The strange duality conjecture for generic curves
The strange duality conjecture for generic curves
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DOI：
10.1090/s0894-0347-07-00569-3
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发表时间：
2006-02
影响因子：
3.9
通讯作者：
P. Belkale
中科院分区：
文献类型：
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作者：
P. Belkale
设SUX(r)是在g≥1 / c的连通光滑射影代数曲线X上秩为r且具有平凡行列式的半稳定向量束的模空间。回想一下，如果对于任意子束V， deg(V)/ rk(V)≤deg(E)/ rk(E)，则X上的向量束E称为半稳定。SUX(r)的点对应于具有平凡行列式的半稳定秩r向量束的同构类，直至等价关系。对于任意次为g−1的线束L，定义ΘL = {E∈SUX(r), h(E⊗L)≥1}。这是一个非零的Cartier除数，其相关的线束L = O（ΘL）不依赖于L。已知L产生SUX(r)的Picard群（对此以及L在上同音行列式方面的精确定义见[DN]）。设U∗X(k)是X上半稳定秩k和阶k（g−1）束的模空间。回想一下，在U∗X(k)上存在一个正则非零的（Cartier）因子Θk，其基础集为{F∈U∗X(k), h(X，F) = 0}。M = 0 （Θk）。考虑由张量积给出的自然映射τk，r: SUX(r)× U∗X(k)→U∗X（kr）。由平方定理可知τ∗k，rM同构于lm。正则元素Θkr∈H0（U∗X(kr)，M）和Kunneth定理给出了一个定义良好的映射，直到标量：
Let SUX(r) be the moduli space of semi-stable vector bundles of rank r with trivial determinant over a connected smooth projective algebraic curve X of genus g ≥ 1 over C. Recall that a vector bundle E on X is called semi-stable if for any subbundle V , deg(V )/ rk(V ) ≤ deg(E)/ rk(E). Points of SUX(r) correspond to isomorphism classes of semi-stable rank r vector bundles with trivial determinant up to an equivalence relation. For any line bundle L of degree g−1 onX define ΘL = {E ∈ SUX(r), h(E⊗L) ≥ 1}. This turns out be a non-zero Cartier divisor whose associated line bundle L = O(ΘL) does not depend upon L. It is known that L generates the Picard group of SUX(r) (for this and the precise definition of L in terms of determinant of cohomology see [DN]). Let U∗ X(k) be the moduli space of semi-stable rank k and degree k(g−1) bundles on X. Recall that on U∗ X(k) there is a canonical non-zero theta (Cartier) divisor Θk whose underlying set is {F ∈ U∗ X(k), h(X,F ) = 0}. Put M = O(Θk). Consider the natural map τk,r : SUX(r)× U∗ X(k) → U∗ X(kr) given by tensor product. From the theorem of the square, it follows that τ∗ k,rM is isomorphic to L M. The canonical element Θkr ∈ H0(U∗ X(kr),M) and the Kunneth theorem give a map well defined up to scalars: