2003/09/03 by Jun-Muk Hwang, S. Ramanan, Hwang, Jun-Muk +1 · 1 citation
Mathematics · #14D20 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.math/0309056
openalex publication_date 2003/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let C be a smooth projective curve of genus g≥ 4 over the complex numbers and \cal 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 \cal SUsC(r,d), there exists a distinguished hypersurface \cal SE consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at E, there exists a distinguished subvariety \cal CE consisting of vectors tangent to Hecke curves in \cal SUsC(r,d) through E. Our main result establishes that the hypersurface \cal SE and the variety \cal CE are dual to each other. As an application of this duality relation, we prove that any surjective morphism \cal SUsC(r,d) → \cal SUsC'(r,d), where C' is another curve of genus g, is biregular. This confirms, for \cal 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 \cal SUsC(r,d).