Matt Bainbridge, Chris Johnson, Chris Judge, Insung Park
Let S be a closed topological surface. Haupt's theorem provides necessary and sufficient conditions for a complex-valued character of the first integer homology group of S to be realized by integration against a complex-valued 1-form that is holomorphic with respect to some complex structure on S. We prove a refinement of this theorem that takes into account the divisor data of the 1-form.
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We compute a closed formula for the class of the closure of the locus of curves in $\overline{\mathcal{M}}_g$ that admit an abelian differential of signature $κ=(k_1,...,k_{g-2})$.
Stable homology of strata of abelian differentials
研究阿贝尔微分流形的同调在简单零点数较大时的稳定性，证明理性的上同调与限制的tautological类一致，且非射影流形的理性的皮卡群为零。
This paper focuses on the interplay between the intersection theory and the Teichmueller dynamics on the moduli space of curves. As applications, we study the cycle class of strata of the Hodge bundle, present an algebraic method to calculate the class of the divisor parameterizing abelian differentials with a non-simple zero, and verify a number of extremal effective divisors on the moduli space of pointed curves in low genus.
Moduli spaces of Abelian and quadratic differentials are stratified by multiplicities of zeroes; connected components of the strata correspond to ergodic components of the Teichmuller geodesic flow. It is known that the strata are not necessarily connected; the connected components were recently classified by M. Kontsevich and the author and by E. Lanneau. The strata can be also viewed as families of flat metrics with conical singularities and with Z/2Z-holonomy. For every connected component of each stratum of Abelian and quadratic differentials we construct an explicit representative which i…
This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected components, distinguished by roots of the cotangent bundle of the surface. In the course of our investigation we also characterize the images of the fundamental groups of strata inside of the mapping class group. The main techniques of proof are mod r winding numbers and a mapping class group-theoretic analogue of the Euclidean algorithm.
$π_1$ of trigonal loci of strata of abelian differentials
本文研究阿贝尔微分分层中三次曲线轨形的结构，利用Segre-Hirzebruch曲面线性系统描述其轨形基本群，特别识别了$E_8$型Artin群的商群。
We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne-Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.
For g>2 we study the cohomology classes in the closure of a stratum of abelian differentials defined by the boundary strata of codimension one. As an application, we find an explicit stratification of the spin moduli space for an odd spin structure consisting of g-1 strata D_j of codimension j-1 such that D_j does not contain a complete subvariety for all j. We also recover some results of Koroktin and Zograf and of Chen using a unified topological approach.
The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian differentials. The main ingredients to study the compactifications of the strata are a version of the plumbing cylinder construction for differential forms and an extension of the parity of the connected components of the strata to the differentials on curves of compact type. We study in detail the compactifications of the hyperelliptic minimal strata and of th…
Fundamental groups of strata of abelian differentials of low codimension
本文证明含足够多简单零点的射影化阿贝尔微分层到带点曲线模空间的包含诱导轨形基本群单射，部分证实 Kontsevich-Zorich 猜想，并将其基本群刻画为标架映射类群。
The boundary of the multi-scale differential compactification of strata of abelian differentials admits an explicit combinatorial description. However, even for low-dimensional strata, the complexity of the boundary requires use of a computer. We give a description of the algorithms implemented in the SageMath package diffstrata to enumerate the boundary components and perform intersection theory on this space. In particular, the package can compute the Euler characteristic of strata using the methods developed by the same authors in [CMZ20].
We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures.
