（Stacky）Hirzebruch曲面上框架层的模空间的Poincare多项式
Poincare polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces
摘要：
利用局部化技术研究了Hirzebruch曲面上框架无挠层的模空间。在Huybrechts-Lehn框架模理论的框架下，讨论了这种模空间的一些一般性质.我们在复曲面作用下对模空间上的不动点进行分类，并利用它来计算后者的Poincare多项式。这意味着我们所考虑的模空间是不可约的。我们还考虑分数第一陈类，这意味着我们将计算扩展到Hirzebruch表面的堆叠变形。从物理的观点，我们的结果提供了N=4的Vafa-Witten理论在P1上线丛的全空间上的分拆函数，这与Ooguri，Strominger和Vafa猜想的黑洞熵计数问题有关.
英文摘要：
We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localization techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincare polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hirzebruch surface. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on total spaces of line bundles on P1, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa.
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