Bumsig Kim
The moduli spaces of stable quasimaps unify various moduli appearing in the study of Gromov-Witten Theory. This note is a survey article on the moduli of stable quasimaps, based on joint papers with Ciocan-Fontanine and Maulik as well as the author's talk at Kinosaki Algebraic Geometry Symposium 2010.
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We construct new compactifications with good properties of moduli spaces of maps from nonsingular marked curves to a large class of GIT quotients. This generalizes from a unified perspective many particular examples considered earlier in the literature.
Using localization methods on the moduli space of stable quasimaps to Grassmannian, we give explicit formulas of $J$-function and double $J$-function of stable quasimaps for complete intersection in Grassmannian.
We analyse the local structure of moduli space of genus one stable quasimaps. Combining it with the p-fields theory developed in \cite{L}, we prove the splitting formula for the virtual cycle of stable quasimaps to complete intersections in $\PP^n$.
For an arbitrary smooth hypersurface X in a projective space, we construct its LG moduli of quasimaps with P fields. Apply Kiem-Li's cosection localization we obtain a virtual fundamental class. We show the class coincides, up to sign, with that of moduli of quasimaps to X. This generalizes Chang-Li's numerical identity to the cycle level, and from Gromov Witten invariants to quasimap invariants.
By using the infinitesimally marking point to break the loop in the localization calculation as Kim and Lho, and Zinger's explicit formulas for double $J$-functions, we obtain a formula for genus one stable quasimaps invariants when the target is a complete intersection Calabi-Yau in projective space, which gives a new proof of Kim and Lho's mirror theorem for elliptic quasimap invariants.
We study the moduli space of twisted quasimaps from a fixed smooth projective curve to a Nakajima's quiver variety and the moduli space of $δ$-stable framed twisted quiver bundles with moment map relations. We show that they carry symmetric obstruction theories and when $δ$ is large enough, they exactly coincide. These results generalize works of D.E. Diaconescu about the ADHM quiver, in the framework of the quasimap theory of I. Ciocan-Fontanine, D. Maulik and the author.
By the reduced component in a moduli space of stable quasimaps to n-dimensional projective space we mean the closure of the locus in which the domain curves are smooth. As in the moduli space of stable maps, we prove the reduced component is smooth in genus 2, degree greater or equal to 3. Then we prove the virtual fundamental cycle of the moduli space of stable quasimaps to a complete intersection X in the projective space of genus 2, degree greater or equal to 3 is explicitly expressed in terms of the fundamental cycle of the reduced component of the projective space and virtual cycles of lo…
Strictly Semistable Quasimaps on the Wall
针对射影空间目标，构建了严格半稳定对象出现的代数栈，并分析其Θ-分层变化，通过非交换局部化发展了ε-稳定拟映射不变量的K-理论墙交叉公式。
We construct new "virtually smooth" modular compactifications of spaces of maps from nonsingular curves to smooth projective toric varieties. They generalize Givental's compactifications, when the complex structure of the curve is allowed to vary and markings are included, and are the toric counterpart of the moduli spaces of stable quotients introduced by Marian, Oprea, and Pandharipande to compactify spaces of maps to Grassmannians. A brief discussion of the resulting invariants and their (conjectural) relation with Gromov-Witten theory is also included.
We prove an equivalence between the Bryan--Steinberg theory of $π$-stable pairs on $Y = \mathcal{A}_{m-1} \times \mathbb{C}$ and the theory of quasimaps to $X = \mathrm{Hilb}(\mathcal{A}_{m-1})$, in the form of an equality of K-theoretic equivariant vertices. In particular, the combinatorics of both vertices are described explicitly via box counting. Then we apply the equivalence to study the implications for sheaf-counting theories on $Y$ arising from 3d mirror symmetry for quasimaps to $X$, including the Donaldson--Thomas crepant resolution conjecture.
Wall-crossing for quasimaps to GIT stack bundles
本文定义到 GIT 栈丛的 ε-稳定拟映射，并研究 ε 变化时 ε-拟映射理论的穿墙行为。
In this paper, we prove a wall-crossing formula for $ε$-stable quasimaps to GIT quotients conjectured by Ciocan-Fontanine and Kim, for all targets in all genera, including the orbifold case. We prove that stability conditions in adjacent chambers give equivalent invariants, provided that both chambers are stable. In the case of genus-zero quasimaps with one marked point, we compute the invariants in the left-most stable chamber in terms of the small $I$-function. Using this we prove that the quasimap $J$-functions are on the Lagrangian cone of the Gromov--Witten theory. The proofs are based on…
K-moduli wall crossing for quasimaps to a projective variety
该研究发展射影簇拟映射的K-模壁跨越理论，构造多 regime 模空间，关联稳定映射、拟映射与紧化空间，用于椭圆曲面紧化间的模插值。
K-moduli of quasimaps and on quasi-projectivity of moduli of K-stable Calabi-Yau fibrations over curves
本文研究了K-稳定的Calabi-Yau纤维模空间的准射影性，通过构造具有特定log Fano条件的K-模空间，并探讨了其与准映射K-模空间的关系。
The contraction morphism between maps and quasimaps to toric varieties
本文针对光滑射影环面簇构造了从稳定映射模空间闭子叠到拟映射模空间的收缩态射，证明了 Fano 情况下的满射性，并基于基点次数概念揭示了拟映射的确定条件。
B. Kim and the first author proved a result comparing the virtual fundamental classes of the moduli spaces of stable quasimaps and stable LG-quasimaps by studying localized Chern characters for 2-periodic complexes. In this paper, we study a K-theoretic analogue of the localized Chern character map and show that for a Koszul 2-periodic complex it coincides with the cosection localized Gysin map by Y.-H. Kiem and J. Li. As an application we compare the virtual structure sheaves of the moduli space of stable quasimaps and stable LG-quasimaps.
