Xiaotao Sun, Mingshuo Zhou
We prove moduli spaces of semistable parabolic bundles and generalized parabolic sheaves with fixed determinant on a smooth projective curve are globally $F$-regular type.
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We prove that moduli spaces of semistable parabolic bundles and generalized parabolic sheaves (GPS) with a fixed determinant on a smooth projective curve are globally F-regular type. As an application, we prove vanishing theorems on the moduli spaces of semistable parabolic sheaves on a singular curve, which combining with Factorization theorems in [24] and [25] give two recurrence relations among dimensions of spaces of generalized theta functions. By using of these recurrence relations, we prove an explicit formula (Verlinde formula) for the dimension of spaces of generalized theta functions…
We prove that the moduli spaces of parabolic symplectic/orthogonal bundles on a smooth curve are globally F regular type. As a consequence, all higher cohomology of theta line bundle vanish. During the proof, we develop a method to estimate codimension, and consider the infinite grassmannians for parabolic $G$ bundles.
We prove that globally F-regular $F$-sandwiches of degree $p$ of a projective space are toric varieties.
We prove that a projective surface of globally $F$-regular type defined over a field of characteristic zero is of Fano type.
In this paper, we show that every invertible subsheaf of the cotangent bundle of a smooth globally $F$-regular threefold of characteristic $p>3$ has Iitaka dimension less than or equal to one.
In this paper, Gotzmann's Regularity Theorem is established for globally generated coherent sheaves on projective space. This is used to extend Gotzmann's explicit construction to the Quot scheme. The Gotzmann representation is applied to bound the second Chern class of a rank 2 globally generated coherent sheaf in terms of the first Chern class.
We define a moduli space of stable regular singular parabolic connections of spectral type on smooth projective curves and show the smoothness of the moduli space and give a relative symplectic structure on the moduli space. Moreover, we define the isomonodromic deformation on this moduli space and prove the geometric Painlevé property of the isomonodromic deformation.
In this paper, we show that projective globally $F$-regular threefolds, defined over an algebraically closed field of characteristic $p\geq 11$, are rationally chain connected.
On rational chain connectedness of globally +-regular varieties
本文研究了全局$+$-正则簇的有理链连通性，证明了其在特定条件下的连通性，并引入了强全局$+$-正则的概念。
Let $X$ be a smooth variety over an algebraically closed field of positive characteristic. An $F$-sandwich of $X$ is a normal variety $Y$ through which the relative Frobenius morphism of $X$ factors as $F:X\rightarrow Y \rightarrow X$. In this paper, we give a classification of globally F-regular $F$-sandwiches of Hirzebruch surfaces.
We prove that the moduli space of mathematical instanton bundles on ${\Bbb P}^3$ with $c_2=5$ is smooth.
Gareth Jones asked during the 2014 SIGMAP conference for examples of regular dessins with nonabelian fields of moduli. In this paper, we first construct dessins whose moduli fields are nonabelian Galois extensions of the form $\mathbb{Q}(ζ_p,\sqrt[p]{q})$, where $p$ is an odd prime and $ζ_p$ is a $p$th root of unity and $q\in\mathbb{Q}$ is not a $p$th power, and we then show that their regular closures have the same moduli fields. Finally, in the special case $p=q=3$ we give another example of a regular dessin with moduli field $\mathbb{Q}(ζ_3,\sqrt[3]{3})$ of degree $2^{19}\cdot3^4$ and genus…
It is shown that there are globally defined Lagrangian distributions on the stable loci of derived Quot-stacks of coherent sheaves on Calabi--Yau four-folds. Dividing by these distributions produces perfectly obstructed smooth stacks with globally defined $-1$-shifted potentials, whose derived critical loci give back the stable loci of smooth stacks of sheaves in global Darboux form.
Fundamental group schemes of globally F-regular varieties
该研究证明正特征代数闭域上连通正规整体F-正则射影簇的连通拟挠子覆盖塔稳定，其正则轨迹的Nori基本群概型有限且线性约化，有限性是Xu和Braun相关定理的正特征类比。
A component of the moduli space M_g(Y,b) of stable maps from genus g curves to a variety Y is said to be regular if it is generically smooth and of the expected dimension provided by deformation theory. In this note we prove existence of regular components of M_g(Y,b) when Y is a product of projective spaces. This result can also be seen as a general position theorem for Brill-Noether loci in M_g.
