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For each $d <\\eta(X)$, where $\\eta(X)$ is the gonality of $X$, the symmetric product $\\text{Sym}^d(X)$ embeds into $\\text{Pic}^d(X)$ by sending an effective divisor of degree $d$ to the corresponding holomorphic line bundle. Therefore, the restriction of the flat K\\\"ahler metric on $\\text{Pic}^d(X)$ is a K\\\"ahler metric on $\\text{Sym}^d(X)$. We investigate this K\\\"ahler metric on $\\text{Sym}^d(X)$. In particular, we estimate it's Bergman kernel. 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Morye, Indranil Biswas, Tathagata Sengupta","submitted_at":"2016-08-07T12:01:48Z","abstract_excerpt":"Let $X$ be a compact connected Riemann surface of genus $g$, with $g \\geq 2$. For each $d <\\eta(X)$, where $\\eta(X)$ is the gonality of $X$, the symmetric product $\\text{Sym}^d(X)$ embeds into $\\text{Pic}^d(X)$ by sending an effective divisor of degree $d$ to the corresponding holomorphic line bundle. Therefore, the restriction of the flat K\\\"ahler metric on $\\text{Pic}^d(X)$ is a K\\\"ahler metric on $\\text{Sym}^d(X)$. We investigate this K\\\"ahler metric on $\\text{Sym}^d(X)$. In particular, we estimate it's Bergman kernel. 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