Georg Schumacher, Hajime Tsuji
By means of analytic methods the quasi-projectivity of the moduli space of algebraically polarized varieties with a not necessarily reduced complex structure is proven including the case of non-uniruled polarized varieties.
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Proper splittings and projectivity for good moduli spaces
本文证明良好模空间在适当一般有限覆盖后具有分裂，并推广Kollár引理给出射影性判据。
Projectivity of the moduli of equidimensional branchvarieties
本文证明等维 branchvarieties 模空间的射影性，解决 Alexeev–Knutson 公开问题，并据此构造满足半稳定条件、带丰富线性级数的约化等维簇射影模空间。
We prove that every proper subspace of the moduli space of stable surfaces with fixed volume over an algebraically closed field of characteristic p>5 is projective. As a consequence we also deduce that the same moduli space is projective over Z[1/30] modulo two conjectural local properties of the moduli functor.
We prove a strengthening of Kollár's Ampleness Lemma and use it to prove that any proper coarse moduli space of stable log-varieties of general type is projective. We also prove subadditivity of log-Kodaira dimension for fiber spaces whose general fiber is of log general type.
Projectivity of Moduli Spaces in the Higher-Rank DT/PT Correspondence
研究光滑射影三维流形上DT/PT墙交叉处模空间的射影性，通过构造全局生成的线丛并分析其分离点的能力，建立高秩PT稳定对象的模空间的射影性。
Projectivity of good moduli spaces of vector bundles on stacky curves
本文利用良模空间存在性判据证明栈曲线上半稳定向量丛模栈具有射影良模空间，并给出线丛无基点性的有效界，从而为抛物丛模空间提供了新的有效构造。
Projectivity of the moduli space of vector bundles on a curve
本文针对亏格 $g\geq 2$ 曲线上半稳定向量丛模空间的射影性，结合好模空间与行列式线丛技术给出现代证明路径。
We prove that, for a natural class of Bridgeland stability conditions on $\mathbb{P}^1\times\mathbb{P}^1$ and the blow-up of $\mathbb{P}^2$ at a point, the moduli spaces of Bridgeland semistable objects are projective. Our technique is to find suitable regions of stability conditions with hearts that are (after "rotation") equivalent to representations of a quiver. The helix and tilting theory is well-behaved on Del Pezzo surfaces and we conjecture that this program (begun in arXiv:1203.0316) runs successfully for all Del Pezzo surfaces, and the analogous Bridgeland moduli spaces are projectiv…
The unbearable projectivity of being a good moduli space of Gorenstein curves of genus one with non-colliding markings
本文分类了亏格一且标记不碰撞的Gorenstein曲线栈中的开放子栈，证明其好模空间恰为Bozlee--Kuo--Neff定义的栈，并给出射影性数值判据，发现当标记数趋于无穷时几乎从不射影。
We construct moduli spaces of semistable objects on an Enriques surface for generic Bridgeland stability condition and prove their projectivity. We further generalize classical results about moduli spaces of semistable sheaves on an Enriques surface to their Bridgeland counterparts. Using Bayer and Macrì's construction of a natural nef divisor varying with the stability condition, we begin a systematic exploration of the relation between wall-crossing on the Bridgeland stability manifold and the minimal model program for these moduli spaces. We give three applications of our machinery to obtai…
The moduli space of Higgs bundles can be defined as a quotient of an infinite-dimensional space. Moreover, by the Kuranishi slice method, it is equipped with the structure of a normal complex space. In this paper, we will use analytic methods to show that the moduli space is quasi-projective. In fact, following Hausel's method, we will use the symplectic cut to construct a normal and projective compactification of the moduli space, and hence prove the quasi-projectivity. The main difference between this paper and Hausel's is that the smoothness of the moduli space is not assumed.
On the projectivity of compactified universal Jacobians
该研究对稳定带点曲线模栈上射影的紧化泛雅可比空间开展分类工作。
We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wall-crossing for Bridgeland-stability and the minimal model program for the moduli space. We give three applications of our method for classical moduli spaces of sheaves on a K3 surfa…
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。
We prove that a normal variety contains finitely many maximal quasi-projective open subvarieties. As a corollary, we obtain the following generalization of the Chevalley-Kleiman projectivity criterion : a normal variety is quasi-projective if and only if every finite subset is contained in an affine open subvariety. The proof builds on a strategy of Wlodarczyk, using results of Boissiere, Gabber and Serman.