We prove Kollár's injectivity theorem for globally $F$-regular varieties.
Let X be a smooth projective connected curve of genus $g \ge 2$ and let I be a finite set of points of X. Fix a parabolic structure on I for rank r vector bundles on X. Let $M^{par}$ denote the moduli space of parabolic semistable bundles and let $L^{par}$ denote the parabolic determinant bundle. In this paper we show that the n-th tensor power line bundle ${L^{par}}^n$ on the moduli space $M^{par}$ is globally generated, as soon as the integer n is such that $n \ge [\frac{r^2}{4}]$. In order to get this bound, we construct a parabolic analogue of the Quot scheme and extend the result of Popa …
We prove that normal projective surfaces of dense globally $F$-split type (resp. globally $F$-regular type) are of Calabi-Yau type (resp. Fano 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。
Threefolds of globally $F$-regular type with nef anti-canonical divisor
本研究针对Schwede和Smith的猜想特例，证明具nef反典范除子的光滑复射影三维簇若为整体$F$-正则型则是弱Fano簇，推进了相关代数几何猜想的验证。
We prove that every globally $F$-regular variety is log Fano. In other words, if a prime characteristic variety $X$ is globally $F$-regular, then it admits an effective $\bQ$-divisor $Δ$ such that $-K_X - Δ$ is ample and $(X, Δ)$ has controlled (Kawamata log terminal, in fact globally $F$-regular) singularities. A weak form of this result can be viewed as a prime characteristic analog of de Fernex and Hacon's new point of view on Kawamata log terminal singularities in the non-$\bQ$-Gorenstein case. We also prove a converse statement in characteristic zero: every log Fano variety has globally $…
We give an example of a regular dessin d'enfant whose field of moduli is not an abelian extension of the rational numbers, namely it is the field generated by a cubic root of 2. This answers a previous question. We also prove that the underlying curve has non-abelian field of moduli itself, giving an explicit example of a quasiplatonic curve with non-abelian field of moduli. In the last section, we note that two examples in previous literature can be used to find other examples of regular dessins d'enfants with non-abelian field of moduli.
Given a vector bundle $E$ of rank $r$ and degree $d$ on a curve $C$ of genus $g$, one can associate to $E$ in a natural way several other vector bundles. For example, one can take wedge powers of $E$. If $E$ is generated by global sections, the kernel of the evaluation map of sections is again a vector bundle. Also, new vector bundles can be produced by taking elementary transformations centered at a fixed point. Under suitable conditions on degree and rank, these constructions can be carried out globally. While all this processes seem quite elementary, very little is known about the resulting…
Let (S,H) be a polarized K3 surface, $E$ be a coherent sheaf on S and W be a linear subspace in the space of global sections H^0(S,E). If we are lucky, there is an exact sequence 0 -> W tensor O -> E -> E' -> 0, which gives a correspondence between moduli spaces of sheaves of different ranks on S. We used this correspondence in the first part of the paper in order to establish some properties of Brill-Noether loci in the moduli spaces. We allow E to be either locally free, torsion free of rank one, or a line bundle with support on a curve, thus studying simultaneously Brill-Noether special vec…
This paper solves the global moduli problem for regular holonomic D-modules with normal crossing singularities on a nonsingular complex projective variety. This is done by introducing a level structure (which gives rise to ``pre-D-modules''), and then introducing a notion of (semi-)stability and applying Geometric Invariant Theory to construct a coarse moduli scheme for semistable pre-D-modules. A moduli is constructed also for the corresponding perverse sheaves, and the Riemann-Hilbert correspondence is represented by an analytic morphism between these moduli spaces.
Let p: Y--> X be a surjection between schemes projective over the algebraic closure of a finite field. Let L be a line bundle on X such that p^*(L) is globally generated. I give a natural necessary and sufficient condition under which some positive tensor power of L is globally generated. I apply this to give a sufficient condition for semi-ampleness of nef line bundles on the moduli space of stable pointed curves, in positive characteristic.
