Corpus ID: 237525358

Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms

@inproceedings{Winsor2021DynamicsOT,
  title={Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms},
  author={Karl Winsor},
  year={2021}
}
Let Sg be a closed oriented surface of genus g, and let ΩMg(κ) be a stratum of the moduli space of holomorphic 1-forms of genus g. We show that the absolute period foliation of ΩMg(κ) is ergodic on the area-1 locus, and we give an explicit full measure set of dense leaves, subject to a mild constraint on κ. For an algebraically generic cohomology class φ ∈ H(Sg;C), we show that the space of holomorphic 1-forms in ΩMg(κ) representing φ is connected. On the other hand, in the non-generic case, we… Expand
1 Citations

Figures from this paper

DENSE REAL REL FLOW ORBITS AND ABSOLUTE PERIOD LEAVES
We show the existence of a dense orbit for real Rel flows on the area-1 locus of every connected component of every stratum of holomorphic 1-forms with at least 2 distinct zeros. For this purpose, weExpand

References

SHOWING 1-10 OF 40 REFERENCES
Moduli spaces of isoperiodic forms on Riemann surfaces
This paper describes the intrinsic geometry of a leaf A(L) of the absolute period foliation of the Hodge bundle ΩMg: its singular Euclidean structure, its natural foliations and its discretizedExpand
Connected components of the moduli spaces of Abelian differentials with prescribed singularities
Consider the moduli space of pairs (C,ω) where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describeExpand
Foliations of Hilbert modular surfaces
The Hilbert modular surface XD is the moduli space of Abelian varieties A with real multiplication by a quadratic order of discriminant D > 1. The locus where A is a product of elliptic curvesExpand
A criterion for density of the isoperiodic leaves in rank 1 affine invariant suborbifolds
We define on any affine invariant orbifold M a foliation F^M that generalises the isoperiodic foliation on strata of the moduli space of translation surfaces and study the dynamics of its leaves inExpand
The field of definition of affine invariant submanifolds of the moduli space of abelian differentials
The field of definition of an affine invariant submanifold M is the smallest subfield of R such that M can be defined in local period coordinates by linear equations with coefficients in this field.Expand
DENSE REAL REL FLOW ORBITS AND ABSOLUTE PERIOD LEAVES
We show the existence of a dense orbit for real Rel flows on the area-1 locus of every connected component of every stratum of holomorphic 1-forms with at least 2 distinct zeros. For this purpose, weExpand
Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants
A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segmentExpand
Periods of Abelian Differentials With Prescribed Singularities
We give a necessary and sufficient condition for a representation of the fundamental group of a closed surface of genus at least 2 to $\mathbb C$ to be the holonomy of a translation surface with aExpand
Isoperiodic dynamics in rank 1 affine invariant orbifolds
Let M be a rank 1 affine invariant orbifold in a stratum of the moduli space of flat surfaces. We show that the leaves of the M-isoperiodic foliation are either all closed or all dense. In the secondExpand
Dynamics of SL 2 ( R ) over moduli space in genus two
This paper classifies orbit closures and invariant measures for the natural action of SL2(R) on ΩM2, the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two.
...
1
2
3
4
...