Hecke curves and Hitchin discriminant

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