Toroidal compactifications of the moduli spaces of Drinfeld modules
本文通过吹胀 Pink 的 Satake 紧化并结合形式模理论与迭代 Tate 一致化，构造了秩 d、带级 N 结构的 Drinfeld 模空间的对数正则环形紧化，并证明其中存在正则紧化。
Let X be a projective irreducible smooth algebraic variety. A "fine moduli space" of sheaves on X is a family F of coherent sheaves on X parametrized by an integral variety M such that : F is flat on M; for all distinct points x, y of M the sheaves F_x, F_y on X are not isomorphic and F is a complete deformation of F_x; F has an obvious local universal property. We define also a "fine moduli space defined locally", where F is replaced by a family (F_i), where F_i is defined on an open subset U_i of M, the U_i covering M. This paper is devoted to the study of such fine moduli spaces. We first g…
We introduce dessins d'enfants from the various existing points of view: As topological covering spaces, as surfaces with triangulations, and as algebraic curves with functions ramified over three points. We prove Belyi's theorem that such curves are defined over number fields, and define the action of the absolute Galois group Gal($\overline Q/Q$) on dessins d'enfants. We prove that several kinds of dessins d'enfants are defined over their field of moduli: regular dessins, dessins with no nontrivial automorphisms and dessins with one face. In the last part, we give two examples of regular des…
We prove the existence of fine moduli spaces of simple coherent sheaves on families of irreducible curves. Our proof is based on the existence of a universal upper bound of the Castelnuovo-Mumford regularity of such sheaves, which we provide.
This paper proposes the use of $F$-split and globally $F$-regular conditions in the pursuit of BAB type results in positive characteristic. The main technical work comes in the form of a detailed study of threefold Mori fibre spaces over positive dimensional bases. As a consequence we prove the main theorem, which reduces birational boundedness for a large class of varieties to the study of prime Fano varieties.
Moduli spaces of framed logarithmic and parabolic connections on a Riemann surface
该研究构造了黎曼曲面上带标架的对数联络与抛物联络的模空间，证明其具有自然代数辛结构，并给出抛物联络模空间正则函数代数超越次数的上界。
We show that if $X$ is an abelian variety of dimension $g \geq 1$ and ${\mathcal E}$ is an M-regular coherent sheaf on $X$, the Castelnuovo-Mumford regularity of ${\mathcal E}$ with respect to an ample and globally generated line bundle ${\mathcal O}(1)$ on $X$ is at most $g$, and that equality is obtained when ${\mathcal E}^{\vee}(1)$ is continuously globally generated. As an application, we give a numerical characterization of ample semihomogeneous vector bundles for which this bound is attained.
In the geometric version of the Langlands correspondence, irregular singular point connections play the role of Galois representations with wild ramification. In this paper, we develop a geometric theory of fundamental strata to study irregular singular connections on the projective line. Fundamental strata were originally used to classify cuspidal representations of the general linear group over a local field. In the geometric setting, fundamental strata play the role of the leading term of a connection. We introduce the concept of a regular stratum, which allows us to generalize the conditio…
We combine the idea of Chang and Ran [Invent. Math. 76 (1984), 41-54] of using monads of vector bundles on the projective 3-space to prove the unirationality of the moduli spaces of curves of low genus with our classification of globally generated vector bundles with small first Chern class $c_1$ on the projective 3-space to get an alternative argument for the unirationality of the moduli spaces of curves of degree at most 13 (based on the general framework of Chang and Ran).
Some components of the moduli space of Koszul Artin-Schelter regular algebras of dimension four
通过计算已知四维Koszul Artin-Schelter正则代数族的霍奇上同调和柯达依-斯宾塞映射，结合映射满射、双射的性质分析，确定了该类代数模栈的若干分支。
In the first part of this paper, we discuss the notion of irreducibility of cycles in the moduli spaces of n-marked rational tropical curves. We prove that Psi-classes and vital divisors are irreducible, and that locally irreducible divisors are also globally irreducible for n \leq 6. In the second part of the paper, we show that the locus of point configurations in (\R^2)^n in special position for counting rational plane curves (defined in two different ways) can be given the structure a tropical cycle of codimension 1. In addition, we compute explicitly the weights of this cycle.
