The number of smooth varieties in an MMP on a 3-fold of Fano type

TL;DR

This paper establishes a bound on the number of smooth varieties encountered during a minimal model program on a Fano type threefold, linking it to a specific cohomology dimension, and explores a partial converse to Kodaira vanishing.

Contribution

It introduces a bound on the count of smooth varieties in a D-MMP on Fano type threefolds based on cohomology, and proves a partial converse to Kodaira vanishing for such divisors.

Findings

Bound on smooth varieties during D-MMP is 1 + h^1(X, 2D).

Partial converse to Kodaira vanishing for movable divisors on Fano type threefolds.

Cohomological conditions influence the MMP process.

Abstract

In this paper, we prove that for a threefold of Fano type $X$ and a movable $\mathbb{Q}$-Cartier Weil divisor $D$ on $X$, the number of smooth varieties that arise during the running of a $D$-MMP is bounded by $1 + h^1(X, 2D)$. Additionally, we prove a partial converse to the Kodaira vanishing theorem for a movable divisor on a threefold of Fano type.