GIT and moduli with a twist

ℂ*翻转链的模空间链：关于光滑曲线上的O扭秩2约束框架Hitchin对的模空间

A chain of $\mathbb{C}^{*}$-flips of the moduli spaces of $\mathcal{O}$-twisted rank 2 constrained framed Hitchin pairs on a smooth curve

研究光滑曲线上的O扭秩2约束框架Hitchin对模空间的ℂ*翻转链，证明其到秩2框架模块模空间翻转链的覆盖映射。

The moduli space of twisted canonical divisors

The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in the moduli space of Deligne-Mumford stable pointed curves which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the moduli spaces of canonical divisors. In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus g curves are o…

Moduli of twisted sheaves

We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that the spaces are asympotically geometrically irreducible, normal, generically smooth, and l.c.i. over the base. We also develop general tools necessary for these results: the theory of associated points and purity of sheaves on Artin stacks, t…

Towards an enumerative geometry of the moduli space of twisted curves and r-th roots

The enumerative geometry of r-th roots of line bundles is the subject of Witten's conjecture and occurs in the calculation of Gromov-Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In math.AG/0603687, we construct a compact stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford's formula for the Chern character of the Hodge bundle to the direct image of the uni…

Moduli of fibered surface pairs from twisted stable maps

We use the theory of twisted stable maps to Deligne-Mumford stacks to construct compactifications of the moduli space of pairs $(X \to C, S + F)$ where $X \to C$ is a fibered surface, $S$ is a sum of sections, $F$ is a sum of marked fibers, and $(X, S + F)$ is a stable pair in the sense of the minimal model program. This generalizes the work of Abramovich-Vistoli, who compactified the moduli space of fibered surfaces without marked fibers. Furthermore, we compare our compactification to Alexeev's space of stable maps and the KSBA compactification of stable pairs. As an application, we describe…

扭曲的阿林金变换与Kuznetsov组件上模空间的导出范畴

Twisted Arinkin transforms and derived categories of moduli spaces on Kuznetsov components

本文研究了K3表面和Kuznetsov组件上模空间的导出范畴，通过扭曲导出等价性推广了Donagi-Pantev的结果，并回答了Mattei和Meinsma的问题。

模栈与向量丛扭曲模空间的Torelli定理

Torelli Theorem for moduli stacks of vector bundles on tame Deligne Mumford curves

本文证明了向量丛模空间的两个Torelli定理：从固定行列式模栈恢复曲线，及从扭曲模空间恢复曲线和Brauer类，方法涉及特殊线性线性化、Hitchin构造和Kummer覆盖等。

扭曲自同态的模空间与欧拉示性数

Moduli Spaces of Twisted Endomorphisms and Euler Characteristics

本文为（拟）射影簇上的层定义了扭曲自同态的模空间，在自同构有限阶时构造非交换代数层并建立等价描述，进而计算其希尔伯特方案的欧拉示性数，推广了单位自同构的结果。

Generalized twisted cubics on a cubic fourfold as a moduli space of stable objects

We revisit the work of Lehn-Lehn-Sorger-van Straten on twisted cubic curves in a cubic fourfold not containing a plane in terms of moduli spaces. We show that the blow-up $Z'$ along the cubic of the irreducible holomorphic symplectic eightfold $Z$, described by the four authors, is isomorphic to an irreducible component of a moduli space of Gieseker stable torsion sheaves or rank three torsion free sheaves. For a very general such cubic fourfold, we show that $Z$ is isomorphic to a connected component of a moduli space of tilt-stable objects in the derived category and to a moduli space of Bri…

Dimension theory of the moduli space of twisted $k$-differentials

In this note we extend the dimension theory for the spaces $\widetilde{\mathcal H}_g^k(μ)$ of twisted $k$-differentials defined by Farkas and Pandharipande in [FP15] to the case $k>1$. In particular, we show that the intersection $\widetilde{\mathcal H}_g^k(μ) \cap \mathcal{M}_{g,n}$ is a union of smooth components of the expected dimensions for all $k\geq 0$. We also extend a conjectural formula from [FP15] for a weighted fundamental class of $\widetilde{\mathcal H}_g^k(μ)$ and provide evidence in low genus.

