scholarly journals diffstrata—A Sage Package for Calculations in the Tautological Ring of the Moduli Space of Abelian Differentials

2021 ◽  
pp. 1-21
Author(s):  
Matteo Costantini ◽  
Martin Möller ◽  
Jonathan Zachhuber
Keyword(s):  
Moduli Space ◽  
Abelian Differentials ◽  
Tautological Ring

On unipotent flows in ℋ(1,1)

2009 ◽  
Vol 30 (2) ◽  
pp. 379-398 ◽  
Author(s):  
KARIANE CALTA ◽  
KEVIN WORTMAN
Keyword(s):  
Moduli Space ◽  
Probability Measures ◽  
Specific Class ◽  
Horocycle Flow ◽  
Abelian Differentials ◽  
Unipotent Flows ◽  
Genus Two

AbstractWe study the action of the horocycle flow on the moduli space of abelian differentials in genus two. In particular, we exhibit a classification of a specific class of probability measures that are invariant and ergodic under the horocycle flow on the stratum ℋ(1,1).

scholarly journals A large deviations bound for the Teichmüller flow on the moduli space of abelian differentials

2010 ◽  
Vol 31 (4) ◽  
pp. 1043-1071 ◽  
Author(s):  
VÍTOR ARAÚJO ◽  
ALEXANDER I. BUFETOV
Keyword(s):  
Dynamical Systems ◽  
Large Deviations ◽  
Moduli Space ◽  
Geodesic Flow ◽  
Large Deviation ◽  
Suspension Flows ◽  
Abelian Differentials ◽  
Quantitative Recurrence ◽  
Teichmuller Geodesic Flow ◽  
Teichmüller Flow

AbstractLarge deviation rates are obtained for suspension flows over symbolic dynamical systems with a countable alphabet. We use a method employed previously by the first author [Large deviations bound for semiflows over a non-uniformly expanding base. Bull. Braz. Math. Soc. (N.S.)38(3) (2007), 335–376], which follows that of Young [Some large deviation results for dynamical systems. Trans. Amer. Math. Soc.318(2) (1990), 525–543]. As a corollary of the main results, we obtain a large deviation bound for the Teichmüller flow on the moduli space of abelian differentials, extending earlier work of Athreya [Quantitative recurrence and large deviations for Teichmuller geodesic flow. Geom. Dedicata119 (2006), 121–140].

scholarly journals Spaces of Abelian Differentials and Hitchin’s Spectral Covers

2019 ◽  
Author(s):  
M Bertola ◽  
D Korotkin
Keyword(s):  
Moduli Space ◽  
Period Matrix ◽  
Tau Function ◽  
Abelian Differentials ◽  
Second Derivatives ◽  
Topological Recursion ◽  
Derivatives Of ◽  
Prime Form

Abstract Using the embedding of the moduli space of generalized $GL(n)$ Hitchin’s spectral covers to the moduli space of meromorphic Abelian differentials we study the variational formulæ of the period matrix, the canonical bidifferential, the prime form and the Bergman tau function. This leads to residue formulæ which generalize the Donagi–Markman formula for variations of the period matrix. The computation of second derivatives of the period matrix reproduces the formula derived in [2] using the framework of topological recursion.

scholarly journals THE MODULI SPACE OF TWISTED CANONICAL DIVISORS

2016 ◽  
Vol 17 (3) ◽  
pp. 615-672 ◽  
Author(s):  
Gavril Farkas ◽  
Rahul Pandharipande
Keyword(s):  
Open Subset ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Fundamental Class ◽  
Tautological Ring ◽  
Zeros And Poles

The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in $\overline{{\mathcal{M}}}_{g,n}$ which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the moduli spaces of canonical divisors.In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus $g$ curves are of pure codimension $g$ in $\overline{{\mathcal{M}}}_{g,n}$. In addition to the closure of the canonical divisors on nonsingular curves, the moduli spaces have virtual components. In the Appendix A, a complete proposal relating the sum of the fundamental classes of all components (with intrinsic multiplicities) to a formula of Pixton is proposed. The result is a precise and explicit conjecture in the tautological ring for the weighted fundamental class of the moduli spaces of twisted canonical divisors.As a consequence of the conjecture, the classes of the closures of the moduli spaces of canonical divisors on nonsingular curves are determined (both in the holomorphic and meromorphic cases).

Typical properties of periodic Teichmüller geodesics: Lyapunov exponents

2021 ◽  
pp. 1-29
Author(s):  
URSULA HAMENSTÄDT
Keyword(s):  
Growth Rate ◽  
Vector Bundle ◽  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Moduli Space ◽  
Abelian Differentials ◽  
The Matrix ◽  
Teichmüller Flow

Abstract Consider a component ${\cal Q}$ of a stratum in the moduli space of area-one abelian differentials on a surface of genus g. Call a property ${\cal P}$ for periodic orbits of the Teichmüller flow on ${\cal Q}$ typical if the growth rate of orbits with property ${\cal P}$ is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle $V\to {\cal Q}$ , the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur–Veech measure.

scholarly journals Affine invariant submanifolds with completely degenerate Kontsevich–Zorich spectrum

2016 ◽  
Vol 38 (1) ◽  
pp. 10-33 ◽  
Author(s):  
DAVID AULICINO
Keyword(s):  
Finite Number ◽  
Moduli Space ◽  
Lyapunov Spectrum ◽  
Affine Invariant ◽  
Invariant Submanifold ◽  
Teichmüller Curve ◽  
Abelian Differentials ◽  
Invariant Submanifolds ◽  
Teichmuller Curves ◽  
Teichmüller Curves

We prove that if the Lyapunov spectrum of the Kontsevich–Zorich cocycle over an affine $\text{SL}_{2}(\mathbb{R})$-invariant submanifold is completely degenerate, i.e. if $\unicode[STIX]{x1D706}_{2}=\cdots =\unicode[STIX]{x1D706}_{g}=0$, then the submanifold must be an arithmetic Teichmüller curve in the moduli space of Abelian differentials over surfaces of genus three, four, or five. As a corollary, we prove that there is at most a finite number of such Teichmüller curves.

Sign in / Sign up

Export Citation Format

Share Document