Hodge theorem for the logarithmic de Rham complex via derived intersections
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RESEARCH
Hodge theorem for the logarithmic de Rham complex via derived intersections Márton Hablicsek* * Correspondence:
[email protected] Leiden University, Leiden, Netherlands
Abstract In a beautiful paper, Deligne and Illusie (Invent Math 89(2):247–270, 1987) proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. Kato (in: Igusa (ed) ALG analysis, geographic and numbers theory, Johns Hopkins University Press, Baltimore, 1989) generalized this result to logarithmic schemes. In this paper, we use the theory of twisted derived intersections developed in Arinkin et al. (Algebraic Geom 4:394–423, 2017) and the author of this paper to give a new, geometric interpretation of the Hodge theorem for the logarithmic de Rham complex.
1 Introduction Let X be a smooth proper variety over an algebraically closed field k of characteristic 0. The algebraic de Rham complex is defined as the complex d
d
1 → ΩX/k − → ... ΩX• := 0 → OX −
where d is the usual differential on the algebraic forms. The de Rham cohomology of X is defined as the hypercohomology of the de Rham complex, ∗ (X) = R∗ Γ (X, ΩX• ). HdR
The Hodge-to-de Rham spectral sequence p,q
p
p+q
E1 = H q (X, ΩX ) ⇒ HdR (X) is given by the stupid filtration on the de Rham complex whose associated graded terms p are the ΩX . In their celebrated paper [9], Deligne and Illusie proved the degeneration of the Hodgeto-de Rham spectral sequence holds in positive characteristics by showing the following result. Theorem 1.1 [9] Let X be a smooth proper scheme over a perfect field k of positive characteristic p ≥ dim X. Assume that X lifts to the ring W2 (k) of second Witt vectors of k. Then the Hodge-to-de Rham spectral sequence for X degenerates at E1 . Then they showed that the corresponding result in characteristic 0 follows from a standard reduction argument.
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Similar results can be obtained in the logarithmic setting. Consider a smooth proper scheme X over a perfect field k of characteristic p > dim X and a reduced normal crossing divisor D on X. Around each point of the divisor, the
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