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Crystalline aspects of geography of low dimensional varieties I: numerology

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Abstract

This is a modest attempt to study, in a systematic manner, the structure of low dimensional varieties in positive characteristics using p-adic invariants. The main objects of interest in this paper are surfaces and threefolds. There are many results we prove in this paper and not all can be listed in this abstract. Here are some of the results. We prove inequalities related to the Bogomolov–Miyaoka–Yau inequality: in Corollary 4.7 that \(c_1^2\hbox {\ {\char 054}\ }\max (5c_2+6b_1,6c_2)\) holds for a large class of surfaces of general type. In Theorem 4.17 we prove that for a smooth, projective, Hodge–Witt, minimal surface of general type (with additional assumptions such as slopes of Frobenius on \(H^2_\mathrm{cris}(X)\) are \(\hbox {\ {\char 062}\ }{1}/{2}\)) that \(c_1^2\hbox {\ {\char 054}\ }5c_2\). We do not assume any lifting, and novelty of our method lies in our use of slopes of Frobenius and the slope spectral sequence. We also construct new birational invariants of surfaces. Applying our methods to threefolds, we characterize Calabi–Yau threefolds with \(b_3=0\). We show that for any Calabi–Yau threefold \(b_2\hbox {\ {\char 062}\ }c_3/2-1\) and that threefolds which lie on the line \(b_2=c_3/2-1\) are precisely those with \(b_3=0\) and threefolds with \(b_2=c_3/2\) are characterized as Hodge–Witt rigid (included are rigid Calabi–Yau threefolds which have torsion-free crystalline cohomology and whose Hodge–de Rham spectral sequence degenerates).

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  1. Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA

    Kirti Joshi

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Correspondence to Kirti Joshi.

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In memoriam Torsten Ekedahl and Michel Raynaud.

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Joshi, K. Crystalline aspects of geography of low dimensional varieties I: numerology. European Journal of Mathematics 6, 1111–1175 (2020). https://doi.org/10.1007/s40879-020-00416-x

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