Baily--Borel compactifications of period images and the b-semiampleness conjecture

TL;DR

This paper proves the semi-ampleness of certain Hodge bundles and constructs functorial compactifications of period map images, advancing the understanding of moduli spaces of Calabi--Yau varieties.

Contribution

It establishes the semi-ampleness of Griffiths bundles and Hodge bundles for Calabi--Yau variations, and proves the b-semiampleness conjecture using o-minimal GAGA and Kollár's work.

Findings

Constructed functorial compactifications of period map images.

Proved Griffiths bundle is semiample in certain cases.

Confirmed the b-semiampleness conjecture for Calabi--Yau moduli.

Abstract

We address two questions related to the semiampleness of line bundles arising from Hodge theory. First, we prove there is a functorial compactification of the image of a period map of a polarizable integral pure variation of Hodge structures for which the Griffiths bundle extends amply. In particular the Griffiths bundle is semiample. We prove more generally that the Hodge bundle of a Calabi--Yau variation of Hodge structures is semiample subject to some extra conditions, and as our second result deduce the b-semiampleness conjecture and the existence of a functorial Hodge-theoretic compactification of moduli spaces of polarized Calabi--Yau varieties. The semiampleness results (and the construction of the Baily--Borel compactifications) crucially use o-minimal GAGA, and the deduction of the b-semiampleness conjecture uses work of Ambro and results of Koll\'ar on the geometry of minimal lc centers to verify the extra conditions.