On the strange duality conjecture for abelian surfaces II

Bolognese, Barbara; Marian, Alina; Oprea, Dragos; Yoshioka, Kota

doi:10.1090/jag/685

Authors: Barbara Bolognese, Alina Marian, Dragos Oprea and Kota Yoshioka
Journal: J. Algebraic Geom. 26 (2017), 475-511
DOI: https://doi.org/10.1090/jag/685
Published electronically: November 10, 2016
MathSciNet review: 3647791
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Abstract | References | Additional Information

Abstract: In the prequel to this paper, two versions of Le Potier’s strange duality conjecture for sheaves over abelian surfaces were studied. A third version is considered here. In the current setup, the isomorphism involves moduli spaces of sheaves with fixed determinant and fixed determinant of the Fourier-Mukai transform on one side, and moduli spaces where both determinants vary, on the other side. We first establish the isomorphism in rank 1 using the representation theory of Heisenberg groups. For product abelian surfaces, the isomorphism is then shown to hold for sheaves with fiber degree $1$ via Fourier-Mukai techniques. By degeneration to product geometries, the duality is obtained generically for a large number of numerical types. Finally, it is shown in great generality that the Verlinde sheaves encoding the variation of the spaces of theta functions are locally free over moduli.

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Additional Information

Barbara Bolognese
Affiliation: Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, Massachusetts 02115
Address at time of publication: The Fields Institute for Research in Mathematical Sciences, 222 College Street, Room 402, Toronto, Ontario M5T 3J1, Canada
Email: bbologne@fields.utoronto.ca

Alina Marian
Affiliation: Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston, Massachusetts 02115
MR Author ID: 689212
Email: a.marian@neu.edu

Dragos Oprea
Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093-0112
MR Author ID: 734182
Email: doprea@math.ucsd.edu

Kota Yoshioka
Affiliation: Department of Mathematics, Faculty of Science, Kobe University, Kobe 657-8501, Japan
MR Author ID: 352192
Email: yoshioka@math.kobe-u.ac.jp

Received by editor(s): November 4, 2014
Published electronically: November 10, 2016
Additional Notes: The second author was supported by NSF grant DMS 1303389 and a Sloan Foundation Fellowship. The third author was supported by NSF grants DMS 1001486, DMS 1150675 and a Sloan Foundation Fellowship. The fourth author was supported by Grant-in-Aid for Scientific Research 22340010, JSPS
The authors gratefully acknowledge correspondence with Emanuele Macrì
Article copyright: © Copyright 2016 University Press, Inc.