Hwang, Jun-Muk ; Ramanan, S.
(2004)
*Hecke curves and Hitchin discriminant*
Annales Scientifiques de l'École Normale Supérieure, 37
(5).
pp. 801-817.
ISSN 0012-9593

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S00129...

Related URL: http://dx.doi.org/10.1016/j.ansens.2004.07.001

## Abstract

Let C be a smooth projective curve of genus g≥4 over the complex numbers and SU^{s}_{C}(r,d) be the moduli space of stable vector bundles of rank r with a fixed determinant of degree d. In the projectivized cotangent space at a general point E of SU^{s}_{C}(r,d), there exists a distinguished hypersurface S_{E} consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at E, there exists a distinguished subvariety C_{E} consisting of vectors tangent to Hecke curves in SU^{s}_{C}(r,d) through E. Our main result establishes that the hypersurface S_{E} and the variety C_{E} are dual to each other. As an application of this duality relation, we prove that any surjective morphism SU^{s}_{C}(r,d)→SU^{s}_{C}(r,d), where C' is another curve of genus g, is biregular. This confirms, for SU^{s}_{C}(r,d), the general expectation that a Fano variety of Picard number 1, excepting the projective space, has no non-trivial self-morphism and that morphisms between Fano varieties of Picard number 1 are rare. The duality relation also gives simple proofs of the non-abelian Torelli theorem and the result of Kouvidakis-Pantev on the automorphisms of SU^{s}_{C}(r,d).

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ID Code: | 45159 |

Deposited On: | 25 Jun 2011 07:56 |

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