论文标题
虚拟Segre和Verlinde的投影表面数
Virtual Segre and Verlinde numbers of projective surfaces
论文作者
论文摘要
最近，Marian-Oprea-Pandharipande建立了（对与希尔伯特（Hilbert）方案相关的Segre数字的概括）的猜想。扩展了约翰逊的工作，他们提供了Segre和Verlinde数字之间的猜想。对于具有全体形态2形式的表面，我们提出了其结果的猜想概括，以归于任何等级的稳定滑轮的模量空间。 使用Mochizuki的公式，我们得出了一个通用函数，该函数以Seiberg-witten不变性词和Hilbert Hilbert of Points of Points of Points of seiberg-witten不变性的形式表达虚拟Segre和Verlinde数量的表面数。我们证明，某些规范的虚拟Segre和Verlinde数量的一般类型表面是拓扑不变的，我们在示例中验证了我们的猜想。 我们的猜想中的功率序列是代数函数，为此我们在几种情况下找到表达式，并且在某些GALOIS作用下被排列。我们的猜想暗示了Mariño-Moore对更高等级唐纳森不变的猜想的代数类似物。对于$ 3 $和$ 4 $的排名，我们就Seiberg-witten不变性人士获得了唐纳森不变的明确表达方式。
Recently, Marian-Oprea-Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of any rank. Using Mochizuki's formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg-Witten invariants and intersection numbers on products of Hilbert schemes of points. We prove that certain canonical virtual Segre and Verlinde numbers of general type surfaces are topological invariants and we verify our conjectures in examples. The power series in our conjectures are algebraic functions, for which we find expressions in several cases and which are permuted under certain Galois actions. Our conjectures imply an algebraic analog of the Mariño-Moore conjecture for higher rank Donaldson invariants. For ranks $3$ and $4$, we obtain explicit expressions for Donaldson invariants in terms of Seiberg-Witten invariants.