We prove that the nonvarying strata of abelian and quadratic differentials in [CM1, CM2] have trivial tautological rings and are affine varieties. We also prove that strata of $k$-differentials of infinite area are affine varieties for all $k$. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A …
Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials
该研究建立了Masur-Veech体积、面积Siegel-Veech常数与阿贝尔微分层相交数的联系，算出大亏格极限及所有层的鞍连接Siegel-Veech常数，且结论适用于自旋和超椭圆分量。
We study the decomposition of a stratum $\mathcal H(κ)$ of abelian differentials into regions of differentials that share a common $L^\infty$-Delaunay triangulation. In particular, we classify the infinitely many adjacencies between these isodelaunay regions, a phenomenon whose observation is attributed to Filip in work of Frankel. This classification allows us to construct a finite simplicial complex with the same homotopy type as $\mathcal H(κ)$, and we outline a method for its computation. We also require a stronger equivariant version of the traditional Nerve Lemma than currently exists in…
Independence of tautological classes and cohomological stability for strata of differentials
本文研究了微分分层的tautological环生成关系及上同调稳定性，证明了在特定条件下不存在低次关系，并推测在大g时tautological环非平凡。
For $g \ge 5$, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building off of work of the first author (arXiv:1901.05482), we find that the non-hyperelliptic components are classified by an invariant known as an $r$-spin structure. This is accomplished by computing a certain monodromy group valued in the mapping class group. To do this, we determine explicit finite generating sets for all $r$-spin stabilizer subgroup…
In this paper we consider the large genus asymptotics for Masur-Veech volumes of arbitrary strata of Abelian differentials. Through a combinatorial analysis of an algorithm proposed in 2002 by Eskin-Okounkov to exactly evaluate these quantities, we show that the volume $ν_1 \big( \mathcal{H}_1 (m) \big)$ of a stratum indexed by a partition $m = (m_1, m_2, \ldots , m_n)$ is $\big( 4 + o(1) \big) \prod_{i = 1}^n (m_i + 1)^{-1}$ as $2g - 2 = \sum_{i = 1}^n m_i$ tends to $\infty$. This confirms a prediction of Eskin-Zorich and generalizes some of the recent results of Chen-Moeller-Zagier and Sauva…
This paper investigates the relationship between strata of abelian differentials and various mapping class groups afforded by means of the topological monodromy representation. Building off of prior work of the authors, we show that the fundamental group of a stratum surjects onto the subgroup of the mapping class group which preserves a fixed framing of the underlying Riemann surface, thereby giving a complete characterization of the monodromy group. In the course of our proof we also show that these "framed mapping class groups" are finitely generated (even though they are of infinite index)…
Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected components, due to spin and hyperelliptic structures. We prove that the spin parity can be distinguished on the boundary of the log compactification. Moreover, combining the techniques of log geometry and admissible covers, we introduce log twisted hyperelliptic differentials, and…
On the Space of Ergodic Measures for the Horocycle Flow on Strata of Abelian Differentials
本文研究平移曲面分层$\mathcal{H}(2)$上的水平流，证明存在一列支撑在周期轨道上的遍历测度弱收敛到非遍历不变测度，并由此得出某些点的水平流轨道不向任何不变测度等分布。
The volumes of strata of Abelian or quadratic differentials play an important role in the study of dynamics on flat surfaces, related to dynamics in polygonal billiards. This article reviews all known ways to compute volumes in the quadratic case and provides explicit values of volumes of the strata of meromorphic quadratic differentials with at most simple poles in all low dimensions.
The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched covering constructions, i.e. every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of abelian differentials. This result implies finiteness of algebraically primitive Teichmuller curves in all hyperelliptic components for genus greater than two. A classification of all GL(2…
Boundary of equisymmetric loci of Riemann surfaces with abelian symmetry
本文研究了亏格$g\geq 2$黎曼曲面模空间中具有阿贝尔自同构群的等对称轨迹，确定了其边界最大维数层的拓扑类型，并用具有固定边数的树描述了超椭圆和循环$p$-边作用边界处的所有拓扑层。
We classify GL(2,R) invariant point markings over components of strata of Abelian differentials. Such point markings exist only when the component is hyperelliptic and arise from marking Weierstrass points or two points exchanged by the hyperelliptic involution. We show that these point markings can be used to determine the holomorphic sections of the universal curve restricted to orbifold covers of subvarieties of the moduli space of Riemann surfaces that contain a Teichmuller disk. The finite blocking problem is also solved for translation surfaces with dense GL(2,R) orbit.
We compute many new classes of effective divisors in $\overline{\mathcal{M}}_{g,n}$ coming from the strata of abelian differentials and efficiently reproduce many known results obtained by alternate methods. Our method utilises maps between moduli spaces and the degeneration of abelian differentials.
We study the transcendence of periods of abelian differentials, both at the arithmetic and functional level, from the point of view of the natural bi-algebraic structure on strata of abelian differentials. We characterise geometrically the arithmetic points, study their distribution, and prove that in many cases the bi-algebraic curves are the linear ones.
In this paper, we study the translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces. We compute the number of connected components of the corresponding strata of the moduli space. We show that in genus greater than or equal to two, one has up to three components with a similar description as the one of Kontsevich and Zorich for the moduli space of Abelian differentials. In genus one, one can obtain an arbitrarily large number of connected components that are easily distinghished by a simple topological invariant.
Finite isoresidual covers in strata of $k$-differentials
本研究针对黎曼球面上本原$k$-微分分层的$k$-剩余映射分歧覆盖，推广阿贝尔微分的已知结果，通过多尺度紧化相交计算等方法给出纤维基数公式，并应用于锥球面度量研究。
We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmueller curves with various geometrically defined divisors on moduli spaces of curves.
