Hecke curves and Hitchin discriminant

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).