We give a moduli-theoretic treatment of the existence and properties of moduli spaces of semistable quiver representations, avoiding methods from geometric invariant theory. Using the existence criteria of Alper--Halpern-Leistner--Heinloth, we show that for many stability functions, the stack of semistable representations admits an adequate moduli space, and prove that this moduli space is proper over the moduli space of semisimple representations. We construct a natural determinantal line bundle that descends to a semiample line bundle on the moduli space and provide new effective bounds for …
In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space $\bar{\mathcal{M}}$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on $\bar{\mathcal{M}}$ and extends the canonical Weil-Petersson current on the moduli space parametrizing smooth Kahler-Einstein Fano manifolds $\mathcal{M}$. As a consequence, we show that the CM line bundle is nef and big on $\bar{\mathcal{M}}$ and its restriction on $\mathcal{M}$ is ample.
We generalize a compactification technique due to C. Simpson in the context of $\mathbb{G}_m$-actions over the ground field of complex numbers, to the case of a universally Japanese base ring. We complement this generalized compactification technique so that it can sometimes yield projectivity results for these compactifications. We apply these projectivity results to the Hodge, de Rham, and Dolbeault moduli spaces for curves, with special regards to ground fields of positive characteristic.
We consider the problem of deciding if a group is the fundamental group of a smooth connected complex quasi-projective (or projective) variety using Alexander-based invariants. In particular, we solve the problem for large families of Artin-Tits groups. We also study finiteness properties of such groups and exhibit examples of hyperplane complements whose fundamental groups satisfy $\text{F}_{k-1}$ but not $\text{F}_k$ for any $k$.
Quasi-Projectivity of Weighted Right-Angled Artin Groups
本文分类了作为光滑复拟射影簇基本群的加权右角Artin群，证明它们恰为有限生成自由群与形如$\langle a,b\mid [a,b]^m=1\rangle$的群的有限直积。
Let H be a semisimple algebaric group and let X be a smooth projective curve defined over an algebraically closed field k. In the first part of this paper we show that the moduli of semistable principal H-bundles exists once given a "low-height" representation of H. We also show the projectivity of the moduli space if p > ψ, where ψis a representation theoritic index. The projectivity is a consequence of a semistable reduction theorem. The irreducibility of the moduli space of semistable H-bundles for simple and simply connected group is obtained as a consequence.
We show that every smooth toric variety (and many other algebraic spaces as well) can be realized as a moduli space for smooth, projective, polarized varieties. Some of these are not quasi--projective. This contradicts a recent paper (Quasi--projectivity of moduli spaces of polarized varieties, Ann. of Math. 159 (2004) 597--639.).
We prove some semipositivity theorems for singular varieties coming from graded polarizable admissible variations of mixed Hodge structure. As an application, we obtain that the moduli functor of stable varieties is semipositive in the sense of Kollár. This completes Kollár's projectivity criterion for the moduli spaces of higher-dimensional stable varieties.
We discuss properties of complex algebraic orbifold groups, their characteristic varieties, and their abelian covers. In particular, we deal with the question of (quasi)-projectivity of orbifold groups. We also prove a structure theorem for the variety of characters of normal-crossing quasi-projective orbifold groups. Finally, we extend Sakuma's formula for the first Betti number of abelian covers of orbifold fundamental groups. Several examples are presented, including a compact orbifold group which is not projective and a Zariski pair of plane projective curves that can be told by considerin…
We study birational projective models of ${\mathcal M}_{2,2}$ obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of ${\mathcal Z}$-stable curves $\overline{\mathcal M}_{2,2}({\mathcal Z})$ defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space $\overline{M}_{2,2}({\mathcal Z})$.
The present paper studies the structure of characteristic varieties of fundamental groups of graph manifolds. As a consequence, a simple proof of Papadima's question is provided on the characterization of algebraic links that have quasi-projective fundamental groups. The type of quasi-projective obstructions used here are in the spirit of Papadima's original work.
We generalize the notion of multi-Gieseker semistability for coherent sheaves, introduced by Greb, Ross, and Toma, to quiver sheaves for a quiver $Q$. We construct coarse moduli spaces for semistable quiver sheaves using a functorial method that realizes these as subschemes of moduli spaces of representations of a twisted quiver, depending on $Q$, with relations. We also show the projectivity of the moduli space in the case when $Q$ has no oriented cycles. Finally, we investigate the parameter dependence of the moduli.
