{"paper":{"title":"On the Kaehler metrics over ${mathrm{Sym}^{d}(X)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.DG","authors_text":"Anilatmaja Aryasomayajula, Archana S. 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. We also prove that any holomorphic automorphism of $\\text{Sym}^d(X)$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.02207","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}