Proper modifications of generalized $p-$K\"ahler manifolds

TL;DR

This paper investigates how generalized p-K"ahler properties of complex manifolds are preserved under proper modifications, such as blow-ups, with specific results for different p-values and examples.

Contribution

It establishes conditions under which generalized p-K"ahler properties are preserved during modifications, including blow-ups at points and along submanifolds, and discusses balanced manifolds and cohomology classes.

Findings

Generalized p-K"ahler properties are conserved under point blow-ups.

For p=1, properties are preserved under blow-ups along submanifolds.

The class of compact generalized balanced manifolds is closed under modifications.

Abstract

In this paper, we consider a proper modification $f : \tilde M \to M$ between complex manifolds, and study when a generalized $p-$K\"ahler property goes back from $M$ to $\tilde M$. When $f$ is the blow-up at a point, every generalized $p-$K\"ahler property is conserved, while when $f$ is the blow-up along a submanifold, the same is true for $p=1$. For $p=n-1$, we prove that the class of compact generalized balanced manifolds is closed with respect to modifications, and we show that the fundamental forms can be chosen in the expected cohomology class. We get some partial results also in the non-compact case; finally, we end the paper with some examples of generalized $p-$K\"ahler manifolds.