Hecke curves and Hitchin discriminant

By Jun-Muk Hwang, S. Ramanan. 2003

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Abstract Let C be a smooth projective curve of genus g ⩾ 4 over the complex numbers and SU C s ( 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 C s ( 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 C s ( 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 C s ( r , d ) → SU C ′ s ( r , d ) , where C ′ is another curve of genus g, is biregular. This confirms, for SU C s ( 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 C s ( r , d ) .

Subfields: Geometry and Topology, History and Philosophy of Science, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7d533f03a5310d8948565bd6255413da7e6c9b65