Hecke curves and Hitchin discriminant

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


Let C be a smooth projective curve of genus g≥4 over the complex numbers and SUsC(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 SUsC(r,d), there exists a distinguished hypersurface SE consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at E, there exists a distinguished subvariety CE consisting of vectors tangent to Hecke curves in SUsC(r,d) through E. Our main result establishes that the hypersurface SE and the variety CE are dual to each other. As an application of this duality relation, we prove that any surjective morphism SUsC(r,d)→SUsC(r,d), where C' is another curve of genus g, is biregular. This confirms, for SUsC(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 SUsC(r,d).

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