Twisted bundles and admissible covers

The cherry on top of this stacky paper is the following: for any g>1 we give a finite group G such that the moduli space of connected admissible G-covers of genus g is a smooth, fine moduli space, which is a Galois cover of the moduli space of stable curves. The proof relies on methods introduced by Looijenga and Pikaart-De Jong, and on the theory of twisted G-covers, a theory announced without proofs in math.AG/9811059, section 3, and developed in the first few sections of this paper. This includes a moduli description of the normalization of the Harris-Mumford space of admissible covers, and…

Moduli spaces of framed instanton bundles on CP^3 and twistor sections of moduli spaces of instantons on C^2

We show that the moduli space $M$ of holomorphic vector bundles on $CP^3$ that are trivial along a line is isomorphic (as a complex manifold) to a subvariety in the moduli of rational curves of the twistor space of the moduli space of framed instantons on $\R^4$, called the space of twistor sections. We then use this characterization to prove that $M$ is equipped with a torsion-free affine connection with holonomy in $Sp(2n,\C)$.

曲线上扭曲等变G-丛的模空间

Moduli spaces of twisted equivariant G-bundles over a curve

本文构造了紧黎曼曲面上多稳定(θ,c)-扭曲Γ-等变G-丛的粗模空间，推广了Ramanathan的G-丛模空间构造，并给出非连通群Ĝ-丛模空间的GIT构造。

到光滑对的扭曲映射模空间

Moduli spaces of twisted maps to smooth pairs

本文研究任意亏格下到光滑对的扭曲映射模空间，通过组合/热带信息分类其不可约分支，计算不同根参数下态射的次数并证明其多项式性，进而联系并推广了Tseng–You关于orbifold Gromov–Witten不变量的结果。

Twisted stability and Fourier-Mukai transform

In this paper, we consider the preservation of stability by using the notion of Twisted stability. As applications, (1) we show that moduli spaces of vector bundles on K3 and abelian surfaces are irreducible, (2) we compute Hodge polynomials of some moduli spaces of vector bundles on Enriques surfaces and on elliptic surfaces with a section.

On A-twisted moduli stack for curves from Witten's gauged linear sigma models

Witten's gauged linear sigma model [Wi1] is one of the universal frameworks or structures that lie behind stringy dualities. Its A-twisted moduli space at genus 0 case has been used in the Mirror Principle [L-L-Y] that relates Gromov-Witten invariants and mirror symmetry computations. In this paper the A-twisted moduli stack for higher genus curves is defined and systematically studied. It is proved that such a moduli stack is an Artin stack. For genus 0, it has the A-twisted moduli space of [M-P] as the coarse moduli space. The detailed proof of the regularity of the collapsing morphism by Ju…

正特征曲线上扭曲正合微分形式的模空间

The Moduli Space of Twisted Exact Differential Forms on Curves in Positive Characteristic

本文在正特征域上引入参数化光滑标记曲线与非零正合微分形式的模栈，并计算其维数。

An Elementary GIT Construction of the Moduli Space of Stable Maps

This paper provides a GIT construction of the Moduli Space of Stable Maps as a GIT quotient of the Graph Space by SL(2,C). As a corollary, we get a birational map from the 0-pointed Moduli Space to a projective variety.

Quiver moduli and small desingularizations of some GIT quotients

We construct small desingularizations of moduli spaces of semistable quiver representations for indivisible dimension vectors using deformations of stabilites and a dimension estimate for nullcones. We apply this construction to several classes of GIT quotients.

Counting twisted Higgs bundles

We count invariants of the moduli spaces of twisted Higgs bundles on a smooth projective curve.

Stable pairs with a twist and gluing morphisms for moduli of surfaces

We propose an alternative definition for families of stable pairs $(X,D)$ over a possibly non-reduced base when $D$ is reduced, by replacing $(X,D)$ with an appropriate orbifold pair $(\mathcal X,\mathcal D)$. This definition of a stable family ends up being equivalent to previous ones, but has the advantage of being more amenable to the tools of deformation theory. Moreover, adjunction for $(\mathcal X,\mathcal D)$ holds on the nose; there is no correction term coming from the different. This leads to the existence of functorial gluing morphisms for families of stable surfaces and functorial …

A weight two phenomenon for the moduli of rank one local systems on open varieties

The twistor space of representations on an open variety maps to a weight two space of local monodromy transformations around a divisor component at infinty. The space of $σ$-invariant sections of this slope-two bundle over the twistor line is a real 3 dimensional space whose parameters correspond to the complex residue of the Higgs field, and the real parabolic weight of a harmonic bundle.

三次扭曲曲线的空间

The space of twisted cubics

本文研究了射影n空间中三次扭曲曲线空间的Cohen-Macaulay紧化，证明其为Hilbert多项式3t+1的CM曲线的精细模概形，并揭示了其在射影3空间中与Hilbert概形分支的同构关系。

The GIT moduli of semistable pairs consisting of a cubic curve and a line on ${\mathbb P}^{2}$

We discuss the GIT moduli of semistable pairs consisting of a cubic curve and a line on the projective plane. We study in some detail this moduli and compare it with another moduli suggested by Alexeev. It is the moduli of pairs (with no specified semi-abelian action) consisting of a cubic curve with at worst nodal singularities and a line which does not pass through singular points of the cubic curve. Meanwhile, we make a comparison between Nakamura's compactification of the moduli of level three elliptic curves and these two moduli spaces.

