# MathSciDoc: An Archive for Mathematician ∫

#### Algebraic Geometrymathscidoc:1912.43647

arXiv preprint math/0401367, 2004.1
In [LL-Y1, III: Sec. 5.4] on mirror principle, a method was developed to compute the integral \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d for a flag manifold $X=\Fl_ {r_1,..., r_I}({\Bbb C}^ n)$ via an extended mirror principle diagram. This method turns the required localization computation on the augmented moduli stack $\bar {\cal M} _ {0, 0}(\CP^ 1\times X)$ of stable maps to a localization computation on a hyper-Quot-scheme $\HQuot ({\cal E}^ n)$. In this article, the detail of this localization computation on $\HQuot ({\cal E}^ n)$ is carried out. The necessary ingredients in the computation, notably, the \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d -fixed-point components and the distinguished ones \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d in $\HQuot ({\cal E}^ n)$, the \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d -equivariant Euler class of \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d in $\HQuot ({\cal E}^ n)$, and a push-forward formula of cohomology classes involved in the problem from the total space of a restrictive flag manifold bundle to its base manifold are given. With these, an exact expression of \int_ {X} ^{st} e^{H\cdot t}\cap {\mathbf 1} _d is obtained. Comments on the Hori-Vafa conjecture are given in the end.
@inproceedings{chien-hao2004s^1-fixed-points,