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Strata of $$k$$-differentials. (English) Zbl 1440.14148
The 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.

##### MSC:
 14H15 Families, moduli of curves (analytic) 30F30 Differentials on Riemann surfaces 32J05 Compactification of analytic spaces
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