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o-minimal GAGA and a conjecture of Griffiths

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Abstract

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 algebraic spaces are algebraic. Applying this to period maps, we conclude that the images of period maps are quasi-projective and that the restriction of the Griffiths bundle is ample.

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Notes

  1. It is important to include the natural scheme structure on the equivalence relation.

  2. Here we use \(|\cdot |\) to denote the vanishing locus as a definable topological space—that is, forgetting the sheaf of functions—rather than the underlying topological space as in Sect. 2.1.

  3. That is, the resulting map \(\text {Hom}(S,R)\rightarrow \text {Hom}(S,X)\times \text {Hom}(S,X)\) is the inclusion of a set-theoretic equivalence relation.

  4. That is, the image of the exceptional locus in \(X^{\textrm{def}}\).

  5. Note that Artin uses the notation

  6. Note that the analytifications of \({\mathcal {O}}_W,I\) as \({\mathcal {O}}_{W'}\)-modules are naturally the analytifications as \({\mathcal {O}}_W\)-modules.

  7. Recall this means that \(R\rightarrow {\mathcal {O}}_Z\) is a homomorphisms of sheaves of rings and that J with its induced ideal structure is of square zero.

  8. That is, a locally liftable map satisfying Griffiths transversality on the regular locus.

  9. The proof in the algebraic space case is the same as that of varieties, as it relies on the existence and uniqueness of the analytic extension and ordinary GAGA.

  10. Strictly speaking, pulling back from the stack. Alternatively, one can take a definable cover by simply-connected opens, lift to \(\Omega \), pull back and glue.

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Acknowledgements

J.T. would like to thank Vivek Shende, Jonathan Pila, and Ryan Keast for useful conversations. B.B. would like to thank Valery Alexeev and Johan de Jong for useful conversations. Y.B. would like to thank Olivier Benoist, Patrick Brosnan, and Wushi Goldring for useful conversations. The authors would also like to thank Ariyan Javanpeykar for useful remarks, specifically regarding Sect. 7.1. This paper, and in particular Sect. 2 owes a lot to the works of Peterzil and Starchenko, who initiated the study of o-minimal complex geometry. B.B. was partially supported by NSF Grant DMS-1702149. The authors are indebted to the referees for their careful reading and for greatly improving the exposition.

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, USA

    Benjamin Bakker

  2. Department of Mathematics, University of Bordeaux, Talence, France

    Yohan Brunebarbe

  3. Department of Mathematics, University of Toronto, Toronto, Canada

    Jacob Tsimerman

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  1. Benjamin Bakker
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Correspondence to Yohan Brunebarbe.

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Bakker, B., Brunebarbe, Y. & Tsimerman, J. o-minimal GAGA and a conjecture of Griffiths. Invent. math. 232, 163–228 (2023). https://doi.org/10.1007/s00222-022-01166-1

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