2 00 2 Opers and Theta Functions
نویسنده
چکیده
We construct maps from moduli spaces of vector bundles on a Riemann surface X to opers on X, using nonabelian theta functions. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle and projective connection, and map to opers via generalized Hitchin maps. For vector bundles off the theta divisor, the Szegö kernel gives a natural construction of matrix oper. The determinant of the Szegö kernel thus defines maps from SLn–bundles off the theta divisor to the affine spaces of opers. These maps are naturally described as quotients of the theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides explicit coordinates on the moduli space. The finiteness results give low–dimensional parametrizations of Jacobians (in P for generic curves), described by 2Θ functions or second logarithmic derivatives of theta.
منابع مشابه
N ov 2 00 2 OPERS AND THETA FUNCTIONS
We construct natural maps (the Klein and Wirtinger maps) from moduli spaces of vector bundles on an algebraic curve X to affine spaces, as quotients of the nonabelian theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides canonical explicit...
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