Moduli of $\mathbb{Q}$-Gorenstein pairs and applications
该研究提出基于$\boldsymbol{\rm{Q}}$-稳定对概念构造$\boldsymbol{\rm{Q}}$-Gorenstein对模空间的框架，证明其具备紧合性与射影粗模空间，可避免小系数下稳定对模空间的病态，并给出稳定对模空间射影性的新证明。
This paper lays the foundation for determining the Kodaira dimension of the projectivized strata of Abelian differentials with prescribed zero and pole orders in large genus. We work with the moduli space of multi-scale differentials constructed in [BCGGM2] which provides an orbifold compactification of these strata. We establish the projectivity of the moduli space of multi-scale differentials, describe the locus of canonical singularities, and compute a series of effective divisor classes. Moreover, we exhibit a perturbation of the canonical class which allows the corresponding pluri-canonic…
We establish P=W and PI=WI conjectures for character varieties with structural group $\mathrm{GL}_n$ and $\mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-étale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and…
Lagrangian families of Bridgeland moduli spaces from Gushel-Mukai fourfolds
本文证明Gushel-Mukai簇与其超平面截面的Kuznetsov分量间拉回与推出函子保持Bridgeland稳定性，并由此构造GM四维簇模空间的8维Lagrangian族及证明GM三维簇模空间的射影性。
The Self-Projecting Grassmannian
本文提出自投影格拉斯曼流形及其相关matroid结构，研究其与经典模空间的关系及计算实现空间。
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。
Chow ring of moduli spaces of quasi-polarised K3 surfaces in lower genus
本文研究低亏格拟极化 K3 曲面模空间，证明亏格不超过 5 时其周环由重言类构成。
We study a compactification of the moduli space of theta characteristics, giving a modular interpretation of the geometric points and describing the boundary stratification. This space is different from the moduli space of spin curves. The modular description and the boundary stratification of the new compactification are encoded by a tropical moduli space. We show that this tropical moduli space is a refinement of the moduli space of spin tropical curves. We describe explicitly the induced decomposition of its cones.
Quasi-Projective Moduli for Polarized klt Good Minimal Models
本文利用Gabber定理证明局部稳定族的直接像弱正性，进而应用Viehweg放大准则，展示极化klt极小模型模空间的正则化是准射影的。
In this paper, we present explicit toric construction of moduli space of quasi maps from $\mathbb{P}^1$ with two marked points to $\mathbb{P}^1 \times \mathbb{P}^1$, which was first proposed by Jinzenji and prove that it is a compact orbifold. We also determine its Chow ring and compute its Poincaré polynomial for some lower degree cases.
Moduli spaces of stable sheaves over quasi-polarized surfaces, and the relative Strange Duality morphism
本文利用 Alper 的好模空间理论，在拟极化射影曲面栈上构造稳定层的相对模空间，并将相对奇异对偶态射推广至拟极化 K3 曲面轨迹。
In this paper, we construct toric data of moduli space of quasi maps of degree $d$ from P^{1} with two marked points to weighted projective space P(1.1,1,3). With this result, we prove that the moduli space is a compact toric orbifold. We also determine its Chow ring. Moreover, we give a proof of the conjecture proposed by Jinzenji that a series of intersection numbers of the moduli spaces coincides with expansion coefficients of inverse function of -log(j(tau)).
Quasimaps to moduli spaces of sheaves
本文发展了曲面层模空间的拟映射理论，证明其模空间正常且带完美障碍理论，并自然同构于三维簇上的层模空间；利用纠缠尾理论建立墙交叉公式，联系Gromov-Witten与Donaldson-Thomas理论，并在del Pezzo曲面上计算了希尔伯特概形的公式。
In this paper, we study the moduli space of quasi-polarized complex K3 surfaces of degree 6 and 8 via geometric invariant theory. The general members in such moduli spaces are complete intersections in projective spaces and we have natural GIT constructions for the corresponding moduli spaces and we show that the K3 surfaces with at worst ADE singularities are GIT stable. We give a concrete description of boundary of the compactification of the degree 6 case via the Hilbert-Mumford criterion. We compute the Picard group via Noether-Lefschetz theory and discuss the connection to the Looijenga's…
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 study the moduli space $\mathcal{F}_{T_1}$ of quasi-trielliptic K3 surfaces of type I, whose general member is a smooth bidegree $(2,3)$-hypersurface of $\mathbb{P}^1\times \mathbb{P}^2$. Such moduli space plays an important role in the study of the Hassett-Keel-Looijenga program of the moduli space of degree $8$ quasi-polarized K3 surfaces. In this paper, we consider several natural compactifications of $\mathcal{F}_{T_1}$, such as the GIT compactification and arithmetic compactifications. We give a complete analysis of GIT stability of $(2,3)$-hypersurfaces and provide a concrete descript…
We introduce a derived enhancement of the moduli space of sections defined by Chang-Li, and we compute its tangent complex. Special cases of this moduli space include stable maps and stable quasi-maps. As an application, we prove that G-theoretic stable map and quasi-map invariants of projective spaces are equal.
