arXiv subjects
Publications and source records attributed to Xuanchun Lu.
We introduce virtual classes in the sense of Behrend-Fantechi, explaining how these directly generalize Fulton's localized top Chern class. We derive Siebert's formula for a virtual class on any Deligne-Mumford stack with a perfect obstruction theory having a global resolution.
For a DM-type morphism of algebraic stacks $f \colon X \to Y$ with a relative perfect obstruction theory $\mathbb{E} \to \mathbb{L}_{X/Y}$ having a global resolution, we give a formula for the induced virtual pullback in terms of $\mathbb{E}$ and $f$. We apply our formula to prove functoriality properties of the virtual fundamental class in Gromov-Witten theory.
We prove a version of genus-0 mirror symmetry for toric Fano surfaces at the level of big quantum cohomology. The mirror Landau-Ginzburg model $(\check{X}, W_{\infty})$ of such a surface $X$ can be defined in terms of counts of Maslov index two tropical or holomorphic discs with bulk insertions. While these mirror LG models are not well-defined, the Jacobian ring $\operatorname{\mathsf{Jac}}(W_{\infty})$ of the superpotential $W_{\infty}$ is well-defined. We show that $\operatorname{\mathsf{Jac}}(W_{\infty})$ is isomorphic over the Novikov base to the big quantum cohomology of $X$ using tropical methods, providing a purely algebraic analogue of the cobordism argument used in works of Fukaya-Oh-Ohta-Ono.