Finite Dimensional Representations of the Projective Modular Group

The ring of projective invariants of eight points on the line via representation theory

The ring of projective invariants of eight ordered points on the line is a quotient of the polynomial ring on V, where V is a fourteen-dimensional representation of S_8, by an ideal I_8, so the modular fivefold (P^1)^8 // GL(2) is Proj(Sym* (V)/I_8). We show that there is a unique cubic hypersurface S in PV whose equation s is skew-invariant, and that the singular locus of S is the modular fivefold. In particular, over Z[1/3], the modular fivefold is cut out by the 14 partial derivatives of s. Better: these equations generate I_8. In characteristic 3, the cubic s is needed to generate the idea…

Sheaves on weighted projective planes and modular forms

We give an explicit description of toric sheaves on the weighted projective plane $\mathbb{P}(a,b,c)$ viewed as a toric Deligne-Mumford stack. The integers $(a,b,c)$ are not necessarily chosen coprime or mutually coprime allowing for gerbe and root stack structures. As an application, we describe the fixed point locus of the moduli scheme of stable rank 1 and 2 torsion free sheaves on $\mathbb{P}(a,b,c)$ with fixed $K$-group class. Summing over all $K$-group classes, we obtain explicit formulae for generating functions of the topological Euler characteristics. In the case of stable rank 2 loca…

平面射影曲线的模不等式

On modular inequalities for plane projective curves

本文通过组合Aomoto复形构造引入平面曲线补集的模不等式，用于研究扭曲Alexander多项式，特别是Milnor纤维的特征多项式，并给出正特征域上共振非平凡性的判据及根重数下界。

Rigid ideal sheaves and modular forms

Let $X$ be a complex smooth quasi-projective surface acted upon by a finite group $G$ such that the quotient $X/G$ has singularities only of ADE type. We obtain an explicit expression for the generating series of the Euler characteristics of the zero-dimensional components in the moduli space of zero-dimensional subschemes on $X$ invariant under the action of $G$. We show that this generating series (up to a suitable rational power of the formal variable) is a holomorphic modular form.

Canonical models of modular curves and the Galois action on CM-points

We use the theory of canonical models of Shimura varieties to describe the projective limit of the curves Y(N), all N, and its automorphism group. In particular we prove that the Galois group of Q(CM) over Q is an extension of a certain abelian group by a 2-element group, where Q(CM) stands for the the extension of Q by all the CM-points on all the curves Y(N).

通过稳定对的模拟射对应

Modular Serre Correspondence via stable pairs

本文通过稳定对的模空间与Gieseker模空间和希尔伯特方案的映射，推广了Serre对应，并展示了墙穿越对模空间连通性的影响。

Lifts of projective congruence groups

We show that noncongruence subgroups of SL_2(Z) projectively equivalent to congruence subgroups are ubiquitous. More precisely, they always exist if the congruence subgroup in question is a principal congruence subgroup Gamma(N) of level N>2, and they exist in many cases also for Gamma_0(N). The motivation for asking this question is related to modular forms: projectively equivalent groups have the same spaces of cusp forms for all even weights whereas the spaces of cusp forms of odd weights are distinct in general. We make some initial observations on this phenomenon for weight 3 via geometri…

Rank 3 rigid representations of projective fundamental groups

Let X be a smooth complex projective variety with basepoint x. We prove that every rigid integral irreducible representation $π_1(X,x)\to SL (3,{\mathbb C})$ is of geometric origin, i.e., it comes from some family of smooth projective varieties. This partially generalizes an earlier result by K. Corlette and the second author in the rank 2 case and answers one of their questions.

The modular group for the total ancestor potential of Fermat simple elliptic singularities

In a series of papers \cite{KS,MR}, Krawitz, Milanov, Ruan, and Shen have verified the so-called Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for simple elliptic singularities $E_N^{(1,1)}$ ($N=6,7,8$). As a byproduct it was also proved that the orbifold Gromov--Witten invariants of the orbifold projective lines $\mathbb{P}^1_{3,3,3}$, $\mathbb{P}^1_{4,4,2}$, and $\mathbb{P}^1_{6,3,2}$ are quasi-modular forms on an appropriate modular group. While the modular group for $\mathbb{P}^1_{3,3,3}$ is $Γ(3)$, the modular groups in the other two cases were left unknown. The goal of this paper is …

