# Calabi-Yau threefolds fibred by high rank lattice polarized K3 surfaces

## TL;DR

This paper classifies Calabi-Yau threefolds fibred by high-rank lattice polarized K3 surfaces, showing they are determined by a map to a moduli space and constructing explicit examples with computed Hodge numbers.

## Contribution

It establishes a classification framework for such Calabi-Yau threefolds via the generalized functional invariant and provides explicit constructions for each case.

## Key findings

- Finite possibilities for polarizing lattices in smooth Calabi-Yau threefolds
- Complete determination of K3 families by maps to moduli space
- Explicit examples with computed Hodge numbers

## Abstract

We study threefolds fibred by K3 surfaces admitting a lattice polarization by a certain class of rank 19 lattices. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the appropriate K3 moduli space, which we call the generalized functional invariant. Then we show that if the threefold total space is a smooth Calabi-Yau, there are only finitely many possibilities for the polarizing lattice and the form of the generalized functional invariant. Finally, we construct explicit examples of Calabi-Yau threefolds realizing each case and compute their Hodge numbers.