Images of toric variety and amplified endomorphism of weak Fano threefolds

TL;DR

This paper investigates the properties of weak Fano threefolds with Picard rank 2, showing they are toric under certain conditions and exploring the implications of amplified endomorphisms, thereby extending existing conjectures.

Contribution

It proves that certain weak Fano threefolds with Picard rank 2 are toric, confirming special cases of conjectures by Ochetta-Wisniewski and Fakhrudding et al.

Findings

Weak Fano 3-folds of Picard rank 2 do not satisfy Bott vanishing.

Any smooth projective 3-fold of Picard rank 2 with nef -K_X and image of a toric variety is toric.

Weak Fano 3-folds with an int-amplified endomorphism are toric.

Abstract

We show that some important classes of weak Fano $3$-folds of Picard rank $2$ do not satisfy Bott vanishing. Using this we show that any smooth projective $3$-fold $X$ of Picard rank $2$ with $-K_X$ nef which is the image of a projective toric variety is toric. This proves a special case of a conjecture by Ochetta-Wisniewski, extending a corresponding previous work for Fano $3$-folds. We also show that a weak Fano $3$-fold of Picard rank $2$ having an int-amplified endomorphism is toric. This proves a special case of a conjecture by Fakhrudding, Meng, Zhang and Zhong, extending corresponding previous work for Fano $3$-folds.