Relative periods of abelian differentials in a prescribed stratum

TL;DR

The paper characterizes which relative period data $\chi\in H^1(S,Z,\mathbb{C})$ arise from abelian differentials in a given connected component of a stratum of the moduli space, establishing the conditions $V(\chi)>0$ and a lattice-compatibility constraint when $\chi(H_1(S,\mathbb{Z}))$ is a lattice. It achieves this by analyzing the action of the mapping class group on relative cohomology, reducing the problem to constructing explicit translation surfaces, notably via branched covers of tori. The main constructive framework reduces realizability to torus-cover data, with a case-split by hyperelliptic, odd, even, and remaining components, and a genus-two treatment completing the argument. The results generalize prior work and answer Filip’s question by providing a unified criterion for the image of the relative period map across all strata components, with concrete constructions for each case.

Abstract

We characterise the elements of $H^1(S, Z, \mathbb C)$, where $S$ is a closed surface and $Z\subset S$ is a finite set, that arise as the relative periods of an abelian differential in a given connected component of a stratum of their moduli space. This generalises a theorem obtained independently by Bainbridge, Johnson, Judge and Park and the author, and answers a question of Filip.

Figures (11)

• Figure 1: Surface in $\mathcal{H}^{\text{hyp}}(4)$ with $d = 5$.
• Figure 2: Surface in $\mathcal{H}^{\text{hyp}}(3, 3)$ with $d = 4$.
• Figure 3: Square-tiled surface in $\mathcal{H}^{\text{hyp}}(3, 3)$ with $d = 8$.
• Figure 4: Fixed points of the hyperelliptic involution.
• Figure 5: Surface in $\mathcal{H}^\mathrm{odd}(6)$ with $d = 7$.

Theorems & Definitions (21)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 3.1
• Proposition 3.2
• proof
• proof
• Lemma 3.3