The local invariants of a meromorphic Abelian differential on a Riemann surface of genus $g$ are the orders of zeros and poles, and the residues at the poles. The main result of this paper is that with few exceptions, every pattern of orders and residues can be obtain by an Abelian differential. These exceptions are two families in genus zero when the orders of the poles are either all simple or all nonsimple. Moreover, we even show that the pattern can be realized in each connected component of strata. Finally we give consequences of these results in algebraic and flat geometry. The main ingr…
We prove that square-tiled surfaces having fixed combinatorics of horizontal cylinder decomposition and tiled with smaller and smaller squares become asymptotically equidistributed in any ambient linear $GL(\mathbb R)$-invariant suborbifold defined over $\mathbb Q$ in the moduli space of Abelian differentials. Moreover, we prove that the combinatorics of the horizontal and of the vertical decompositions are asymptotically uncorrelated. As a consequence, we prove the existence of an asymptotic distribution for the combinatorics of a "random" interval exchange transformation with integer lengths…
In two fundamental classical papers, Masur and Veech have independently proved that the Teichmueller geodesic flow acts ergodically on each connected component of each stratum of the moduli space of quadratic differentials. It is therefore interesting to have a classification of the ergodic components. Veech has proved that these strata are not necessarily connected. In a recent work, Kontsevich and Zorich have completely classified the components in the particular case where the quadratic differentials are given by the global square of Abelian differentials. Here we are interested in the comp…
Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and Zorich. Similar work for strata of quadratic differentials has been done in Masur and Zorich, although in that case the connected components were not distinguished. We classify the configurations for quadratic differentials on the Riemann sphere and on hyperelliptic connected components of the moduli space of quadratic differentials. We show that, in genera greate…
Monodromy kernels for strata of translation surfaces
本文证明平移曲面分层中非超椭圆分量$\mathcal{H}(3,1)$和$\mathcal{H}^{nh}(4)$的单值映射核包含秩2非阿贝尔自由群，从而未能验证与映射类群的可公度性猜想。
In the first part we extend the construction of the smooth normal-crossing divisors compactification of projectivized strata of abelian differentials given by Bainbridge, Chen, Gendron, Grushevsky and Moeller to the case of k-differentials. Since the generalized construction is closely related to the original one, we mainly survey their results and justify the details that need to be adapted in the more general context. In the second part we show that the flat area provides a canonical hermitian metric on the tautological bundle over the projectivized strata of finite area k-differentials whos…
Every GL(2,R)-orbit in hyperelliptic components of strata of abelian differentials in genus greater than two is either closed, dense, or contained in a locus of branched covers.
Periodic orbits in the thin part of strata
该研究针对满足条件的闭定向曲面上的二次或阿贝尔微分模空间层的连通分支，证明了模空间紧子集补集内泰希米勒流周期轨道数的渐近增长率下界为层复维数减一。
Translation surfaces with poles correspond to meromorphic differentials on compact Riemann surfaces. They appear in compactifications of strata of the moduli space of Abelian differentials and in the study of stability conditions. Such structures have different geometrical and dynamical properties than usual translation surfaces. In particular, they always have infinite area and can have a finite number of saddle connections. We provide a combinatorial characterization of the strata for which there can be an infinite number of saddle connections and give lower and upper bounds for the number o…
Coarse density of subsets of $M_g$
本文研究亏格 $g$ 黎曼曲面模空间 $\mathcal{M}_g$ 的代数子簇在 Teichmüller 度量或 Thurston 度量下的粗稠密性，证明其粗稠密当且仅当为整个模空间，并应用于阿贝尔微分空间中的轨道闭包投影。
We classify the GL(2,R)-invariant subvarieties M in strata of Abelian differentials for which any two M-parallel cylinders have homologous core curves. This answers a question of Mirzakhani and Wright. As a corollary we show that outside of an explicit list of exceptions, if M is a GL(2,R)-invariant subvariety, then the Kontsevich-Zorich cocycle has nonzero Lyapunov exponents in the symplectic orthogonal of the projection of the tangent bundle of M to absolute cohomology.
In every connected component of every stratum of Abelian differentials, we construct square-tiled surfaces with one vertical and one horizontal cylinder. We show that for all but the hyperelliptic components this can be achieved in the minimum number of squares necessary for a square-tiled surface in that stratum. For the hyperelliptic components, we show that the number of squares required is strictly greater and construct surfaces realising these bounds. Using these surfaces, we demonstrate that pseudo-Anosov homeomorphisms optimising the ratio of Teichmüller to curve graph translation lengt…
Stratification and realization of singularity data on the Loch Ness monster
本文为无限型驯服平移曲面引入基于奇点数据的分层理论，证明Loch Ness monster上有限锥角奇点个数为有限是实现障碍，且所有其他奇点数据均可实现，从而给出该曲面分层的完整刻画。
Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy-Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmueller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely the linear involutions. These maps are related to general measured foliations o…
The Hurwitz problem for abelian differentials
研究固定亏格g≥2的阿贝尔微分平移群最大阶t(g)的取值，证明当不存在正则origamis时t(g)=2(g-1)，并给出存在时t(g)的精确形式，分类相关分层并构造无限族亏格。
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