[Paper Review] Period Mappings and Ampleness of the Hodge line bundle

Mark Green, Phillip Griffiths+2 more|arXiv (Cornell University)|Aug 31, 2017

Algebraic Geometry and Number Theory36 references10 citations

TL;DR

This paper proposes a Hodge-theoretic completion of the period map, conjectured to be a compact complex analytic variety. When the period map's image has dimension ≤2, the augmented Hodge line bundle extends to an ample line bundle on the completion, proving it is a projective algebraic compactification analogous to the Satake-Baily-Borel compactification.

ABSTRACT

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. The conjecture is proved when the image of the period map has dimension 1,2. Assuming the conjecture holds, we then prove that the augmented Hodge line bundle extends to an ample line bundle on the completion. In particular, the completion is a projective algebraic variety that compactifies the image, analogous to the Satake-Baily-Borel compactification.

Motivation & Objective

• To develop a Hodge-theoretic completion of the period map that extends the image of the period map into a compact complex analytic variety.
• To conjecture that this completion admits the structure of a compact complex analytic space.
• To establish that the augmented Hodge line bundle extends to an ample line bundle on the completion under the conjecture.
• To show that the completion is a projective algebraic variety, generalizing the Satake-Baily-Borel compactification in low-dimensional cases.
• To provide a Hodge-theoretic framework for compactifying moduli spaces of polarized Hodge structures.

Proposed method

• Define a completion of the period map using Hodge-theoretic data, particularly focusing on limiting mixed Hodge structures.
• Conjecture that this completion is a compact complex analytic variety, based on geometric and analytic properties of the period domain.
• Use the theory of augmented Hodge line bundles to analyze their extension properties across the boundary of the completion.
• Prove the conjecture in the case where the image of the period map has dimension 1 or 2, relying on analytic and algebraic geometry techniques.
• Apply results from Hodge theory and complex geometry to show that the extended Hodge line bundle is ample on the completion.
• Establish that the completion is projective by leveraging the ampleness of the Hodge line bundle and standard algebraic geometry criteria.

Experimental results

Research questions

• RQ1 Can the period map be completed via a Hodge-theoretic construction to yield a compact complex analytic variety?
• RQ2 Does the augmented Hodge line bundle extend to an ample line bundle on this completion?
• RQ3 Is the completion a projective algebraic variety when the period map's image has dimension ≤2?
• RQ4 How does this Hodge-theoretic completion compare to the classical Satake-Baily-Borel compactification?
• RQ5 What conditions ensure the ampleness of the Hodge line bundle on the compactified period domain?

Key findings

• The Hodge-theoretic completion of the period map is conjectured to be a compact complex analytic variety.
• The conjecture is proven true when the image of the period map has dimension 1 or 2.
• Under this conjecture, the augmented Hodge line bundle extends to an ample line bundle on the completion.
• The completion is shown to be a projective algebraic variety, providing a Hodge-theoretic compactification.
• The construction generalizes the Satake-Baily-Borel compactification in low-dimensional cases.
• The ampleness of the Hodge line bundle on the completion ensures the existence of a projective embedding.