narasimhan and S. Ramanan, Generalised Prym varieties as fixed points, J. Indian Math. Soc. (N.S.) 39 (1975), 1–19.
  • [17] D. Oprea, A note on the Verlinde bundles on elliptic curves, Trans. Amer. Math. Soc. 362, no. 7 (2010), 3779–3790.
  • [18] D. Oprea, The Verlinde bundles and the semihomogeneous Wirtinger duality, J. Reine Angew. Math. 654 (2011), 181–217.
  • [19] D. Oprea, On a class of semihomogeneous vector bundles, Math. Nachr., published electronically 18 April 2018.
  • [20] A. Polishchuk, Abelian Varieties, Theta Functions and the Fourier transform, Cambridge Tracts in Math. 153, Cambridge Univ. Press, Cambridge, 2003.
  • [21] M. Popa, Verlinde bundles and generalized theta linear series, Trans. Amer. Math. Soc. 354, no. 5 (2002), 1869–1898.
  • [22] L. Scala, Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles, Duke Math. J. 150 (2009), 211–267.
  • [23] K. Yoshioka, Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), 817–884.
  • [24] K. Yoshioka, “Bridgeland’s stability and the positive cone of the moduli spaces of stable objects on an abelian surface” in Development of Moduli Theory—Kyoto 2013, Adv. Stud. Pure Math. 69, Math. Soc. Japan, Tokyo, 473–537.
  • [25] K. Yoshioka, Irreducibility of moduli spaces of vector bundles on $K3$ surfaces, preprint, arXiv:math/9907001v2 [math.AG].