论文标题
gopakumar-vafa类型的全态符号符号4倍
Gopakumar-Vafa type invariants of holomorphic symplectic 4-folds
论文作者
论文摘要
使用降低的Gromov-witten理论，我们定义了新的不变式，这些不变式捕获了4倍4倍的曲线的列出几何形状。不变的人类似于gopakumar和vafa的BPS计数3倍的Calabi-yau，Klemm和Pandharipande的Calabi-yau 4倍，Pandharipande和Zinger，calabi-yau 5倍。 我们猜想我们的不变式是整数，并以减少$ 4 $维度的唐纳森 - 托马斯（Donaldson-Thomas）不变的一维稳定滑轮的折叠解释。我们检查了两个$ k3 $表面的产品以及$ \ mathbb {p}^2 $的cotangent捆绑包。 Modulo是猜想的全态异常方程，我们还计算出$ k3 $表面的两个点的希尔伯特方案。这产生了一个猜想的公式，用于在非常一般的Hyperkähler$ 4 $ fold $ k3^{[2]} $ - 类型的孤立属$ 2 $ $ 2 $最低学位的曲线。该公式可以被视为经典Yau-Zaslow公式的$ 4 $维度类似物，涉及$ K3 $表面上的理性曲线计数。 在计算过程中，我们还为$ k3 $表面和广义的kummer品种的两种点方案的Chern类别捆绑包的富士式捆绑的富士式捆绑套装和广义品种。
Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds, Klemm and Pandharipande for Calabi-Yau 4-folds, Pandharipande and Zinger for Calabi-Yau 5-folds. We conjecture that our invariants are integers and give a sheaf-theoretic interpretation in terms of reduced $4$-dimensional Donaldson-Thomas invariants of one-dimensional stable sheaves. We check our conjectures for the product of two $K3$ surfaces and for the cotangent bundle of $\mathbb{P}^2$. Modulo the conjectural holomorphic anomaly equation, we compute our invariants also for the Hilbert scheme of two points on a $K3$ surface. This yields a conjectural formula for the number of isolated genus $2$ curves of minimal degree on a very general hyperkähler $4$-fold of $K3^{[2]}$-type. The formula may be viewed as a $4$-dimensional analogue of the classical Yau-Zaslow formula concerning counts of rational curves on $K3$ surfaces. In the course of our computations, we also derive a new closed formula for the Fujiki constants of the Chern classes of tangent bundles of both Hilbert schemes of points on $K3$ surfaces and generalized Kummer varieties.
