Localizing Virtual Cycles by Cosections
Young-Hoon Kiem, Jun Li
我们表明，一个完美的障碍理论的障碍层的余割局部化的虚拟循环的非满射轨迹的余割。给出了局部化Gysin映射和局部化虚圈的代数构造。这些结构的各种应用进行了讨论。
We show that a cosection of the obstruction sheaf of a perfect obstruction theory localizes the virtual cycle to the non-surjective locus of the cosection. We give algebraic constructions of localized Gysin maps and localized virtual cycles. Various applications of these constructions are discussed.
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We construct a cosection localized virtual structure sheaf when a Deligne-Mumford stack is equipped with a perfect obstruction theory and a cosection of the obstruction sheaf.
Torus localization and wall crossing for cosection localized virtual cycles
自1995年由李田和贝尔德-范泰奇提出以来，虚基本类理论在代数几何中发挥了关键作用，定义了重要的不变量，如Gromov-Witten不变量和Donaldson-Thomas不变量。虚拟基本类的处理方法很多，如环面定位、退化、虚拟拉回、余割定位等。通常，将这些方法结合起来是非常有效的。本文证明了余割局部化虚环的虚拉回、环面局部化和过壁公式。
Recently Oh-Thomas constructed a virtual cycle $[X]^{\mathrm{vir}}\in A_*(X)$ for a quasi-projective moduli space $X$ of stable sheaves or complexes over a Calabi-Yau 4-fold against which DT4 invariants may be defined as integrals of cohomology classes. In this paper, we prove that the virtual cycle localizes to the zero locus $X(σ)$ of an isotropic cosection $σ$ of the obstruction sheaf $Ob_X$ of $X$ and construct a localized virtual cycle $[X]^{\mathrm{vir}}_{\mathrm{loc}}\in A_*(X(σ))$. This is achieved by further localizing the Oh-Thomas class which localizes Edidin-Graham's square root Eu…
We establish cosection localization and vanishing results for virtual fundamental classes of derived manifolds, combining the theory of derived differential geometry by Joyce with the theory of cosection localization by Kiem-Li. As an application, we show that the stable pair invariants of hyperkähler fourfolds, defined by Cao-Maulik-Toda, are zero.
Cosection localization via shifted symplectic geometry
本文从导出代数几何视角研究枚举几何，证明了拟光滑栈的余截面局部化虚拟圈与退化轨迹虚拟拉格朗日圈的等价性，建立了移位拉格朗日乘数法以推导量子 Lefschetz 定理，并研究了规范线性 sigma 模型的增强以得到相对虚拟圈。
Let $\mathcal{E}$ be a locally free sheaf of rank $r$ on a smooth projective surface $S$. The Quot scheme $\mathrm{Quot}^{l}_{S}(\mathcal{E})$ of length $l$ coherent sheaf quotients of $\mathcal{E}$ is a natural higher rank generalization of the Hilbert scheme of $l$ points of $S$. We study the virtual intersection theory of this scheme. If $C\subset S$ is a smooth canonical curve, we use cosection localization to show that the virtual fundamental class of $\mathrm{Quot}^{l}_{S}(\mathcal{E})$ is $(-1)^{l}$ times the fundamental class of the smooth subscheme $\mathrm{Quot}^{l}_{C}(\mathcal{E}\v…
Localizing virtual fundamental cycles for semi-perfect obstruction theories
最近H.- L. Chang和J. Li将虚基类理论推广到半完全阻塞理论的背景下。一个半完全阻塞理论只需要一个具有相容条件的完全阻塞理论的局部存在。本文将环面局部化、余割局部化及其组合推广到半完全阻塞理论的情形。作为应用，我们证明了Jiang-Thomas的虚符号欧拉示性数理论不需要代数几何中的技术准光滑性假设.
Cosection Localization for D-Manifolds and $(-2)$-Shifted Symplectic Derived Schemes, Revisited
本文延续作者先前工作，为具有满射余截面的导出流形构造约化虚基本类，为$(-2)$-移位辛导出概形构造余截面局部化虚基本类，并利用Oh-Thomas结果证明代数与微分几何构造同调等价，应用于Calabi-Yau四维流形Donaldson-Thomas理论中约化不变量的构造与整性。
Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and $K$-theoretic invariants for many moduli stacks of interest, including $K$-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the $K$-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory…
Witten's top Chern class via cosection localization
对于Landau Ginzburg空间（[C^n/G]，W），在所有扇区都很窄的情况下，我们通过余割局部虚循环将Witten的顶级Chern类构造为代数循环。我们验证了所有的公理类。我们推导出一个明确的公式，在自由的情况下，这样的类。我们证明了这个构造等价于Polishchuk-Vaintrob，Chiodo和Fan-Jarvis-Ruan的先验构造.
We generalize the results of Chang-Li, Kim-Oh and Chang-Li on the moduli of $p$-fields to the setting of (quasi-)maps to complete intersections in arbitrary smooth Deligne-Mumford stacks with projective coarse moduli. In particular, we show that the virtual cycle of stable (quasi-)maps to a complete intersection can be recovered by the cosection localized virtual cycle of the moduli of $p$-fields of the ambient space.
For a compact quasi-smooth derived scheme M with (-1)-shifted cotangent bundle N, there are at least two ways to localise the virtual cycle of N to M via torus and cosection localisations, introduced by Jiang-Thomas. We produce virtual cycles on both the projective completion and projectivisation and show the ones on the former push down to Jiang-Thomas cycles and the one on the latter computes the difference. Using similar ideas we give an expression for the difference of the quintic and t-twisted quintic GW invariants of Guo-Janda-Ruan.
We introduce the notion of log R-maps, and develop a proper moduli stack of stable log R-maps in the case of a hybrid gauged linear sigma model. Two virtual cycles (canonical and reduced) are constructed for these moduli stacks. The main results are two comparison theorems relating the reduced virtual cycle to the cosection localized virtual cycle, as well as the reduced virtual cycle to the canonical virtual cycle. This sets the foundation of a new technique for computing higher genus Gromov-Witten invariants of complete intersections.
Virtual Riemann-Roch Theorems for Almost Perfect Obstruction Theories
本文在几乎完美障碍理论框架下证明虚拟黎曼-罗赫定理，涵盖非等变与等变情形及余截面局部化版本，推广并简化了既有结果，为K-理论不变量提供坚实基础。
A Hopf index for isotropic sections of orthogonal bundles
将Hopf指数推广到带有二次型的丛的各向同性截面，得到八个类似物，应用于余截面局部化虚拟循环和DT^4虚拟循环。
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.
Virtual signed Euler characteristics
本文利用导出代数几何中的(-1)-位移余切丛构造拟光滑空间的五种虚拟带符号欧拉示性数，并证明其中两种相等且形变不变，另外三种相等但非形变不变。
Towards Logarithmic GLSM: The r-spin case
本文利用稳定对数映射为规范线性 sigma 模型建立对数紧化基础，通过 Witten 的 r-spin 类构造正常模空间并恢复其虚拟圈，为应用 Gromov-Witten 理论工具铺路。
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