Line bundles on rigid varieties and Hodge symmetry


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.

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  1. 1.

    Note that the perfectness assumption of the residue field here is not essential, as Hodge numbers doesn’t change under ground field extension.

  2. 2.

    Although the way to write this extension is non-canonical, this number \(r-k\) only depends on X

  3. 3.

    Here and elsewhere in this paper, we use \(\mathbb {C}_K \;{:=}\; \widehat{\overline{K}}\) to denote the completion of an algebraic closure of K.

  4. 4.

    In our situation where the ground field has characteristic 0, one can drop the smoothness in this definition.

  5. 5.

    The proof of the analogous result for abelian varieties given in loc. cit. extends with almost no change to the setting of abeloid varieties, except that one has to use the rigid geometry version of rigidity lemma (c.f. [16, Lemma 7.1.2]).

  6. 6.

    The authors would like to thank Professor Michael Temkin for pointing this reference to us in a private communication.

  7. 7.

    This Proposition was essentially proved by W. Cherry in [3] with the same proof (he only mentions abelian varieties instead of general abeloids, but the proof is identical in the abeloid case). We thank the referee for pointing this out to us.

  8. 8.

    Note that the concept of overconvergent is called conservative in loc. cit.


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The second named author would like to thank his advisor Professor Johan de Jong for many helpful discussions during the preparation of this paper. He would also like to thank Zijian Yao, Dingxin Zhang and Yang Zhou for discussing things related to this paper. We thank various anonymous referees heartily for providing many valuable comments and suggestions concerning previous drafts of this article.

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Hansen, D., Li, S. Line bundles on rigid varieties and Hodge symmetry. Math. Z. 296, 1777–1786 (2020).

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