Framed mapping class groups and the monodromy of strata of Abelian differentials

Vanishing cycles, plane curve singularities, and framed mapping class groups

Let f be an isolated plane curve singularity with Milnor fiber of genus at least 5. For all such f, we give (a) an intrinsic description of the geometric monodromy group that does not invoke the notion of the versal deformation space, and (b) an easy criterion to decide if a given simple closed curve in the Milnor fiber is a vanishing cycle or not. With the lone exception of singularities of type $A_n$ and $D_n$, we find that both are determined completely by a canonical framing of the Milnor fiber induced by the Hamiltonian vector field associated to f. As a corollary we answer a question of …

Invariant Subvarieties of Minimal Homological Dimension, Zero Lyapunov Exponents, and Monodromy

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.

扭曲全纯1-形式的模空间与膨胀曲面分层上的不变测度

Invariant measures on moduli spaces of twisted holomorphic 1-forms and strata of dilation surfaces

针对黎曼曲面上扭曲全纯1-形式的模空间，通过映射类群系数上同调的新颖计算，构造了分层上的SL(2,R)-不变Lebesgue类测度，即Masur-Veech测度的类似物。

低余维数阿贝尔微分层的基本群

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。

关于正辫子、单值群与框架

On Positive Braids, Monodromy Groups and Framings

本文为每个正辫子构造辫子单值群，推广孤立平面曲线奇点的几何单值群，并在闭包为纽结时将其等同于带框架映射类群，从而得到良定义的纽结不变量，进而证明不可约奇点的几何单值群由亏格和Arf不变量决定。

Monodromy groups of $\mathbb{C}\mathrm{P}^1$-structures on punctured surfaces

For a punctured surface $S$, we characterize the representations of its fundamental group into $\mathrm{PSL}_2 (\mathbb{C})$ that arise as the monodromy of a meromorphic projective structure on $S$ with poles of order at most two and no apparent singularities. This proves the analogue of a theorem of Gallo-Kapovich-Marden concerning $\mathbb{C}\mathrm{P}^1$-structures on closed surfaces, and settles a long-standing question about characterizing monodromy groups for the Schwarzian equation on punctured spheres. The proof involves a geometric interpretation of the Fock-Goncharov coordinates of t…

$k$-微分分层中的有限等剩余覆盖

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。

Isoresidual fibration and resonance arrangements

The stratum $\mathcal{H}(a,-b_{1},\dots,-b_{p})$ of meromorphic $1$-forms with a zero of order $a$ and poles of orders $b_{1},\dots,b_{p}$ on the Riemann sphere has a map, the isoresidual fibration, defined by assigning to any differential its residues at the poles. We show that above the complement of a hyperplane arrangement, the resonance arrangement, the isoresidual fibration is an unramified cover of degree $\frac{a!}{(a+2-p)!}$. Moreover, the monodromy of the fibration is computed for strata with at most three poles and a system of generators and relations is given for all strata. These …

Moduli space of meromorphic differentials with marked horizontal separatrices

We study framed translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces, for which a horizontal separatrix is marked for each pole or zero. Such geometric structures naturally appear when studying flat geometry surfaces "near" the Deligne-Mumford boundary.We compute the number of connected components of the corresponding strata, and give a simple topological invariant that distinguishes them. In particular we see that for $g>0$, there are at most two such components, except in the hyperelliptic case.