On the maps from X(4p) to X(4)

We study pullbacks of modular forms of weight 1 from the modular curve X(4) to the modular curve X(4p), where p is an odd prime. We find the extent to which such modular forms separate points on X(4p). Our main result is that these modular forms give rise to a morphism F from the quotient of X(4p) by a certain involution i to projective space, such that F is a projective embedding of X(4p)/i away from the cusps. We also report on computer calculations regarding products of such modular forms, going up to weight 4 for p <= 13, and up to weight 3 for p <= 23, and make a conjecture about these pr…

The modular variety of hyperelliptic curves of genus three

The modular variety of non singular and complete hyperelliptic curves with level-two structure of genus 3 is a 5-dimensional quasi projective variety which admits several standard compactifications. The first one, X, comes from the realization of this variety as a sub-variety of the Siegel modular variety of level two and genus three .We will be to describe the equations of X in a suitable projective embedding and its Hilbert function. It will turn out that X is normal. A further model comes from geometric invariant theory using so-called semistable degenerated point configurations in (P^1)^8 …

模曲线的施瓦茨映射

Schwarz maps for modular curves

解决F. 克莱因提出的经典问题，通过不变射影几何和皮卡 - 维西奥理论，为奇数\(N\geq3\)及偶数\(N\geq4\)构造模曲线典范模型及相关线性常微分方程，实现模变换群几何化，还给出\(X(9)\)相关方程实例。

Stable reduction of modular curves

We determine the stable reduction at $p$ of all three point covers of the projective line with Galois group ${\rm SL}_2(p)$. As a special case, we recover the results of Deligne and Rapoport on the reduction of the modular curves $X_0(p)$ and $X_1(p)$. Our method does not use the fact that modular curves are moduli spaces. Instead, we rely on results of Raynaud and the authors which describe the stable reduction of three point covers whose Galois group is strictly divisible by $p$.

Multiple zeta-values, Galois groups, and geometry of modular varieties

We discuss the following two problems: 1) The properties of the multiple zeta-values and their generalizations, multiple polylogarithms at N-th roots of unity; 2) The action of the absolute Galois group on the pro-l-completion of the fundamental group of the projective line without zero, infinity, and all N-th roots of unity; and a surprising connection of these problems with the geometry and topology of modular varieties for GL_m.

射影表示 I. 环上的射影直线

Projective Representations I. Projective lines over rings

本文研究含单位元的环上的射影直线在域上的射影空间中的表示，基于双模构造，并探讨远距离点与非远距离点与互补子空间的对应关系。

Projective representations of fundamental groups of quasiprojective varieties: a realization and a lifting result

We discuss two results about projective representations of fundamental groups of quasiprojective varieties. The first is a realization result which, under a nonresonance assumption, allows to realize such representations as monodromy representations of flat projective logarithmic connections. The second is a lifting result: any representation as above, after restriction to a Zariski open set and finite pull-back, can be lifted to a linear representation.

Picard modular forms and the cohomology of local systems on a Picard modular surface

We formulate a detailed conjectural Eichler-Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes $\mathrm{GU}(2,1,\mathbb{Q}(\sqrt{-3}))$. The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula. Along the way we prove new results on characteristic polynomials of Frobenius …

射影表示 II. 广义链几何

Projective Representations II. Generalized chain geometries

本文研究广义链几何的射影表示，探讨链映射为直纹面的条件，并确定某类几何的所有双射态射。

拟射影簇是箭图表示的完全正合子范畴的格拉斯曼流形

Quasi-projective varieties are Grassmannians for fully exact subcategories of quiver representations

本文通过限制Reineke同构到完全正合子范畴，证明拟射影簇可表示为箭图格拉斯曼流形。

半投射表示与扭表示群

Semi-Projective Representations and Twisted Representation Groups

本文将Schur的投射表示群概念推广到代数闭域上的半投射表示，并给出计算机算法，可生成给定有限群在复数域平凡或共轭作用下的所有扭表示群。

Holonomy representation of quasi-projective leaves of codimension one foliations

We prove that a representation of the fundamental group of a quasi-projective manifold into the group of formal diffeomorphisms of one variable either is virtually abelian or, after taking the quotient by its center, factors through an orbicurve.

