The Kodaira Dimension of Lefschetz Fibrations

TL;DR

This paper proves the equivalence of complex and symplectic Kodaira dimensions for certain 4-manifolds and computes the Kodaira dimension for various Lefschetz fibrations, advancing understanding in complex and symplectic geometry.

Contribution

It establishes the equality of complex and symplectic Kodaira dimensions for smooth 4-manifolds and provides calculations for numerous Lefschetz fibrations.

Findings

Complex and symplectic Kodaira dimensions are equal for smooth 4-manifolds.

Calculated Kodaira dimensions for many Lefschetz fibrations.

Enhanced understanding of the geometric classification of 4-manifolds.

Abstract

In this note, we verify that the complex Kodaira dimension $\kappa^h$ equals the symplectic Kodaira dimension $\kappa^s$ for smooth 4-manifolds with complex and symplectic structures. We also calculate the Kodaira dimension for many Lefschetz fibrations.