量子Lefschetz原理的范畴化
A categorification of the quantum Lefschetz principle
AI总结：
针对经典量子Lefschetz公式适用条件受限的问题，本文借助导出几何给出无特殊假设的几何层面一般表述，推导拟凝聚层∞-范畴下的公式范畴化版本，还证明Manolache虚拟拉回可构造为导出拉回并用于恢复经典公式。
摘要：
量子Lefschetz公式解释了稳定映射模叠的虚拟基本类（或结构层）在从外围目标概形过渡到截面的零轨迹时的行为规律，但它仅在特殊假设（亏格$0$、截面正则性及丛的凸性）下成立。本文利用导出几何，在几何层面给出了一个去除这些假设的一般性结论。通过研究导出零轨迹的结构层，我们在拟凝聚层的$\boldsymbol{\text{∞}}$-范畴中推导出该公式的一个范畴化版本。我们还证明了Manolache虚拟拉回可构造为导出拉回，并在满足经典量子Lefschetz公式的假设条件时，利用其恢复了该经典公式。
英文摘要：
The quantum Lefschetz formula explains how virtual fundamental classes (or structure sheaves) of moduli stacks of stable maps behave when passing from an ambient target scheme to the zero locus of a section. It is only valid under special assumptions (genus $0$, regularity of the section and convexity of the bundle). In this paper, we give a general statement at the geometric level removing these assumptions, using derived geometry. Through a study of the structure sheaves of derived zero loci we deduce a categorification of the formula in the $\infty$-categories of quasi-coherent sheaves. We also prove that Manolache's virtual pullbacks can be constructed as derived pullbacks, and use them to recover the classical Quantum Lefschetz formula when its hypotheses are satisfied.
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