论文标题
Orbifold Hirzebruch-riemann-Roch用于商人的DELIGNE-MUMFORD堆栈和均等的模量理论$ k3 $表面
Orbifold Hirzebruch-Riemann-Roch for quotient Deligne-Mumford stacks and equivariant moduli theory on $ K3 $ surfaces
论文作者
论文摘要
我们研究了商人Hirzebruch-Riemann-Roch（HRR）定理用于商deligne-Mumford堆栈，探索其与有限基团的表示理论的关系，并通过Orbifold Mukai配对得出了新的Orbifold HRR公式。作为第一个应用程序，我们使用此公式来计算$ g $ equivariant模量的尺寸，该尺寸是$ k3 $ surface $ x $在其符号自动形态自动形态群体的有限亚组$ g $下的尺寸。然后，我们应用Orbifold HRR公式来复制$ x $的固定点的数量，而无需使用lefschetz固定点公式。我们证明，在某些温和条件下，$ x $上的稳定滑轮的模量模量空间是不可减至的符号歧管变形，与$ x $的希尔伯特方案相当于Gieseker和Bridgeland Moduli空间之间的$ x $点，以及派生的McKay对应关系。
We study the orbifold Hirzebruch-Riemann-Roch (HRR) theorem for quotient Deligne-Mumford stacks, explore its relation with the representation theory of finite groups, and derive a new orbifold HRR formula via an orbifold Mukai pairing. As a first application, we use this formula to compute the dimensions of $ G $-equivariant moduli spaces of stable sheaves on a $ K3 $ surface $ X $ under the action of a finite subgroup $ G $ of its symplectic automorphism group. We then apply the orbifold HRR formula to reproduce the number of fixed points on $ X $ when $ G $ is cyclic without using the Lefschetz fixed point formula. We prove that under some mild conditions, equivariant moduli spaces of stable sheaves on $ X $ are irreducible symplectic manifolds deformation equivalent to Hilbert schemes of points on $ X $ via a connection between Gieseker and Bridgeland moduli spaces, as well as the derived McKay correspondence.