On quasimap invariants of moduli spaces of Higgs bundles
本文计算 Higgs SL_2-丛模空间及素秩情形的亏格 1 奇数阶 quasimap/Gromov–Witten 不变量，验证相关猜想，并给出次数与秩互素时的通用计算方案。
We study a toric degeneration of the Cox ring of the moduli of principal $SL_m(\mathbb{C})$ bundles on the projective line, with quasi parabolic data given by the the stabilizer of the highest weight vector in $\mathbb{C}^m$ and its dual $\bigwedge^{m-1}(\mathbb{C}^m)$. The affine semigroup algebra resulting from this degeneration is described using the $K-$Pieri rule from Kac-Moody representation theory. Along the way we give a proof of the $K-$Pieri rule which utilizes the classical Pieri rule and elements of commutative algebra, and we describe a relationship between the Cox ring and a clas…
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…
Quasimaps to moduli spaces of sheaves on a $K3$ surface
该研究构造到 K3 曲面层模空间拟映射障碍理论的满射余截面，建立约化穿墙公式，并据此证明 Igusa 尖点形式猜想的 Hilbert 概形部分及两类 DT 对应。
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。
A tale of two moduli spaces: logarithmic and multi-scale differentials
本文证明了对数微分与多尺度微分在全局留数条件下等价，并建立了它们的（粗糙）模栈同构，同时通过显式爆破描述了橡胶空间和多尺度空间，提出了扭曲Hodge丛中的细化双分岔圈公式。
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.
We show that the moduli spaces of stable sheaves on projective schemes admit certain non-commutative structures, which we call quasi NC structures, generalizing Kapranov's NC structures. The completion of our quasi NC structure at a closed point of the moduli space gives a pro-representable hull of the non-commutative deformation functor of the corresponding sheaf developed by Laudal, Eriksen, Segal and Efimov-Lunts-Orlov. We also show that the framed stable moduli spaces of sheaves have canonical NC structures.
Relative forms of real algebraic varieties and examples of quasi-projective surfaces with algebraic moduli of real forms
本文提出一个框架以精确化“由簇参数化的簇的实形式族”概念，并构造了首个具有任意大代数模维度的非同构实形式的拟射影实曲面例子。
Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective surface that maps to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of the surface. As a result, we can describe the fibration induced by the moduli map quite explicitly. A refined affirmativ…
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.
Coherent Sheaves on Ribbons and their Moduli
本文研究了带状结构上凝聚层的性质及其Simpson模空间，给出了半稳定拟局部自由层存在的充要条件与轨迹闭包维数，并猜想性描述了模空间的不可约分支。
Quasi-BPS categories for K3 surfaces
本文引入K3曲面的拟BPS范畴，构造半稳定对象模叠导出范畴的半正交分解，证明穿墙等价性，并研究约化拟BPS范畴，证明其拓扑K-理论恢复K3曲面的BPS不变量，将其视为非交换超kähler流形。
We give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the theorem to (quasi-compact and separated) algebraic spaces. Using more advanced technology (and assuming flatness) we also give a proof of the folklore result that the group of autoequivalences of the category of quasi-coherent sheaves consists of automorphisms of the underlying space and twists by line bundles. We apply our strategy to prove analogous statement…
We give a short and "classical" proof of Borcherds' theorem that the moduli space of Enriques surfaces is quasi-affine. The use of the Borcherds' product is replaced in our proof by an application of the Grothendieck-Riemann-Roch theorem.
Logarithmic Quasimaps
本文利用对数几何构造了任意亏格下相对于简单正规交叉除子的拟映射的Deligne-Mumford栈模空间，并证明其具有预期维数的虚拟基本类，所得数值不变量与Battistella-Nabijou理论一致。
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