The Beauville-Narasimhan-Ramanan correspondence for twisted Higgs $V$-bundles and components of parabolic $\text{Sp}(2n,\mathbb{R})$-Higgs moduli Spaces

We generalize the classical Beauville-Narasimhan-Ramanan correspondence to the case of parabolic Higgs bundles with regular singularities and Higgs $V$-bundles. Using this correspondence along with Bott-Morse theoretic techniques we provide an exact component count for moduli spaces of maximal parabolic $\text{Sp}\left( 2n,\mathbb{R} \right)$-Higgs bundles with fixed parabolic structure.

粗模空间中的有理点与扭表示

Rational points in coarse moduli spaces and twisted representations

本文在一般设定下研究Azumaya代数中结合代数的Schur表示的模空间与模叠，阐明扭表示构成的模叠的结构，并通过非阿贝尔上同调解释经典粗模空间有理点的几何起源。

Compactification by GIT-stability of the moduli space of abelian varieties

The moduli space $\cM_g$ of nonsingular projective curves of genus $g$ is compactified into the moduli $\bcM_g$ of Deligne-Mumford stable curves of genus $g$. We compactify in a similar way the moduli space of abelian varieties by adding some mildly degenerating limits of abelian varieties. A typical case is the moduli space of Hesse cubics. Any Hesse cubic is GIT-stable in the sense that its $\SL(3)$-orbit is closed in the semistable locus, and conversely any GIT-stable planar cubic is one of Hesse cubics. Similarly in arbitrary dimension, the moduli space of abelian varieties is compactified…

Fano varieties of cubic fourfolds containing a plane

We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 surface. The moduli spaces of complexes and of sheaves are related by wall-crossing in the derived category of twisted sheaves on the corresponding K3 surface.

Twisted Gromov-Witten r-spin potential and Givental's quantization

The universal curve p:C->\Mbar over the moduli space \Mbar of stable r-spin maps to a target Kähler manifold X carries a universal spinor bundle L->C. Therefore the moduli space \Mbar itself carries a natural K-theory class Rp_*L. We introduce a twisted r-spin Gromov-Witten potential of X enriched with Chern characters of Rp_*L. We show that the twisted potential can be reconstructed from the ordinary r-spin Gromov-Witten potential of X via an operator that assumes a particularly simple form in Givental's quantization formalism.

A GIT Construction of Moduli Spaces of Stable Maps in Positive Characteristic

In a previous paper, the author and David Swinarski constructed the moduli spaces of stable maps, \bar M_g,n(P^r,d), via geometric invariant theory (GIT). That paper required the base field to be the complex numbers, a restriction which this paper removes: here the coarse moduli spaces of stable maps are constructed via GIT over a more general base.

K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space IV: the structure of invariant

A holomorphic torsion invariant of K3 surfaces with involution was introduced by the second-named author. In this paper, we completely determine its structure as an automorphic function on the moduli space of such K3 surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds' conject…

Non-Fine Moduli Spaces of Sheaves on K3 Surfaces

In general, if M is a moduli space of stable sheaves on X, there is a unique alpha in the Brauer group of M such that a pi_M^* alpha^{-1}-twisted universal sheaf exists on X times M. In this paper we study the situation when X and M are K3 surfaces, and we identify alpha in terms of Mukai's map between the cohomology of X and of M (defined by means of a quasi-universal sheaf). We prove that the derived category of sheaves on X and the derived category of alpha-twisted sheaves on M are equivalent. This suggests a conjecture which describes, in terms of Hodge isometries of lattices, when derived…

$p$-adic 非阿贝尔 Hodge 理论：通过模堆的曲线情形

$p$-adic non-abelian Hodge theory for curves via moduli stacks

本文通过模堆的典范扭转，将曲线上的 $p$-adic 非阿贝尔 Hodge 理论几何化，并恢复 Faltings 对应为同胚，同时将表示堆映射到半稳定 Higgs 丛的开子堆。

Verra fourfolds, twisted sheaves and the last involution

We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of $K3^{[2]}$-type admitting non symplectic involutions with invariant lattices $U(2)\oplus D_4(-1)$ or $U(2)\oplus E_8(-2)$. This complements the results obtained in [Mongardi and Wandel 2015], [Bossiere et al 2016], and the results from [arXiv:1603.00403] about the geometry of IHS fou…

