Michail Savvas
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.
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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。
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.
Since its introduction in 1995 by Li-Tian and Behrend-Fantechi, the theory of virtual fundamental class has played a key role in algebraic geometry, defining important invariants such as the Gromov-Witten invariant and the Donaldson-Thomas invariant. Quite a few methods for handling the virtual fundamental classes were discovered such as torus localization, degeneration, virtual pullback and cosection localization. Often combining these methods turns out to be quite effective. In this paper, we prove virtual pullback, torus localization and wall crossing formulas for cosection localized virtua…
Localizing Virtual Cycles by Cosections
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。
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…
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 construct virtual fundamental classes in all intersection theories including Chow theory, K-theory and algebraic cobordism for quasi-projective Deligne-Mumford stacks with perfect obstruction theories and prove the virtual pullback formula, the virtual torus localization formula and cosection localization principle.
Recently H.-L. Chang and J. Li generalized the theory of virtual fundamental class to the setting of semi-perfect obstruction theory. A semi-perfect obstruction theory requires only the local existence of a perfect obstruction theory with compatibility conditions. In this paper, we generalize the torus localization, the cosection localization and their combination, to the setting of semi-perfect obstruction theory. As an application, we show that the Jiang-Thomas theory of virtual signed Euler characteristic works without the technical quasi-smoothness assumption from derived algebraic geometr…
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.
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…
We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension 0 quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension 0 and 1 quotients on surfaces (using also universality, torus localization, and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa-Witten Euler characteristics of the moduli of bundles, is defined and studied. Co…
We show that a strong vanishing conjecture for $n$-fold Massey products holds for fields of virtual cohomological dimension at most $1$ using a theorem of Haran. We also prove the same for PpC fields, using results of Haran--Jarden. Finally we construct a pro-$2$ group which satisfies the weak Massey vanishing property for every $n\geq3$, but does not satisfy the strong Massey vanishing property for $n=4$.
Virtual classes via vanishing cycles
本文提出了一种利用消没圈反常层为拟光滑导出概形构造虚拟基类的新方法，并证明其与现有类一致，同时探讨了$-2$-移位辛导出概形的DT4虚拟类构造方法。
Relative virtual localization and vanishing of tautological classes on moduli spaces of curves
证明了稳定相对映射的模空间的一个局部化公式。作为应用，我们证明了稳定点曲线的模空间上的余维i重言式类从至少具有i-g+1亏格0分支的曲线所对应的层消失. (The这个结果的证明是相当简短和幼稚的。）作为结果，我们证明并推广了关于曲线的各种模空间的各种结构和定理（由于Getzler，Ionel，Faber，Looijenga，Pandharipande，Diaz等人）。这个定理似乎是这些结果背后的几何内容;其余的是简单的图形组合。该定理还表明了模空间的分层的有理分量的数量的重要性。
In previous work, Mixed-Spin-P field has been introduced and their moduli space $\cal{W}_{g,γ,\bf{d}}$ together with a $\mathbb{C}^*$ action is constructed. Applying virtual localization to their virtual classes $[\cal{W}_{g,γ,\bf{d}}]^{vir}$, polynomial relations among GW and FJRW invariants of Fermat quintics are derived. In this paper, we prove a vanishing of a class of terms in $[(\cal{W}_{g,γ,\bf{d}})^{\mathbb{C}^*}]^{vir}$. This vanishing verifies that in Witten's GLSM only $r$-spin invariants of insertions $2/5$ contribute to the phase transition between GW and FJRW invariants of Fermat…
A Generalized vanishing theorem for Blow-ups of Quasi-smooth Stacks
该研究证明了拟光滑导出Artin栈导出爆破上拟凝聚层的广义消没定理，给出其在K理论、消奇、半正交分解及代数作用范畴化四方面的应用，并提出相关猜想。
Virtual intersections on the Quot-scheme and Vafa-Intriligator formulas
我们构造了一个虚拟的基本类的报价计划参数化的平凡丛的曲线。我们使用虚拟定位公式来计算Quot上的虚拟相交数。因此，我们重新证明的Vafa-Intriligator公式，我们的答案是有效的，即使报价计划是不好的行为。用同样的方法计算更多的交点。最后，我们给出了一个应用于丛的模空间的Pontrjagin环的非零性。
Let $G$ be a reductive affine algebraic group defined over a field $k$ of characteristic zero. In this paper, we study the cotangent complex of the derived $G$-representation scheme $ {\rm DRep}_G(X)$ of a pointed connected topological space $X$. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of $ {\rm DRep}_G(X) $ to the representation homology $ {\rm HR}_*(X,G) := π_*{\mathcal O}[{\rm DRep}_G(X)] $ to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, l…
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