On explicit birational geometry for weak Fano varieties and polarised Calabi-Yau varieties

TL;DR

This paper investigates bounds on anti-canonical volumes and stability indices for weak Fano and polarised Calabi-Yau varieties, advancing understanding of their birational geometry through explicit geometric analysis.

Contribution

It introduces methods to establish bounds on anti-canonical invariants for weak Fano and polarised Calabi-Yau varieties, extending explicit birational geometry techniques.

Findings

Derived lower bounds for anti-canonical volume.

Established upper bounds for anti-canonical stability index.

Applied methods to polarised Calabi-Yau varieties.

Abstract

Given a natural number $l$ and a weak Fano $n$-fold $X$ with $\operatorname{dim}\overline{\varphi_{-lK_X}(X)}\geq n-1$, we study the lower bound of the anti-canonical volume and the upper bound of the anti-canonical stability index. The method can also be used to give similar bounds for polarised Calabi-Yau varieties.