Line bundles on rigid varieties and Hodge symmetry
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Mathematische Zeitschrift
Line bundles on rigid varieties and Hodge symmetry David Hansen1 · Shizhang Li2 Received: 24 November 2018 / Accepted: 27 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over non-archimedean fields. Our arguments rely on known structure theorems for the relevant Picard varieties, together with recent advances in p-adic Hodge theory. We also define a rigid analytic Albanese naturally associated with any smooth proper rigid space.
1 Introduction Let K be a p-adic field, i.e. a complete discretely valued extension of Q p with perfect residue field κ.1 Let X be a smooth proper rigid analytic space over K . Among fundamental invariants associated with X are its Hodge numbers h i, j := dim K H j (iX ). In this paper, we study the relationship between the Hodge numbers h 1,0 and h 0,1 of X . d=0 For a compact complex manifold Y , we always have dim H 1 (Y , OY ) ≥ dim H 0 (Y , 1Y ) (c.f. [17, Chapter IV, Sect. 2]). In the rigid analytic setting, Scholze proved that the Hodge–de Rham spectral sequence always degenerates at E 1 , and in particular every global 1-form on X as above is automatically closed (c.f. [12, Theorem 8.4]). One is naturally led to guess that for X as above we always have dim H 1 (X , O X ) ≥ dim H 0 (X , 1X ). In this paper we confirm this inequality assuming that X has a strictly semistable formal model (c.f. [9, Sect. 1]) and provide a geometric interpretation of the difference. Theorem 1.1 (Main Theorem) Under the conditions stated above, we have dim H 1 (X , O X ) ≥ dim H 0 (X , 1X ). Moreover, the difference between two numbers above is the virtual torus rank of the Picard variety of X (to be defined in the next section).
1 Note that the perfectness assumption of the residue field here is not essential, as Hodge numbers doesn’t
change under ground field extension.
B
Shizhang Li [email protected]
1
Max Planck Institute for Mathematics, Bonn, Germany
2
Department of Mathematics, University of Michigan, Ann Arbor, USA
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D. Hansen and S. Li
Remark (1) Possessing strictly semistable reduction is stable under finite étale base extension, hence the theorem is insensitive to finite unramified extensions of the ground field. Since the central fiber of a strictly semistable formal model is generically smooth, by Hensel’s lemma, we may and do assume that X has a K -rational point x : Sp(K ) → X . We will fix this rational point from now on. (2) The proof relies crucially on the assumption that X has a strictly semistable formal model, which we use to determine the structure of the Picard variety of X , c.f. Theorem 2.1 below. We certainly expect that the structure of the Picard variety should be of this shape in general. However, it is also a long standing folklore conjecture that any quasi-compact smooth rigid space potentially admits a strictly semistable formal model. (3) Assuming a result in progress by Conrad–
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