Monodromy representations of meromorphic projective structures

We determine the image of the monodromy map for meromorphic projective structures with poles of orders greater than two. This proves the analogue of a theorem of Gallo-Kapovich-Marden, and answers a question of Allegretti and Bridgeland. Our proof uses coordinates on the moduli space of framed representations arising from the work of Fock and Goncharov.

Vector bundles on projective varieties and representations of quivers

We present equivalences between certain categories of vector bundles on projective varieties, namely cokernel bundles, Steiner bundles, syzygy bundles, and monads, and full subcategories of representations of certain quivers. As an application, we provide decomposability criteria for such bundles.

Regular algebraic surfaces isogenous to a higher product constructed from group representations using projective planes

Regular algebraic surfaces isogenous to a higher product of curves can be obtained from finite groups with ramification structures. We find unmixed ramification structures for finite groups constructed as p-quotients of particular infinite groups with special presentations related to finite projective planes.

射影簇上拟映射的K-模壁跨越理论

K-moduli wall crossing for quasimaps to a projective variety

该研究发展射影簇拟映射的K-模壁跨越理论，构造多 regime 模空间，关联稳定映射、拟映射与紧化空间，用于椭圆曲面紧化间的模插值。

Landau-Ginzburg potentials via projective representations

We interpret the Landau-Ginzburg potentials associated to Gross-Hacking-Keel-Kontsevich's partial compactifications of cluster varieties as F-polynomials of projective representations of Jacobian algebras. Along the way, we show that both the projective and the injective representations of Jacobi-finite quivers with potential are well-behaved under Derksen-Weyman-Zelevinsky's mutations of representations.

三次分支覆盖的复射影直线的数值计算

Numerical calculation of three-point branched covers of the projective line

本文提出一种数值方法来计算复射影直线的三次分支覆盖，通过开发处理Fuchsian三角群及其有限指数子群的算法，计算这些群上的模形式的幂级数展开。

Representation theory and projective geometry

We give an elementary introduction to our papers relating the geometry of rational homogeneous varieties to representation theory. We also describe related work and recent progress.

Modular Form Representation for Periods of Hyperelliptic Integrals

To every hyperelliptic curve one can assign the periods of the integrals over the holomorphic and the meromorphic differentials. By comparing two representations of the so-called projective connection it is possible to reexpress the latter periods by the first. This leads to expressions including only the curve's parameters $λ_j$ and modular forms. By a change of basis of the meromorphic differentials one can further simplify this expression. We discuss the advantages of these explicitly given bases, which we call Baker and Klein basis, respectively.

Vafa-Witten theory and iterated integrals of modular forms

Vafa-Witten (VW) theory is a topologically twisted version of N=4 supersymmetric Yang-Mills theory. S-duality suggests that the partition function of VW theory with gauge group SU(N) transforms as a modular form under duality transformations. Interestingly, Vafa and Witten demonstrated the presence of a modular anomaly, when the theory has gauge group SU(2) and is considered on the complex projective plane P2. This modular anomaly could be expressed as an integral of a modular form, and also be traded for a holomorphic anomaly. We demonstrate that the modular anomaly for gauge group SU(3) invo…

Representations on the cohomology of smooth projective hypersurfaces with symmetries

This paper is concerned with the primitive cohomology of a smooth projective hypersurface considered as a linear representation for its automorphism group. Using the Lefschetz-Riemann-Roch formula, the character of this representation is described on each piece of the Hodge decomposition. A consequence about the existence of smooth symmetric hypersurfaces that are stable under the standard irreducible permutation representation of the symmetric group on homogeneous coordinates is drawn.

Cusps, Congruence Groups and Monstrous Dessins

We study general properties of the dessins d'enfants associated with the Hecke congruence subgroups $Γ_0(N)$ of the modular group $\mathrm{PSL}_2(\mathbb{R})$. The definition of the $Γ_0(N)$ as the stabilisers of couples of projective lattices in a two-dimensional vector space gives an interpretation of the quotient set $Γ_0(N)\backslash\mathrm{PSL}_2(\mathbb{R})$ as the projective lattices $N$-hyperdistant from a reference one, and hence as the projective line over the ring $\mathbb{Z}/N\mathbb{Z}$. The natural action of $\mathrm{PSL}_2(\mathbb{R})$ on the lattices defines a dessin d'enfant s…