Bott-Chern complexity of Kähler pairs

TL;DR

This work introduces Bott-Chern complexity $c_{ m BC}(X,B)$ for compact Kähler pairs and develops a framework linking it to projectivity via the MRC fibration. It proves non-negativity of the fine complexity for Calabi–Yau pairs and provides projectivity criteria in terms of both fine and Bott-Chern complexities, showing that very small values force projectivity or toric structure, and constraining the MRC base. The main Bott-Chern result shows that $c_{ m BC}(X,B)<3$ implies projectivity, while $c_{ m BC}(X,B)=3$ with $X$ non-projective leads to a birational model exhibiting a base K3 surface with controlled $H^{1,1}_{ m BC}$ and toric fibers; this bound is demonstrated to be sharp using singular non-projective K3 surfaces. The paper also furnishes explicit non-projective K3 examples with $c_{ m BC}=3$, highlighting the geometry of Calabi–Yau Kähler pairs beyond the projective setting and extending toric/cluster-type connections to the Kähler realm.

Abstract

We introduce the Bott-Chern complexity of a compact Kähler 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 Kähler 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.

Theorems & Definitions (20)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2: c.f. DH20
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Lemma 2.6