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Curvature on algebraic curves
Abel Castorena (Universidad Nacional Autónoma de México, Campus Morelia)
Aula Beltrami - Mercoledì 20 Luglio 2016 h.14:30
Abstract. Let $C$ be a curve of genus $g$, and let $J(C)$ the Jacobian of $C$. In a canonical way we can restrict any flat metric on $J(C)$ to the curve $C$
using the embedding of the Abel-Jacobi map of $C$ into $J(C)$. We want to describe some (very) elementary results on Gaussian curvature on algebraic curves,
in particular on Hyperelliptic curves and we give examples in genus two where this curvature function is a Morse function. We study the curvature of a metric
coming from any symmetric hermitian positive definite matrix on the tangent space at 0, $T_0(J(C))$ of $J(C)$. I will explain what happen when we consider
the Kahler metric coming from the Poincare Dual of the Theta Divisor of $J(C)$. For this metric we need to consider the Period matrix of C and the Riemann
Bilinear relations to explore when we have a Morse function of the corresponding metric.
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