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Hodge structure on twisted cohomologies and twisted Riemann inequalities. I. (English) Zbl 0956.14020

Let \(x_1,\dots,x_n\) be distinct points on the complex projective line \(\mathbb{P}^1\), and let \(u=\prod^n_{j=1} (t-x_j)^{\alpha_j}\) be a multi-valued function on \(U=\mathbb{P}^1- \{x_1, \dots, x_n\}\), where the exponents \(\alpha_j\) are assumed to satisfy \(\alpha_j\in \mathbb{R}-\mathbb{Z}\), \(\sum\alpha_j=0\). Let \({\mathcal L}= {\mathcal L}_u\) be the local system on \(U\) determined by \(u\) and \(\overline {\mathcal L}\) its complex conjugate. The main result of this paper says that the twisted cohomology group \(H^1(U,{\mathcal L}\oplus \overline{\mathcal L})\) has a polarized \(\mathbb{R}\)-Hodge structure of weight one. The authors also give a natural generalization of this result to the case of Riemann surfaces of arbitrary genus, and prove Riemann’s equalities and inequalities in this setting.

MSC:

14H55 Riemann surfaces; Weierstrass points; gap sequences
14D07 Variation of Hodge structures (algebro-geometric aspects)
14F20 Étale and other Grothendieck topologies and (co)homologies
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