摆动向量丛的受限希钦映射
The restricted Hitchin map of wobbly vector bundles
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中文总结 AI 辅助
研究光滑射影曲线\(C\)上稳定秩\(2\)向量丛\(V\)的受限希钦映射\(h_V\)，通过分析一般摆动向量丛的幂零扭曲自同态，证明\(h_V\)一般有限并计算其度数，还得出在特定条件下\(h_V\)像的相关结论。
中文摘要 AI 辅助
本文研究光滑射影曲线\(C\)上稳定秩\(2\)向量丛\(V\)的受限希钦映射\(h_V\)。此映射将一个无迹扭曲自同态\(\varphi: V \to V \otimes K_C\)与其行列式（一个二次微分）相关联。我们证明若\(V\)是一般的摆动向量丛，则其至多有一个幂零扭曲自同态（相差一个标量）。由此表明\(h_V\)一般有限并计算其度数。此外，若\(C\)不是超椭圆的，则\(h_V\)的像包含一个具有简单零点的二次微分，这等价于存在与\(V\)的扭曲自同态相关的光滑谱曲线。
英文摘要
This article studies the restricted Hitchin map $h_V$ of a stable rank 2 vector bundle $V$ on a smooth projective curve $C$. This map associates to a trace-free twisted endomorphism $φ: V \to V \otimes K_C$ its determinant, which is a quadratic differential. We show that if $V$ is a general wobbly vector bundle, then it has a single nilpotent twisted endomorphism, up to scalars. As a consequence we show that $h_V$ is generically finite and we compute its degree. Furthermore, we show that if $C$ is not hyperelliptic, then the image of $h_V$ contains a quadratic differential with simple zeros. This is equivalent to saying that there is a smooth spectral curve associated to a twisted endomorphism of $V$.
