Hecke曲线与Hitchin判别式
Hecke curves and Hitchin discriminant
摘要：
设$C$是复数域上亏格为$g\geq 4$的光滑射影曲线，${\cal SU}^s_C(r,d)$是秩为$r$且具有固定行列式次数的稳定向量丛的模空间$d$ .在投影余切空间中，在${\cal SU}^s_C(r,d)$的一般点$E$处，存在一个由具有奇异谱曲线的余切向量组成的特殊超曲面${\cal S}_E$.在$E$上的射影切空间中，存在一个由${\cal SU}^s_C(r,d)$到$E$中Hecke曲线的切向量组成的特殊子簇${\cal C}_E$.我们的主要结果证明了超曲面${\cal S}_E$和簇${\cal C}_E$是对偶的.作为这个对偶关系的应用，我们证明了任何满射态射${\cal SU}^s_C(r,d) \to {\cal SU}^s_{C'}(r,d)$，其中$C'$是亏格$g$的另一条曲线，都是双正则的.对于${\cal SU}^s_C(r,d)$，这证实了一般的期望，即皮卡德数1的法诺簇，除了投射空间，没有非平凡的自态射，并且皮卡德数1的法诺簇之间的态射是罕见的。对偶关系也给出了非交换Torelli定理和Kouvidakis-Pantev关于${\cal SU}^s_C(r,d)$的自同构的结果的简单证明。
英文摘要：
Let $C$ be a smooth projective curve of genus $g\geq 4$ over the complex numbers and ${\cal SU}^s_C(r,d)$ be the moduli space of stable vector bundles of rank $r$ with a fixed determinant of degree $d$. In the projectivized cotangent space at a general point $E$ of ${\cal SU}^s_C(r,d)$, there exists a distinguished hypersurface ${\cal S}_E$ consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at $E$, there exists a distinguished subvariety ${\cal C}_E$ consisting of vectors tangent to Hecke curves in ${\cal SU}^s_C(r,d)$ through $E$. Our main result establishes that the hypersurface ${\cal S}_E$ and the variety ${\cal C}_E$ are dual to each other. As an application of this duality relation, we prove that any surjective morphism ${\cal SU}^s_C(r,d) \to {\cal SU}^s_{C'}(r,d)$, where $C'$ is another curve of genus $g$, is biregular. This confirms, for ${\cal SU}^s_C(r,d)$, the general expectation that a Fano variety of Picard number 1, excepting the projective space, has no non-trivial self-morphism and that morphisms between Fano varieties of Picard number 1 are rare. The duality relation also gives simple proofs of the non-abelian Torelli theorem and the result of Kouvidakis-Pantev on the automorphisms of ${\cal SU}^s_C(r,d)$.
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