• Corpus ID: 119723989

# o-minimal GAGA and a conjecture of Griffiths

```@article{Bakker2018ominimalGA,
title={o-minimal GAGA and a conjecture of Griffiths},
author={Benjamin Bakker and Yohan Brunebarbe and Jacob Tsimerman},
journal={arXiv: Algebraic Geometry},
year={2018}
}```
• Published 29 November 2018
• Mathematics
• arXiv: Algebraic Geometry
We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure, and prove a GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic spaces. We then combine this with algebraization theorems of Artin to show that proper definable images of complex…
Completion of Period Mappings and Ampleness of the Hodge bundle
• Mathematics
• 2017
We discuss progress towards a conjectural Hodge theoretic completion of a period map. The completion is defined, and we conjecture that it admits the structure of a compact complex analytic variety.
Period Mappings and Ampleness of the Hodge line bundle
• Mathematics
• 2017
We discuss progress towards a conjectural Hodge theoretic completion of a period map. The completion is defined, and we conjecture that it admits the structure of a compact complex analytic variety.
Hyperbolicity of coarse moduli spaces and isotriviality for certain families
In this paper, we prove the Kobayashi hyperbolicity of the coarse moduli spaces of canonically polarized or polarized Calabi-Yau manifolds in the sense of complex \$V\$-spaces (a generalization of
Cohomology of algebraic varieties over non-archimedean fields
• Mathematics
• 2020
We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field \$K\$ based on Hrushovski-Loeser's stable completion. In parallel, we
Hodge loci and atypical intersections: conjectures
We present a conjecture on the geometry of the Hodge locus of a (graded polarizable, admissible) variation of mixed Hodge structure over a complex smooth quasi-projective base, generalizing to this
Fixed points, local monodromy, and incompressibility of congruence covers
• Mathematics
• 2020
We prove a fixed point theorem for the action of certain local monodromy groups on etale covers and use it to deduce lower bounds in essential dimension. In particular, we give more geometric proofs
Picard hyperbolicity of manifolds admitting nilpotent harmonic bundles
• Mathematics
• 2021
For a quasi-compact Kähler manifoldU endowed with a nilpotent harmonic bundlewhoseHiggs eld is injective at one point, we prove thatU is pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic, and
Lectures on the Ax–Schanuel conjecture
• Mathematics
• 2020
Functional transcendence results have in the last decade found a number of important applications to the algebraic and arithmetic geometry of varieties X admitting flat or hyperbolic uniformizations:
Integral points on algebraic subvarieties of period domains: from number fields to finitely generated fields
• Mathematics
• 2019
We show that for a variety which admits a quasi-finite period map, finiteness (resp. non-Zariski-density) of S-integral points implies finiteness (resp. non-Zariski-density) of points over all
The global asymptotic structure of period mappings
• Mathematics
• 2020
. This work is part of a project to construct completions of period mappings Φ : B → Γ \ D . A proper topological SBB-esque completion Φ 0 : B → ℘ 0 is constructed. The ﬁbres of Φ 0 are projective

## References

SHOWING 1-10 OF 42 REFERENCES
Quasi-projective moduli for polarized manifolds
• E. Viehweg
• Mathematics
Ergebnisse der Mathematik und ihrer Grenzgebiete
• 1995
This text discusses two subjects of quite different natures: construction methods for quotients of quasi-projective schemes either by group actions or by equivalence relations; and properties of
Tame topology of arithmetic quotients and algebraicity of Hodge loci
We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized \$\mathbb{Z}\$-variation of Hodge structure \$\mathbb{V}\$ on a smooth complex
On Coverings of Deligne–Mumford Stacks and Surjectivity of the Brauer Map
• Mathematics
• 2003
The paper proves a result on the existence of finite flat scheme covers of Deligne–Mumford stacks. This result is used to prove that a large class of smooth Deligne–Mumford stacks with affine moduli
Criteria for quasi-projectivity
In transcendental algebraic geometry it often happens that one gets complex spaces by analytic operations and one wishes to show that they are actually algebraic. In the case of compact complex
Model Theory with Applications to Algebra and Analysis: Complex analytic geometry in a nonstandard setting
• Mathematics
• 2008
Given an arbitrary o-minimal expansion of a real closed field R, we develop the basic theory of definable manifolds and definable analytic sets, with respect to the algebraic closure of R, along the
Quotients of non-classical flag domains are not algebraic
• Mathematics
• 2013
A flag domain D = G/V for G a simple real non-compact group G with compact Cartan subgroup is non-classical if it does not fiber holomorphically or anti-holomorphically over a Hermitian symmetric
Quotients by Groupoids
• Mathematics
• 1995
We show that if a flat group scheme acts properly, with finite stabilizers, on an algebraic space, then a quotient exists as a separated algebraic space. More generally we show any flat groupid for
Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems
0. Introduction 229 Par t I. Summary of main results 231 1. The geometric situation giving rise to variation of Hodge structure. . . . 231 2. Data given by the variation of Hodge structure 232 3.
Piecewise Weierstrass preparation and division for o-minimal holomorphic functions
Given an o-minimal structure expanding the field of reals, we show a piecewise Weierstrass preparation theorem and a piecewise Weierstrass division theorem for definable holomorphic functions. In the
Quelques espaces de modules d'intersections complètes lisses qui sont quasi-projectifs
For some values of the degrees of the equations, we show, using geometric invariant theory, that the coarse moduli space of smooth complete intersections in \$\mathbb{P}^N\$ is quasi-projective.