On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces

TL;DR

This paper investigates the positivity and base loci of the relative anti-canonical divisor in algebraic fiber spaces, establishing new relations and algebraic proofs for existing conjectures, and exploring the structure of such fiber spaces.

Contribution

It provides new algebraic insights into the base loci of the relative anti-canonical divisor and offers an algebraic proof of a key equality previously proven analytically.

Findings

All base loci are located in the horizontal direction unless empty.

An algebraic proof of Campana--Cao--Matsumura's equality on Hacon--$ m{M^c}$Kernan's question.

Partial progress on the structure of fiber spaces with semi-ample relative anti-canonical divisors.

Abstract

In this paper, we study the relative anti-canonical divisor $-K_{X/Y}$ of an algebraic fiber space $\phi: X \to Y$, and we reveal relations among positivity conditions of $-K_{X/Y}$, certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana--Cao--Matsumura's equality on Hacon--$\rm{M^c}$Kernan's question, whose original proof depends on analytics methods. The third result partially solves the question which asks whether algebraic fiber spaces with semi-ample relative anti-canonical divisor actually have a product structure via the base change by an appropriate finite \'etale cover of $Y$. Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.