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Zbl 1084.14023
Moonen, Ben; Wedhorn, Torsten
Discrete invariants of varieties in positive characteristic. (English)
[J] Int. Math. Res. Not. 2004, No. 72, 3855-3903 (2004). ISSN 1073-7928; ISSN 1687-0247/e

Let $F: X \to S$ be a smooth proper morphism of sheaves in characteristic $p >0$. If the sheaves $R^b f_* \Omega^a_{X/S}$ are locally free and the Hodge-de Rham spectral sequence degenerates at $E_1$, then the de Rham cohomology sheaves $\Cal M = H^m_{\text{dR}} (X/S)$ are locally free $\Cal O_S$-modules equipped with a descending Hodge filtration $C^\bullet$ and an ascending conjugate filtration $D_\bullet$ as well as with $\Cal O_S$-linear isomorphisms $\varphi_i: (\text{gr}^i_C)^{(p)} \to \text{gr}_i^D$ given by the inverse Cartier operator. In this paper the authors give a complete classification of the structures $(\Cal M, C^\bullet, D_\bullet, \varphi_\bullet)$ over an algebraically closed field. The result shows that such structures are essentially combinatorial objects.
[Min Ho Lee (Cedar Falls)]

MSC 2000:: *14F40 De Rham cohomology
14C30 Transcendental methods
14G35 Modular and Shimura varieties
14J28 K3-surfaces, etc.

Keywords: Hodge structures; Cartier operator; de Rham cohomology

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