# Line bundles on rigid varieties and Hodge symmetry

@article{Hansen2017LineBO, title={Line bundles on rigid varieties and Hodge symmetry}, author={David T. Hansen and Shizhang Li}, journal={Mathematische Zeitschrift}, year={2017}, pages={1-10} }

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.

## 12 Citations

Rigid-analytic varieties with projective reduction violating Hodge symmetry

- MathematicsCompositio Mathematica
- 2021

We construct examples of smooth proper rigid-analytic varieties admitting formal models with projective special fibers and violating Hodge symmetry for cohomology in degrees ${\geq }3$. This answers…

RIGID-ANALYTIC VARIETIES WITH PROJECTIVE REDUCTION VIOLATING

- Mathematics
- 2021

We construct examples of smooth proper rigid-analytic varieties admitting formal model with projective special fiber and violating Hodge symmetry for cohomology in degrees ≥ 3. This answers…

Diamantine Picard functors of rigid spaces

- Mathematics
- 2021

For a connected smooth proper rigid space X over a perfectoid ﬁeld extension of Q p , we show that the ´etale Picard functor of X ♦ deﬁned on perfectoid test objects is the diamondiﬁcation of the…

Prismatic cohomology of rigid analytic spaces over de Rham period ring

- Mathematics
- 2021

Inspired by Bhatt-Scholze [BS19], in this article, we introduce prismatic cohomology for rigid analytic spaces with l.c.i singularities, with coefficients over Fontaine’s de Rham period ring B dR .

p-ADIC GEOMETRY

- MathematicsProceedings of the International Congress of Mathematicians (ICM 2018)
- 2019

We discuss recent developments in $p$-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for "compact $p$-adic manifolds" over new…

Hodge symmetry for rigid varieties via log hard Lefschetz.

- Mathematics
- 2020

Motivated by a question of Hansen and Li, we show that a smooth and proper rigid analytic space $X$ with projective reduction satisfies Hodge symmetry in the following situations: (1) the base…

ON THE u ∞ -TORSION SUBMODULE OF PRISMATIC COHOMOLOGY

- Mathematics
- 2022

. We investigate the maximal ﬁnite length submodule of the Breuil–Kisin prismatic cohomology of a smooth proper formal scheme over a p -adic ring of integers. This submodule governs pathology…

On the u^{\infty}-torsion submodule of prismatic cohomology

- Mathematics
- 2022

. We investigate the maximal ﬁnite length submodule of the Breuil–Kisin prismatic cohomology of a smooth proper formal scheme over a p -adic ring of integers. This submodule governs pathology…

A geometric $p$-adic Simpson correspondence in rank one

- Mathematics
- 2022

A major open question in p -adic non-abelian Hodge theory raised by Faltings is which Higgs bundles will correspond to continuous representations under the p -adic Simpson correspondence for a smooth…

The p-adic Corlette–Simpson correspondence for abeloids

- MathematicsMathematische Annalen
- 2022

For an abeloid variety A over a complete algebraically closed field extension K of $$\mathbb {Q}_p$$
Q
p
, we construct a p-adic Corlette–Simpson correspondence, namely an equivalence between…

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