Geometric Langlands Twists of N = 4 Gauge Theory from Derived Algebraic Geometry

We develop techniques for describing the derived moduli spaces of solutions to the equations of motion in twists of supersymmetric gauge theories as derived algebraic stacks. We introduce a holomorphic twist of N=4 supersymmetric gauge theory and compute the derived moduli space. We then compute the moduli spaces for the Kapustin-Witten topological twists as its further twists. The resulting spaces for the A- and B-twist are closely related to the de Rham stack of the moduli space of algebraic bundles and the de Rham moduli space of flat bundles, respectively. In particular, we find the unexpe…

Rational points of quiver moduli spaces

For a perfect field $k$, we study actions of the absolute Galois group of $k$ on the $\bar{k}$-valued points of moduli spaces of quiver representations over $k$; the fixed locus is the set of $k$-rational points and we obtain a decomposition of this fixed locus indexed by elements in the Brauer group of $k$. We provide a modular interpretation of this decomposition using quiver representations over division algebras, and we reinterpret this description using twisted quiver representations. We also see that moduli spaces of twisted quiver representations give different forms of the moduli space…

关于普里姆曲线模空间的有理上同调

On the rational cohomology of moduli spaces of Prym curves

研究扭曲普里姆曲线模空间的低次有理上同调群，通过证明其在特定奇数度消失及二度群为代数，涵盖了经典普里姆曲线模空间等，明确了该模空间低次有理上同调群的性质。

Twisted cyclic quiver varieties on curves

We study the algebraic geometry of twisted Higgs bundles of cyclic type along complex curves. These objects, which generalize ordinary cyclic Higgs bundles, can be identified with representations of a cyclic quiver in a twisted category of coherent sheaves. Referring to the Hitchin fibration, we produce a fibre-wise geometric description of the locus of such representations within the ambient twisted Higgs moduli space. When the genus is 0, we produce a concrete geometric identification of the moduli space as a vector bundle over an associated (twisted) A-type quiver variety; we count the numb…

李代数胚联络、扭曲希格斯丛与模空间的动机

Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces

该研究证明亏格≥2的光滑射影曲线上，秩与次数互素的李代数胚联络模空间的动机类仅依赖线丛次数，与李代数胚结构及线丛本身无关，且等于对应扭曲希格斯丛模空间的动机类，并推广至E多项式。

Involutions on moduli spaces of vector bundles and GIT quotients

Let $C$ be a hyperelliptic curve of genus $g \geq 3$. We give a new description of the theta map for moduli spaces of rank 2 semistable vector bundles with trivial determinant. In orther to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb{P}^1)^{2g}//\operatorname{PGL(2)}$. Then, we use recent results of Kumar to identify the restriction of the theta map to these GIT quotients with some explicit osculating projection. As a corollary of this construction, we obtain a birational equivalence between the ramification locus of th…

Twistor spaces for supersingular K3 surfaces

We develop a theory of twistor spaces for supersingular K3 surfaces, extending the analogy between supersingular K3 surfaces and complex analytic K3 surfaces. Our twistor spaces are obtained as relative moduli spaces of twisted sheaves on universal gerbes associated to the Brauer groups of supersingular K3 surfaces. In rank 0, this is a geometric incarnation of the Artin-Tate isomorphism. Twistor spaces give rise to curves in moduli spaces of twisted supersingular K3 surfaces, analogous to the analytic moduli space of marked K3 surfaces. We describe a theory of crystals for twisted supersingul…

秩为2的扭曲希格斯丛模空间上的流线

Flow lines on the moduli space of rank $2$ twisted Higgs bundles

本文研究紧黎曼曲面上秩为2扭曲希格斯丛模空间中希格斯场L^2范数平方的梯度流线，证明其空间可用割线簇进行代数几何分类，并给出莫尔斯理论紧化的代数解释。

Moduli spaces of weighted pointed stable rational curves via GIT

We construct the Mumford-Knudsen space of n pointed stable rational curves by a sequence of explicit blow-ups from the GIT quotient (P^1)^n//SL(2) with respect to the symmetric linearization O(1,...,1). The intermediate blown-up spaces turn out to be the moduli spaces of weighted pointed stable curves for suitable ranges of weights. As an application, we provide a new unconditional proof of M. Simpson's Theorem about the log canonical models of the Mumford-Knudsen space. We also give a basis of the Picard group of the moduli spaces of weighted pointed stable curves.