We notice that for $0<d\le 6$ the Poincaré polynomial of Simpson moduli space $M_{dm + 1}(\mathbb P_2)$ is divisible by the Poincaré polynomial of the projective space $\mathbb P_{3d-1}$. A somehow regular behaviour of the difference of the Poincaré polynomials of the Hilbert scheme of $\frac{(d-2)(d-1)}{2}$ points on $\mathbb P_2$ and the moduli space of Kronecker modules $N(3; d-2, d-1)$ is noticed for $d=4, 5, 6$.
We study ample stable vector bundles on minimal rational surfaces. We give a complete classification of those moduli spaces for which the general stable bundle is both ample and globally generated. We also prove that if $V$ is any stable bundle, then a large enough direct sum $V^{\oplus n}$ has ample deformations unless there is an obvious numerical reason why it cannot. Previous work in this area has mostly focused on rank two bundles and relied primarily on classical constructions such as the Serre construction. In contrast, we use recent advances in moduli of vector bundles to obtain strong…
We extend results on generic strange duality for K3 surfaces by showing that the proposed isomorphism holds over an entire Noether-Lefschetz divisor in the moduli space of quasipolarized K3s. We interpret the statement globally as an isomorphism of sheaves over this divisor, and also describe the global construction over the space of polarized K3s.
We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global $F$-regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita's conjecture to mixed characteristic.
More regular formal moduli spaces and arithmetic transfer conjectures: the ramified quadratic case
本文针对 p 进域二次分歧扩张上的酉群，构造了带仿射仿射层次的正则形式模空间及其间的对应，提出算术转移猜想，并在最低维情形下给出证明。
We study birational maps among 1) the moduli space of semistable torsion sheaves of Hilbert polynomial $4m+2$ on a smooth quadric surface, 2) the moduli space of semistable torsion sheaves of Hilbert polynomial $m^{2}+3m+2$ on $\mathbb{P}^{3}$, 3) Kontsevich's moduli space of genus zero stable maps of degree 2 to Grassmannian $Gr(2, 4)$. A regular birational morphism from 1) to 2) is described in terms of Fourier-Mukai transform. The map from 3) to 2) is Kirwan's partial desingularization. Also we investigate several geometric properties of 1) by using the variation of moduli spaces of stable …
We define various formal moduli spaces of p-divisible groups which are regular, and morphisms between them. We formulate arithmetic transfer conjectures, which are variants of the arithmetic fundamental lemma conjecture of the third author in the presence of ramification. These conjectures include the AT conjecture of our previous joint paper. We prove these conjectures in low-dimensional cases.
Generators for the cohomology of the moduli space of irregular parabolic Higgs bundles
该研究证明非正则抛物Higgs丛模空间上同调环纯部分的生成元构成，并将类似结果推广到正则完全旗情形下的抛物与强抛物Higgs丛模空间。
We derive topological string amplitudes on local Calabi-Yau manifolds in terms of polynomials in finitely many generators of special functions. These objects are defined globally in the moduli space and lead to a description of mirror symmetry at any point in the moduli space. Holomorphic ambiguities of the anomaly equations are fixed by global information obtained from boundary conditions at few special divisors in the moduli space. As an illustration we compute higher genus orbifold Gromov-Witten invariants for C^3/Z_3 and C^3/Z_4.
We provide a classification of globally generated vector bundles with $c_1 = 5$ on the projective 3-space. The classification is complete (except for one case) but not as detailed as the corresponding classification in the case $c_1 = 4$ from our paper [Memoirs A.M.S., Vol. 253, No. 1209 (2018), also arXiv:1305.3464]. We determine, at least, the pairs of integers $(a , b)$ for which there exist globally generated vector bundles on the projective 3-space with Chern classes $c_1 = 5$, $c_2 = a$, $c_3 = b$ (except for the case $(12 , 0)$ and the complementary case $(13 , 5)$ which remain undecide…
We give a complete description of the two-dimensional moduli spaces of stable Higgs bundles of rank 2 over complex projective line with one irregular singular point, having a regular leading-order term, and endowed with a generic compatible parabolic structure such that the parabolic degree of the Higgs bundle is 0. Our method relies on elliptic fibrations of the rational elliptic surface, an equivalence of categories between irregular Higgs bundles and some sheaves on a ruled surface, and an analysis of stability conditions.
