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The strata do not contain complete varieties. (Les strates ne possèdent pas de variétés complètes.) (French. English summary) Zbl 1452.14023
The moduli space of holomorphic differentials on Riemann surfaces with prescribed numbers of zeros and multiplicities is called a stratum. In this paper the author shows that any stratum of holomorphic differentials does not contain complete algebraic curves (i.e. no compact Riemann surfaces can be embedded in the stratum algebraically). The proof applies the maximum modulus principle in a cute way to the shortest saddle connections joining zeros of the differentials under the induced flat metric. It remains an open question to determine whether the projectivized strata (i.e. parameterizing the underlying effective canonical divisors) can contain a complete algebraic curve or not. Note that if one considers the strata of strictly meromorphic differentials or non-effective canonical divisors, then they do not contain any complete algebraic curve [D. Chen, J. Inst. Math. Jussieu 18, No. 6, 1331–1340 (2019; Zbl 1423.14184)].
MSC:
 14H10 Families, moduli of curves (algebraic) 14H15 Families, moduli of curves (analytic) 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 30F30 Differentials on Riemann surfaces
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References:
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