Betti numbers in three dimensional minimal model program

TL;DR

This paper establishes a bound on the variation of the third Betti number during the minimal model program for smooth projective threefolds, depending solely on the Picard number.

Contribution

It provides a new bound on the third Betti number's variance in the minimal model program, linking it to the Picard number of the threefold.

Findings

Bound on the variance of the third Betti number established.

The bound depends only on the Picard number.

Results apply to smooth projective threefolds.

Abstract

Let $X$ be a smooth projective threefold. We prove that, among the process of minimal model program, the variance of the third Betti number can be bounded by some integer depends only on the Picard number of $X$.