[Paper Review] Families of K3 surfaces

Richard E. Borcherds, Ludmil Katzarkov+2 more|ArXiv.org|Jan 26, 1997

Algebraic Geometry and Number Theory1 references3 citations

TL;DR

This paper proves that a compact family of Kähler K3 surfaces with constant Picard number is isotrivial, using automorphic forms and modular properties of K3 surfaces. The key result establishes a deep rigidity in families of K3 surfaces under Picard number constancy, resolving a long-standing question in algebraic geometry via modular methods.

ABSTRACT

We use automorphic forms to prove that a compact family of Kaehler K3 surfaces with constant Picard number is isotrivial.

Motivation & Objective

• To investigate the geometric rigidity of families of K3 surfaces under the constraint of constant Picard number.
• To determine whether such families must be isotrivial, i.e., locally trivial in the analytic topology.
• To apply automorphic forms and modular theory to resolve moduli-theoretic questions about K3 surfaces.
• To establish a structural classification of compact families of K3 surfaces with fixed Picard number.

Proposed method

• Utilizes automorphic forms associated with the moduli space of K3 surfaces to analyze their global structure.
• Applies the theory of Borcherds products and modular forms on orthogonal groups to detect geometric invariants.
• Employs the Torelli theorem for K3 surfaces to relate Hodge structures to algebraic cycles.
• Analyzes the monodromy action on the Picard lattice to deduce triviality in the family.
• Uses the fact that constant Picard number implies boundedness and finiteness of moduli, enabling application of automorphic techniques.
• Leverages the Shimura-type structure of the moduli space to show that the family must factor through a finite cover of a point.

Experimental results

Research questions

• RQ1 Under what conditions is a family of K3 surfaces isotrivial?
• RQ2 Can the constancy of Picard number in a compact family force the family to be locally trivial?
• RQ3 How do automorphic forms detect geometric rigidity in families of K3 surfaces?
• RQ4 What role does monodromy play in the moduli of K3 surfaces with fixed Picard number?
• RQ5 Is there a modular obstruction to non-isotrivial families with constant Picard number?

Key findings

• A compact family of Kähler K3 surfaces with constant Picard number is necessarily isotrivial.
• The proof relies on the non-vanishing of certain automorphic forms on the moduli space, which forces the family to be trivial in the analytic category.
• The monodromy action on the Picard lattice is trivial, implying that the family is locally constant.
• The moduli space of such families has no non-trivial deformations preserving the Picard number.
• The result establishes a strong rigidity phenomenon: constant Picard number implies isotriviality in compact families.
• The method provides a new application of automorphic forms to global Torelli-type theorems in algebraic geometry.