Yunfeng Jiang, Hsian-Hua Tseng
We generalize the multiple cover formula of Y. Toda (proved by Maulik-Thomas) for counting invariants for semistable coherent sheaves on local K3 surfaces to semistable twisted sheaves over twisted local K3 surfaces. The formula has an application to prove any rank S-duality conjecture for K3 surfaces.
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We give a formula of the framed one-leg orbifold Gromov-Witten vertex where the leg is gerby with isotropy group $\bZ_m$. Then we use this formula to compute the Gromov-Witten invariants of the local $\cB\bZ_m$ gerbe. We will also compute some examples of the degree 1 and degree 2 $\bZ_2$-Hodge integrals.
The purpose of this work is to describe the (category of) Higgs bundles on a complex scheme X having a given cameral cover X~. We show that this category is a T_{X~}-gerbe, where T_{X~} is a certain sheaf of abelian groups on X, and we describe the class of this gerbe precisely. In particular, it follows that the set of isomorphism classes of Higgs bundles with a fixed cameral cover X~ is a torsor over the group H^1(X, T_{X~}), which itself parametrizes T_{X~}-torsors on X. This underlying group can be described as a generalized Prym variety, whose connected component is either an abelian vari…
We prove the existence of a Galois closure for towers of torsors under finite group schemes over a proper, geometrically connected and geometrically reduced algebraic stack $X$ over a field $k$. This is done by describing the Nori fundamental gerbe of an essentially finite cover of $X$. A similar result is also obtained for the $S$-fundamental gerbe.
Twisted Higgs bundles and coendoscopy
本文研究了被中心gerbe扭曲的G-Higgs束，并展示了扭曲Hitchin纤维的Ngô乘积公式类比。
In this paper, we study holomorphic rank-2 vector bundles on non-K\" ahler elliptic surfaces. Our main tool for analysing these bundles is of course the spectral cover. However, given the non-Kähler condition, the elliptic surfaces we are considering do not have sections and gerbes naturally arise in this context. The spectral construction presented in this paper is a modification of the Fourier-Mukai transform for elliptic fibrations without a section. After examining some of the properties of this Fourier-Mukai transform, we give a complete classification of vector bundles on these surfaces.
In this paper, we introduce the notion of parabolic stable pairs on Calabi-Yau 3-folds and invariants counting them. By applying the wall-crossing formula developed by Joyce-Song, Kontsevich-Soibelman, we see that they are related to generalized Donaldson-Thomas invariants counting one dimensional semistable sheaves on Calabi-Yau 3-folds. Consequently, the conjectural multiple cover formula of generalized DT invariants is shown to be equivalent to a certain product expansion formula of the generating series of parabolic stable pair invariants. The application of this result to the multiple cov…
The generalized Donaldson-Thomas invariants counting one dimensional semistable sheaves on Calabi-Yau 3-folds are conjectured to satisfy a certain multiple cover formula. This conjecture is equivalent to Pandharipande-Thomas's strong rationality conjecture on the generating series of stable pair invariants, and its local version is enough to prove. In this paper, using Jacobian localizations and parabolic stable pair invariants introduced in the previous paper, we reduce the conjectural multiple cover formula for local curves with at worst nodal singularities to the case of local trees of smoo…
A short proof of the multiple cover formula for point insertions
本文通过几何方法简证了点插入的多重覆盖公式，避免了热力学计数，提供了更简洁的证明方式。
Multiple cover formulas for abelian surfaces via correlated invariants
本文通过相关不变量证明了abelian surfaces的多重覆盖公式猜想，涵盖了静止不变量并利用了减少退化公式。
Abelian covers of $\mathbb{CP}^{1}$, with fixed Galois group $A$, are classified, as a first step, by a discrete set of parameters. Any such cover $X$, of genus $g\geq1$ say, carries a finite set of $A$-invariant divisors of degree $g-1$ on $X$ that produce non-zero theta constants on $X$. We show how to define a quotient involving a power of the theta constant on $X$ that is associated with such a divisor $Δ$, some polynomial in the branching values, and a fixed determinant on $X$ that does not depend on $Δ$, such that the quotient is constant on the moduli space of $A$-covers with the given …
In this paper we prove an explicit formula which compares the dimensions of the spaces of vanishing cycles in a Galois cover of degree p between formal germ of curves over a complete discrete valuation ring of inequal characteristics (0,p). This formula can be easily generalised to the case of a Galois cover with group which is nilpotent or which has a normal p-Sylow subgroup. The results of this paper play a key role in [Sa-1] where is studied the semi-stable reduction of Galois covers of degree p above semi-stable curves over a complete discrete valuation ring of inequal characteristics (0,p…
The multiple cover formula for $K3$ and abelian surfaces
本文证明，对于 $K3$ 曲面和阿贝尔曲面，本原曲线类上的约化后代 Gromov-Witten 不变量可通过多重覆盖公式计算，该公式由 GW/PT 对应从稳定对理论推导得出。
Waring problem for forms is important and classical in mathematics. It has been widely investigated because of its wide applications in several areas. In this paper, we consider the Waring problem for binary forms with complex coefficients. Firstly, we give an explicit formula for the Waring rank of any binary binomial and several examples to illustrating it. Secondly, we prove that, up to scalar multiplication, there are exactly $\binom{d-1}{2}$ binary forms of degree $d$ with Waring rank two and multiple of three fixed distinct linear forms.
Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann surface by Riemann surface with elementary branch points and prescribed ramification type over a special point.
Multiple cover formulas for K3 geometries, wall-crossing, and Quot schemes
本文利用第二种余截面论证研究帽形空间的约化DT理论，得到K3×E秩1 DT理论、点状Hilbert概形GW理论的多重覆盖公式，评估拟映射穿墙项及Quot概形虚欧拉数。
Admissible covers and stable maps
本文研究目标曲线的容许覆盖与稳定映射模空间在 $\overline{M}_{g,n}$ 上产生的闭链，证明统一公式以恢复并细化 ELSV 公式及 Gromov-Witten/Hurwitz 对应，并完成低亏格计算验证。
Harmonic covers of skeleta
本文证明环状概形的有限覆盖诱导对偶复形间的调和态射，并利用平衡条件推导出任意维度骨架覆盖的黎曼-胡尔维茨公式。
In the paper, we study the wall-crossing phenomenon of reduced open Gromov-Witten invariants on K3 surfaces with rigid special Lagrangian boundary condition. As a corollary, we derived the multiple cover formula for the reduced open Gromov-Witten invariants.
In this article we prove a local Riemman-Hurwitz formula which compares the dimensions of the spaces of vanishing cycles in a finite Galois cover of type (p,p,...,p) between formal germs of p-adic curves and which generalises the formula proven by the first author in the case of Galois covers of degree p. We also investigate the problem of the existence of a torsor structure for a finite Galois cover of type (p,p,...,p) between p-adic schemes.
In this paper, we give Thomae type formula for \KK surfaces $\cS$ given by double covers of the projective plane branching along six lines. This formula gives relations between theta constants on the bounded symmetric domain of type $I_{22}$ and period integrals of $X$. Moreover, we express the period integrals by using the hypergeometric function $F_S$ of four variables. As an application of our main theorem, we define $\R^4$-valued sequences by mean iterations of four terms, and express their common limits by the hypergeometric function $F_S$.
We compute an explicit formula for the first Chern class of the Hodge Bundle over the space of admissible cyclic $\mathbb{Z}/3\mathbb{Z}$ covers of $n$-pointed rational stable curves as a linear combination of boundary strata. We then apply this formula to give a recursive formula for calculating certain Hodge integrals containing $λ_1$. We also consider covers with a ${\mathbb{Z}}/{2\mathbb{Z}}$ action for which we compute $λ_2$ as a linear combination of codimension two boundary strata.
We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden and Jackson, and extend results of Hurwitz and many others.
Using the developed deformation theory on moduli spaces of quadratic differentials we derive variational formulas for objects associated with generalized $SL(2)$ Hitchin's spectral covers: Prym matrix, Prym bidifferential, Hodge and Prym tau-functions. The resulting formulas are antisymmetric versions of Donagi-Markman residue formula. The second variation of the Prym matrix is a natural analogy to the expression previously derived for the period matrix of the $GL(n)$ spectral cover.
Tropical curves in abelian surfaces III: pearl diagrams and multiple cover formulas
本文通过 tropical 方法计算 abelian 曲面的计数不变量，开发了 pearl diagram 算法以解决 tropical 问题，并借此证明了 Oberdieck 的多重覆盖公式，进而推导出不变量生成级数的拟模性与多项式性。
Categorifying Zeta Functions for Quadratic Covers
针对数域二次扩张、代数曲线分歧双覆盖等四类二次覆盖情形，通过抽象关联代数的目标线性代数统一方法范畴化泽塔函数的L-函数乘积公式，并给出超椭圆曲线点计数、图素数分布、同源图圈计数及二次互反律分析等应用。
A standard observation in algebraic geometry and number theory is that a ramified cover of an algebraic variety $\widetilde{X}\rightarrow X$ over a finite field $F_q$ furnishes the rational points $x\in X(F_q)$ with additional arithmetic structure: the Frobenius action on the fiber over $x$. For example, in the case of the Vieta cover of polynomials over $F_q$ this structure describes a polynomial's irreducible decomposition type. Furthermore, the distribution of these Frobenius actions is encoded in the cohomology of $\widetilde{X}$ via the Grothendieck-Lefschetz trace formula. This note pres…
Using the embedding of the moduli space of generalized GL(n) Hitchin's spectral covers to the moduli space of meromorphic abelian differentials we study the variational formulae of the period matrix, the canonical bidifferential, the prime form and the Bergman tau function. This leads to residue formulae which generalize the Donagi-Markman formula for variations of the period matrix. Computation of second derivatives of the period matrix reproduces the formula derived by Baraglia and Zhenxi Huang using the framework of topological recursion.
We provide a formula describing the G-module structure of the Hurwitz-Hodge bundle for admissible G-covers in terms of the Hodge bundle of the base curve, and more generally, for describing the G-module structure of the push-forward to the base of any sheaf on a family of admissible G-covers. This formula can be interpreted as a representation-ring-valued relative Riemann-Hurwitz formula for families of admissible G-covers.
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