Strata of \(k\)-differentials.

*(English)*Zbl 1440.14148The authors extend their work on stratifications of differentials on complex curves given by order of zeroes and poles [Duke Math. J. 167, 2347–2416 (2018; Zbl 1403.14058)] to stratifications of \(k\)-differentials with \(k>1\). Thus for genus \(g \geq 2\), \(k \geq 1\), and a fixed \(n\)-tuple \(\mu=(m_1, \dots, m_n)\) of integers with \(\sum m_i = k(2g-2)\), let \(\Omega^k \mathcal M_g (\mu)\) be the space of tuples \((C, \xi, p_1, \dots, p_n)\) where \(p_i \in C\) are \(n\) distinct marked points on a smooth connected curve and \(\xi \in H^0(C, \omega_C^{\otimes k})\) is a \(k\)-differential satisfying \(\text{ord}_{p_i} \xi = m_i\) for \(1 \leq k \leq n\). The first theorem says that every connected component of \(\Omega^k \mathcal M_g (\mu)\) is a smooth orbifold of dimension \(2g-1+n\) if all \(m_i \geq 0\) (the holomorphic case) and dimension \(2g-2+n\) otherwise: in particular, the connected components of \(\Omega^k \mathcal M_g (\mu)\) are irreducible. These connected components are understood by work of C. Boissy [Comment. Math. Helv. 90, No. 2, 255–286 (2015; Zbl 1323.30060)] extending the case \(k=2\) analyzed by M. Kontsevich and A. Zorich [Invent. Math. 153, No. 3, 631–678 (2003; Zbl 1087.32010)].

Next the authors compactify \(\Omega^k \mathcal M_g (\mu)\) as in their earlier work [loc. cit.]. If the \(m_i \geq 0\), then \(\Omega^k \mathcal M_g (\mu)\) sits inside the Hodge bundle \(\Omega^k \mathcal M_g \to \mathcal M_g\). After pulling back to \(\mathcal M_{g,n}\), the Hodge bundle extends to the Deligne-Mumford stratification \(\overline{\mathcal M}_{g,n}\). Projectivizing and taking the closure of the image gives a compactification \(\mathbb P \Omega^k \overline{\mathcal M}_{g,n} (\mu)\) called the incidence variety compactification (if some \(m_i < 0\) they use the same procedure with a twisted Hodge bundle). The notion of a twisted \(k\)-differential of type \(\mu\) on a nodal curve \(C\) is natural, consisting of a \(k\)-differential on each irreducible component \(C_v\) of \(C\) satisfying compatibility conditions. The second theorem describes precisely which twisted \(k\)-differentials are in \(\mathbb P \Omega^k \overline{\mathcal M}_{g,n} (\mu)\). The authors give two equivalent formulations, one in terms of admissible \(k\)-fold cyclic covers of \(C\) and compatible full orders on dual graphs, the other in terms of the twisted \(k\)-differential itself. The second characterization is more complicated, but allows for direct checking of the conditions. They close with a calculation of the dimension of spaces of twisted differentials of fixed type compatible with a level graph for use in future work on smoothing compactifications.

Next the authors compactify \(\Omega^k \mathcal M_g (\mu)\) as in their earlier work [loc. cit.]. If the \(m_i \geq 0\), then \(\Omega^k \mathcal M_g (\mu)\) sits inside the Hodge bundle \(\Omega^k \mathcal M_g \to \mathcal M_g\). After pulling back to \(\mathcal M_{g,n}\), the Hodge bundle extends to the Deligne-Mumford stratification \(\overline{\mathcal M}_{g,n}\). Projectivizing and taking the closure of the image gives a compactification \(\mathbb P \Omega^k \overline{\mathcal M}_{g,n} (\mu)\) called the incidence variety compactification (if some \(m_i < 0\) they use the same procedure with a twisted Hodge bundle). The notion of a twisted \(k\)-differential of type \(\mu\) on a nodal curve \(C\) is natural, consisting of a \(k\)-differential on each irreducible component \(C_v\) of \(C\) satisfying compatibility conditions. The second theorem describes precisely which twisted \(k\)-differentials are in \(\mathbb P \Omega^k \overline{\mathcal M}_{g,n} (\mu)\). The authors give two equivalent formulations, one in terms of admissible \(k\)-fold cyclic covers of \(C\) and compatible full orders on dual graphs, the other in terms of the twisted \(k\)-differential itself. The second characterization is more complicated, but allows for direct checking of the conditions. They close with a calculation of the dimension of spaces of twisted differentials of fixed type compatible with a level graph for use in future work on smoothing compactifications.

Reviewer: Scott Nollet (Fort Worth)