In this paper we construct the analogue of Dedekind eta-function on the moduli space of polarized CY manifolds. We prove that the L-two norm of eta is the regularized determinants of the Laplacians of the CY metric on (0,1) forms. We construct the analogue of the Baily-Borel Compactification of the moduli space of polarized CY and prove that it has the same properties as the Baily-Borel compactification of the locally symmetric Hermitian spaces. We proved that the compactification constructed in the paper is the minimal.
We consider the problem of finding the limit at infinity (corresponding to the downward Morse flow) of a Higgs bundle in the nilpotent cone under the natural $\mathbb{C}^*$-action on the moduli space. For general rank we provide an answer for Higgs bundles with regular nilpotent Higgs field, while in rank three we give the complete answer. Our results show that the limit can be described in terms of data defined by the Higgs field, via a filtration of the underlying vector bundle.
The moduli space of dormant opers on elliptic curves
本文研究正特征椭圆曲线上休眠opers及休眠Miura opers的模空间，利用李代数正则元显式描述这些对象并证明模空间的连通性，同时探讨向高阶和素数幂特征的推广。
Let G denote a connected reductive algebraic group over an algebraically closed field k and let X denote a projective G x G-equivariant embedding of G. The large Schubert varieties in X are the closures of the double cosets BgB, where B denotes a Borel subgroup of G, and g is in G. We prove that these varieties are globally F-regular in positive characteristic, resp. of globally F-regular type in characteristic 0. As a consequence, the large Schubert varieties are normal and
We prove generic regularity and Uhlenbeck-type compactification theorems for the moduli spaces of PU(2)-monopoles. Generic regularity is NOT obtained in the usual way (by applying Sard theorem to a smooth parameterized moduli space), since the parameterized moduli space can be a priori singular. We explain why, using the standard order 0-perturbations, one cannot get the smoothness of the parameterized moduli space. (The same difficulty arises in the case of ASD-Spin^c moduli spaces, so the definition of the Spin^c-polynomial invariants should be revised.) We show that the singular locus of th…
We prove the following "linkage" theorem: two p-regular graphs of the same genus can be obtained from one another by a finite alternating sequence of one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the linkage theorem to prove that various moduli spaces of tropical curves are connected through codimension one.
In this paper we study the moduli space of the tropicalizations of Riemann surfaces. We first tropicalize a smooth pointed Riemann surface by a graph defined by its (hyperbolic) pair of pants decomposition. Then we can construct the moduli space of tropicalizations based on a fixed regular tropicalization, and compactify it by adding strata parametrizing weighted contractions. We show that this compact moduli space is also Hausdorff. In the end, we compare this moduli space with the moduli space of Riemann surfaces, establishing a partial order-preserving correspondence between the stratificat…
We will introduce a regularization for $p$-adic multiple zeta values and show that the generalized double shuffle relations hold. This settles a question raised by Deligne, given as a project in Arizona winter school 2002. Our approach is to use the theory of Coleman functions on the moduli space of genus zero curves with marked points and its compactification. The main ingredients are the analytic continuation of Coleman functions to the normal bundle of divisors at infinity and definition of a special tangential base point on the moduli space.
We construct a moduli scheme for semistable pre-$\D$-modules with prescribed singularities and numerical data on a smooth projective variety. These pre-$\D$-modules are to be viewed as regular holonomic $\D$-modules with `level structure'. We also construct a moduli scheme for perverse sheaves on the variety with prescribed singularities and other numerical data, and represent the de Rham functor (which gives the Riemann-Hilbert correspondence) by an analytic morphism between the two moduli schemes.
Weak Akizuki-Nakano vanishing theorem for globally $F$-split 3-folds
本文证明了全局 $F$-分裂 3-折上的弱 Akizuki-Nakano 消没定理，并据此研究了全局 $F$-分裂 Fano 3-折的形变及局部完全交全局 $F$-正则 3-折增厚的 Kodaira 消没定理。
We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus $g$, our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with $g$ interior lattice points. It has dimension $2g+1$ unless $g \leq 3$ or $g = 7$. We compute these spaces explicitly for $g \leq 5$.
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.
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