广义 Deligne-Hitchin 扭量空间：构造与性质

Generalized Deligne-Hitchin Twistor Spaces: Construction and Properties

本文通过粘合两个 Hodge 模空间构造广义 Deligne-Hitchin 扭量空间，证明其具有权一全纯截面、半负能量、平衡度量及秩一情形下全纯切丛的稳定性，并研究了相关自同构群。

Moduli space of stable maps to projective space via GIT

We compare the Kontsevich moduli space of genus 0 stable maps to projective space with the quasi-map space when $d=3$. More precisely, we prove that when $d=3$, the obvious birational map from the quasi-map space to the moduli space of stable maps is the composition of three blow-ups followed by two blow-downs. Furthermore, we identify the blow-up/down centers explicitly in terms of the moduli spaces for lower degrees. Using this, we calculate the Betti numbers, the integral Picard group, and the rational cohomology ring. The degree two case is worked out as a warm-up.

仿射非约化 GIT 与带重数箭图表示的模空间

Affine Non-Reductive GIT and moduli of representations of quivers with multiplicities

本文借助射影非约化 GIT 建立仿射簇在带分次幂单根群作用下的显式商与射影完备化，并用于构造带重数箭图半稳定表示的模空间。

A generalization of the double ramification cycle via log-geometry

We give a log-geometric description of the space of twisted canonical divisors constructed by Farkas--Pandharipande. In particular, we introduce the notion of a principal rubber $k$-log-canonical divisor, and we study its moduli space. It is a proper Deligne--Mumford stack admitting a perfect obstruction theory whose virtual fundamental cycle is of dimension $2g-3+n$. In the so-called strictly meromorphic case with $k=1$, the moduli space is of the expected dimension and the push-forward of its virtual fundamental cycle to the moduli space of stable curves equals the weighted fundamental class…

Wall crossing of the moduli spaces of perverse coherent sheaves on a blow-up

We give a remark on the wall crossing behavior of perverse coherent sheaves on a blow-up and stability condition constructed by Toda. We also explain the wall crossing of twisted stability in terms of stability condition.

The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations

In this paper, we study the moduli spaces of parabolic connections with a quadratic differential. We endow these moduli spaces with symplectic structures by using the fundamental 2-forms on the moduli spaces of parabolic connections (which are phase spaces of isomonodromic deformation systems). Moreover, we see that the moduli spaces of parabolic connections with a quadratic differential are equipped with structures of twisted cotangent bundles.

Complex K-theory of moduli spaces of Higgs bundles

We establish an isomorphism of complex $K$-theory of the moduli space $\check{\mathcal{M}}$ of $``SL_n"$-Higgs bundles of degree $d$ and rank $n$ (in the sense of Hausel--Thaddeus) and twisted complex $K$-theory of the orbifold $\hat{\mathcal{M}}$ of $PGL_n$-Higgs bundles of degree $e$, where $(n,d)=(n,e)=1$. Along the way we prove the vanishing of torsion for $H^*(\check{\mathcal{M}})$ and certain twisted complex $K$-theory groups of $\hat{\mathcal{M}}$. We also extend Arinkin's autoduality of compactified Jacobian to a derived equivalence between $SL_n$ and $PGL_n$-Hitchin systems over the e…

Rank One Bridgeland Stable Moduli Spaces on A Principally Polarized Abelian Surface

We compute moduli spaces of Bridgeland stable objects on an irreducible principally polarized complex abelian surface corresponding to twisted ideal sheaves. We use Fourier-Mukai techniques to extend the ideas of Arcara and Bertram to express wall-crossings as Mukai flops and show that the moduli spaces are projective.

GIT Constructions of Moduli Spaces of Stable Curves and Maps

This largely expository paper first gives an introduction to Hilbert stability and its use in Gieseker's GIT construction of $\overline{M}_g$. Then I review recent work in this area--variants for unpointed curves that arise in Hassett's log minimal model program, starting with Schubert's moduli space of pseudostable curves, and constructions for weighted pointed stable curves and for pointed stable maps due to Swinarski and to Baldwin and Swinarski respectively. The focus is on the steps at which new ideas are needed. Finally, I list open problems in the area, particularly some arising in the …

超越扭曲弧：为可约群的 McKay 对应关系

Beyond twisted arcs: a McKay correspondence for reductive groups

本文提出扭曲映射的概念，用于研究Artin堆栈，并通过动机变量替换公式建立线性可约群的McKay对应关系。

II型扭变的模空间

Moduli spaces of type II twists

通过分类十维(1,1)超庞加莱代数中平方零元素在Spin群作用下的轨道，完整分类了平坦背景中IIA超重力的扭变超荷，并结合IIB结果完成II型超重力扭变模空间的描述，讨论了其十一维起源及与弦世界片扭变的关系。