Degenerations of abelian differentials. (English) Zbl 1388.14080

Chen, Dawei

J. Differ. Geom. 107, No. 3, 395-453 (2017).

D. Eisenbud and J. Harris introduced the concept of limit linear series for degenerations of linear series on algebraic curves [Invent. Math. 85, 337–371 (1986; Zbl 0598.14003)]. In the present article, the author pursues an analogous theory for moduli spaces of abelian differentials on compact Riemann surfaces.
For precise statements, let \(\mathcal{M}_g\) be the moduli space of genus \(g\) curves and \(\overline{\mathcal{M}}_g\) its Deligne-Mumford compactification. The Hodge bundle \(\mathcal{H} \rightarrow \mathcal{M}_g\), extends to a rank \(g\) vector bundle \(\overline{\mathcal{H}} \rightarrow\overline{\mathcal{M}}_g\). Inside of \(\mathcal{H}\), let \(\mathcal{H}(\mu) \subset \mathcal{H}\) be the stratum of holomorphic abelian differentials with signature \(\mu\), for \(\mu = (m_1,\dots,m_n)\) a partition of \(2g-2\). In particular, \(\mathcal{H}(\mu)\) parametrizes pairs \((C,\omega)\), where \(C\) is a smooth, connected, compact, complex curve of genus \(g\), and \(\omega\) is a holomorphic abelian differential on \(C\) with divisor of zeros: \[ (\omega)_0 = m_1 p_1 + \dots + m_n p_n, \] for distinct points \(p_1,\dots,p_n \in C\). Furthermore, let \(\overline{\mathcal{H}}(\mu) \subset \overline{\mathcal{H}}\) be the closure of \(\mathcal{H}(\mu) \subset \mathcal{H}\).
In this context, the author’s first result pertains to those strata closures \(\overline{\mathcal{H}}(\mu)\) which contain a given fixed \((C,\omega) \in \overline{\mathcal{H}}\). The author then refines this result by marking the zeros of differentials in \(\mathcal{H}(\mu)\). For example, inside of \(\mathcal{M}_{g,n}\), the moduli space of genus \(g\) curves with \(n\)-marked points, consider \(\mathcal{P}(\mu) \subseteq \mathcal{M}_{g,n}\) the stratum of holomorphic canonical divisors with signature \(\mu\) and its closure \(\overline{\mathcal{P}}(\mu) \subseteq \overline{\mathcal{M}}_{g,n}\) in the Deligne-Mumford compactification of \(\mathcal{M}_{g,n}\). In this setting, the author gives necessary and sufficient conditions for a given marked curve, of compact type and with a single node, to lie in \(\overline{\mathcal{P}}(\mu)\). The author also obtains a version of this result which applies to the strata of meromorphic canonical divisors with a given signature.
The author then studies the connected components for the strata of holomorphic differentials and builds on work of M. Kontsevich and A. Zorich [Invent. Math. 153, No. 3, 631–678 (2003; Zbl 1087.32010)]. Indeed, the author obtains a result which distinguishes the boundary points of the spin components in the locus of curves of pseudocompact type. Finally, the author describes Weierstrass point behaviour for general differentials in \(\mathcal{H}(\mu)\). These results contribute to the study of the nonhyperelliptic components of the strata of holomorphic differentials.

Reviewer: Nathan Grieve (East Lansing)

MSC:

14H10	Families, moduli of curves (algebraic)
14H15	Families, moduli of curves (analytic)
14K20	Analytic theory of abelian varieties; abelian integrals and differentials

Keywords:

abelian differential; translation surface; moduli space of curves; limit linear series; spin structure; admissible cover; Weierstrass point