Brill-Noether duality for moduli spaces of sheaves on K3 surfaces
Brill-Noether duality for moduli spaces of sheaves on K3 surfaces
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K3 表面滑轮模空间的布里尔-诺特对偶性
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发表时间：
1999
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通讯作者：
E. Markman
中科院分区：
文献类型：
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作者：
E. Markman
K3曲面上的层的模空间的分量由格（代数）Mukai格参数化。Mukai格的同构常常提升为分量集合的辛双有理同构。这种双有理同构的一个例子是阿贝尔-雅可比映射，它将2g-2次的K3上的g点的希尔伯特方案与可积系统联系起来：亏格g的超平面截面（曲线）的雅可比行列式的并集。主要结果如下：
1)我们构建了一个分层版本的Mukai基本变换建模后，对偶对斯普林格决议的幂零轨道。它适用于具有分层的全纯辛簇M，其中第一层是P^n丛，但下层是格拉斯曼丛。由此产生的（变换）辛簇W允许由对偶格拉斯曼丛分层。
2)Mukai格的反射群，它平凡地作用于K3曲面的第二上同调，作用于层的模空间（具有"最小“第一陈类）作为双有理分层基本变换。
3)我们得到了一个Picard-Lefschetz型公式，它用代数对应关系确定了全纯辛簇M的上同调环与其分层变换W的同构为杯积.
Components of the Moduli space of sheaves on a K3 surface are parametrized by a lattice; the (algebraic) Mukai lattice. Isometries of the Mukai lattice often lift to symplectic birational isomorphisms of the collection of components. An example of such a birational isomorphism is the Abel-Jacobi map relating the Hilbert scheme of g points on a K3 of degree 2g-2 to an integrable system: the union of Jacobians of hyperplane sections (curves) of genus g. The main results are:
1) We construct a stratified version of a Mukai elementary transformation modeled after dual pairs of Springer resolutions of nilpotent orbits. It applies to a holomorphic-symplectic variety M with a stratification where the first stratum is a P^n bundle, but lower strata are Grassmannian bundles. The resulting (transformed) symplectic variety W admits a stratification by the dual Grassmannian bundles.
2) The group of reflections of the Mukai lattice, which act trivially on the second cohomology of the K3 surface, acts on moduli spaces of sheaves (with ``minimal' first Chern class) as birational stratified elementary transformations.
3) We derive a Picard-Lefschetz type formula identifying the isomorphism of cohomology rings of a holomorphic-symplectic variety M and its stratified transform W as the cup product with an algebraic correspondence.