论文标题
虚拟Morse-Bott索引，成对的模量空间以及对光滑四个manifolds拓扑的应用
Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds
论文作者
论文摘要
我们以前开发了一种用于holomorthic $ \ mathbb {C}^*$在复杂的分析空间上的动作的bialynicki-birula理论的方法，以及用于诱发圈子动作的汉密尔顿函数的虚拟摩尔斯 - 摩托 - 底部指数的概念（请参阅Fefehan，arxiv：2206.14710）。对于圆圈动作在封闭，复杂的Kaehler歧管上的功能，虚拟摩尔斯 - 摩特索引与Bott（1954）和Frankel（1959）的经典Morse-Bott索引相吻合。我们方法中的一个关键原则是，在哈密顿函数的临界点上，虚拟摩尔斯 - 摩特指数的阳性意味着即使临界点是模量空间中的一个单数点，临界点也不是局部最小值。在这一专着中，我们在非亚伯式单孔的模量空间上考虑了我们的方法，在封闭的，复杂的，kaehler的表面上。 We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and we prove that these indices are positive in a setting motivated by the conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type obey the Bogomolov-Miyaoka-Yau inequality.
We previously developed an approach to Bialynicki-Birula theory for holomorphic $\mathbb{C}^*$ actions on complex analytic spaces and the concept of virtual Morse-Bott indices for singular critical points of Hamiltonian functions for the induced circle actions (see Feehan, arXiv:2206.14710). For Hamiltonian functions of circle actions on closed, complex Kaehler manifolds, the virtual Morse-Bott index coincides with the classical Morse-Bott index due to Bott (1954) and Frankel (1959). A key principle in our approach is that positivity of the virtual Morse-Bott index at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum even when that critical point is a singular point in the moduli space. In this monograph, we consider our method in the context of the moduli space of non-Abelian monopoles over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and we prove that these indices are positive in a setting motivated by the conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type obey the Bogomolov-Miyaoka-Yau inequality.