Motivic quasimap wall-crossing for Grassmannians
研究格拉斯曼流形的动机拟映射壁穿越，通过定义对称函数环的\(\mathbb{Q}\)-代数自同构给出公式，其\(\varepsilon \to 0\)极限有特定交换，利用相对商概型几何及对称函数理论证明壁穿越公式。
Higher-genus quasimap wall-crossing via localization
本文通过局部化方法和扭曲图空间，为任意亏格下不同$ε$的$ε$-稳定拟映射模空间虚拟类关联的穿墙公式提供了新证明，并预计可推广至一般规范线性sigma模型。
For a simple algebraic group $G$ we study the space $Q$ of Quasimaps from the projective line $C$ to the flag variety of $G$. We prove that the global Intersection Cohomology of $Q$ carries a natural pure Tate Hodge structure, and compute its generating function. We define an action of the Langlands dual Lie algebra $g^L$ on this cohomology. We present a new geometric construction of the universal enveloping algebra $U(n^L_+)$ of the nilpotent subalgebra of $g^L$. It is realized in the Ext-groups of certain perverse sheaves on Quasimaps' spaces, and it is equipped with a canonical basis number…
Let $C$ be a smooth projective curve of genus 0. Let $\CB$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $α\in\BN[I]$ of positive integers one can consider the space $\CQ_α$ of algebraic maps of degree $α$ from $C$ to $\CB$. This space admits some remarkable compactifications $\CQ^D_α$ (Quasimaps), $\CQ^L_α$ (Quasiflags), $\CQ^K_α$ (Stable Maps) of $\CQ_α$ constructed by Drinfeld, Laumon and Kontsevich respectively. It has been proved that the natural map $π: \CQ^L_α\to \CQ^D_α$ is a small resolution of singularities. The aim of the present not…
A boundedness theorem for principal bundles on curves
本文提出关于曲线上的主丛的有界性定理，通过固定特征标诱导的线丛次数，证明此类主丛集合的有界性，并推导出相关准映射的有界性结论。
This is the first in a pair of papers developing a framework for the application of logarithmic structures in the study of singular curves of genus $1$. We construct a smooth and proper moduli space dominating the main component of Kontsevich's space of stable genus $1$ maps to projective space. A variation on this theme furnishes a modular interpretation for Vakil and Zinger's famous desingularization of the Kontsevich space of maps in genus $1$. Our methods also lead to smooth and proper moduli spaces of pointed genus $1$ quasimaps to projective space. Finally, we present an application to t…
Stable maps to quotient stacks with a properly stable point
本文研究具有射影好模空间的商栈上曲线映射的模空间紧化，提出扩展加权吹胀的双有理操作，将拟映射理论紧化扩展至多类映射与纤维化模空间，并在附录中模块化证明Hassett猜想。
Moduli of surfaces fibered in log Calabi-Yau pairs
本文研究容许 log Calabi-Yau 纤维化的曲面对的模空间，利用稳定约化结果给出 KSBA 紧致化边界的显式描述，并应用于双度除子、K3 曲面和椭圆曲面等情形。
On positivity of CM line bundles on the moduli space of klt good minimal models with $κ=1$
研究了具有κ=1的klt良好极小模型模空间上CM线包的正性，证明了其正则化后为 ample，并构造了数值等价类的模空间以克服先前方法的局限性。
Representations on the cohomology of $\overline{\mathcal{M}}_{0,n}$
本文利用拟映射模空间的壁交叉给出对称群作用于 $\overline{\mathcal{M}}_{0,n}$ 上同调特征标的闭公式，并证明特定上同调群为置换表示，方法推广至 Fulton-MacPherson 紧化及稳定映射模空间。
For any finite group G we define the moduli space of pointed admissible G-covers and the concept of a G-equivariant cohomological field theory (G-CohFT), which, when G is the trivial group, reduce to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that by taking the "quotient" by G, a G-CohFT reduces to a CohFT. We also prove that a G-CohFT contains a G-Frobenius algebra, a G-equivariant generalization of a Frobenius algebra, and that the "quotient" by G agrees with the obvious Frobenius algebra structure on the space of G-invariants, after re…
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