Compactification of strata of abelian differentials.

*(English)*Zbl 1403.14058Let \(g \geq 2\) and \({\mathcal M}_{g}\) be the moduli space of closed Riemann surfaces of genus \(g\). The Hodge bundle \(\Omega{\mathcal M}_{g}\) consists of classes of pairs \((X,\omega)\), where \(X\) is a closed Riemann surface and \(\omega\) is a holomorphic one-form on \(X\) (if \(X\) has trivial group of holomorphic automorphisms, then the fiber over a point \([X] \in {\mathcal M}_{g}\) can be identified with the \(g\)-dimensional complex vector space \(H^{1,0}(X)\) of holomorphic one-forms on \(X\)). If \(\Omega{\mathcal M}_{g}^{*}\) is the complement of the zero section, then the natural action of \({\mathbb C}^{*}={\mathbb C}-\{0\}\) by multiplication on the second factor provides the projectivized Hodge structure \({\mathbb P}\Omega{\mathcal M}_{g}=\Omega{\mathcal M}_{g}^{*}/{\mathbb C}^{*}\). For each tuple \(\mu=(m_{1},\ldots,m_{n})\), where \(m_{i} \geq 1\) are integers such that \(2g-2=m_{1}+\cdots+m_{n}\), there is associated the stratum \({\mathbb P}\Omega{\mathcal M}_{g}(\mu)\) consisiting of those classes for which the holomorphic one-form has \(n\) zeroes, of respective orders \(m_{1},\ldots,m_{n}\). If \(\overline{\mathcal M}_{g}\) denotes the Deligne-Mumford compactification of \({\mathcal M}_{g}\) (obtained by adding stable Riemann surfaces of genus \(g\)), then the Hodge bundle \(\Omega{\mathcal M}_{g}\) extends to a bundle \(\Omega\overline{\mathcal M}_{g}\) of \(\overline{\mathcal M}_{g}\) (by considering stable differentials on stable Riemann surfaces). Again, one may consider the projectivized one \({\mathbb P}\Omega\overline{\mathcal M}_{g}\). In this way, the closure of \({\mathbb P}\Omega{\mathcal M}_{g}(\mu)\) in \({\mathbb P}\Omega\overline{\mathcal M}_{g}\) provides a compactification of it, called the Hodge bundle compactification. Another compactification of \({\mathbb P}\Omega{\mathcal M}_{g}(\mu)\) is obtained by lifting it to the moduli space \({\mathcal M}_{g,n}\) (the \(n\) marked points being the zeroes of the holomorphic forms) and by taking its closure in \(\overline{\mathcal M}_{g,n}\), called the Deligne-Mumford compactification.

In the paper under review, the authors consider another compactification by looking \({\mathbb P}\Omega{\mathcal M}_{g}(\mu)\) inside \({\mathbb P}\Omega\overline{\mathcal M}_{g,n}\), called the incidence variety compactification. This compactification takes cares of the limit pointed stable holomorphic differentials and also takes cares of the limit of the corresponding zeroes. They also provides an explicit characterization of the points in the boundary of such a compactificaction (this is theorem 1.3). There are two natural projection from the incidence variety compactification to the other two compactifications above (this is explained in corollary 1.4). There are many explicative examples and the paper is very nicely written.

In the paper under review, the authors consider another compactification by looking \({\mathbb P}\Omega{\mathcal M}_{g}(\mu)\) inside \({\mathbb P}\Omega\overline{\mathcal M}_{g,n}\), called the incidence variety compactification. This compactification takes cares of the limit pointed stable holomorphic differentials and also takes cares of the limit of the corresponding zeroes. They also provides an explicit characterization of the points in the boundary of such a compactificaction (this is theorem 1.3). There are two natural projection from the incidence variety compactification to the other two compactifications above (this is explained in corollary 1.4). There are many explicative examples and the paper is very nicely written.

Reviewer: Rubén A. Hidalgo (Temuco)

##### MSC:

14H15 | Families, moduli of curves (analytic) |

14H10 | Families, moduli of curves (algebraic) |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

##### Keywords:

abelian differentials; flat surfaces; moduli space of stable curves; Teichmüller dynamics; plumbing
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\textit{M. Bainbridge} et al., Duke Math. J. 167, No. 12, 2347--2416 (2018; Zbl 1403.14058)

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