Bott-Chern complexity of K\"ahler pairs

TL;DR

This paper introduces the Bott-Chern complexity as a new invariant for compact K"ahler pairs, analyzing its properties especially in Calabi-Yau cases, and establishes bounds related to projectivity and singular surfaces.

Contribution

It defines the Bott-Chern complexity for K"ahler pairs and investigates its properties, including bounds and optimality in non-projective Calabi-Yau cases.

Findings

Bott-Chern complexity is non-negative for Calabi-Yau pairs.

Complexity is at least three for non-projective Calabi-Yau pairs.

The minimal complexity value is achieved by certain singular non-projective K3 surfaces.

Abstract

We introduce the Bott-Chern complexity of a compact K\"ahler pair $(X,B)$. This invariant compares $\dim(X)$, $\dim H^{1,1}_{\rm BC}(X)$ and the sum of the coefficients of $B$. When $(X,B)$ is Calabi-Yau, we show that its Bott-Chern complexity is non-negative. We prove that the Bott-Chern complexity of a Calabi-Yau compact K\"ahler pair $(X,B)$ is at least three whenever $X$ is not projective. Furthermore, we show this value is optimal and is achieved by certain singular non-projective K3 surfaces.