平面正特征上Hilbert方案上的可逆顶形式
Invertible top form on the Hilbert scheme of a plane in positive characteristic
AI总结：
本文在正特征下证明了平面Hilbert方案上存在可逆顶微分形式，并利用其局部可和性推广了Harish-Chandra定理。
摘要：
我们证明了在正特征下平面的Hilbert方案上存在可逆的顶微分形式。这暗示了平面对称幂的某些可积性质。这允许定义一个在局部域上monic多项式集合上的函数，可以被视为判别式逆平方根的一种变体。在特征0的情况下，它本质上与这个逆平方根一致，但在一般情况下则有所不同，并且不同于这个逆平方根，它在局部上是可和的。在后续的工作[AGKS]中，我们利用这种局部可和性，以证明正特征下的Harish-Chandra局部可积定理的正特征版本。本文的主要结果在特征零情况下已知。实际上，已知一个更强的结果：在平面的Hilbert方案上存在辛形式。
英文摘要：
We prove that the Hilbert scheme of the plane in positive characteristic admits an invertible top differential form. This implies certain integrability properties of the symmetric powers of the plane. This allows to define a function on the collection of monic polynomials over a local field which can be thought of as a variant of the inverse square root of the discriminant. In characteristic 0 it essentially coincides with this inverse square root, however in general it is quite different, and unlike this inverse square root, it is locally summable. In a sequel work [AGKS] we use this local summability in order to prove the positive characteristic analog of Harish-Chandra's local integrability theorem of characters of representations under certain conditions. The main results of this paper are known in characteristic zero. In fact a stronger result is known: there is a symplectic form on the Hilbert scheme of a plane.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
