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Linear manifolds in the moduli space of one-forms. (English) Zbl 1148.32007

Let \(M_g\) denote the moduli space of curves of genus \(g\). The total space of the Hodge bundle \(\Omega M_g\) over \(M_g\) is the space of pairs \((X,\omega)\) where \(X\) is a Riemann surface of genus \(g\) and \(\omega\) a holomorphic one-form on \(X\). The space \(\Omega M_g\) admits a linear structure, induced by the operation of integration of one-forms over a basis of the first homology relative to the zeros of one-forms. From this linear structure, one deduces a natural action of \(\mathrm{GL}_2^+(\mathbb{R})\) on \(M_g\), and an important question that has been raised in the last few years is the classification of the closures of the orbits of this action. The question is partly motivated by a result by Marina Ratner on the classification of orbit closures in homogeneous manifolds.
The question studied in the paper under review has been answered by C. T. McMullen for the case \(g=2\), in his paper [Ann. Math. (2) 165, 397–456 (2007; Zbl 1131.14027)]. The author analyzes the case \(g\geq 3\). For that purpose, he splits the question of orbit closure into three subquestions: (1) to show that the orbit closures are complex manifolds; (2) to show that these complex manifolds are algebraic; (3) to classify these algebraic manifolds. The author shows the importance of the hyperelliptic loci in this classification problem. The main result is the following:
Theorem: Let \(B\) be an algebraic manifold of the generic stratum \(S\) of \(\Omega M_g\) with linear structure. Then, there are three possibilites: 6mm
(1)
\(B\) is a connected component of \(S\);
(2)
\(g\geq 3\) and \(B\) is the preimage in \(S\) of the hyperelliptic locus in \(M_g\);
(3)
\(B\) parametrizes curves with a Jacobian whose endomorphism ring is strictly larger than \(\mathbb{Z}\).
The author also obtains statements for nongeneric strata, mostly by a case-by-case inspection.
The results obtained in this paper are based on a cohomological description of the tangent bundle to \(\Omega M_g\).
The author also introduces a notion of linear manifold that includes Teichmüller curves, Hilbert modular surfaces, and the ball quotients of Deligne and Mostow. He proposes an explanation for the difference between the linearity of the eigenform bundle over a Hilbert modular surface and the nonlinearity of the Hilbert modular threefold parametrizing Abelian varieties of dimension three with real multiplication by \(\mathbb{Z}[\zeta_7+\zeta^{-1}_7]\). This difference was pointed out by C. T. McMullen in his paper [J. Am. Math. Soc. 16, 857–885 (2003; Zbl 1030.32012)].

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14D07 Variation of Hodge structures (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
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