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

Abstract

In this paper, we study the relative anti-canonical divisor −K X/Y of an algebraic fiber space φ : X → 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-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 étale cover of Y . Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.

The Nonvanishing problem for varieties with nef anticanonical bundle

“…We now come to the following very recent structure theorem from [MW21,EIM20] for varieties with nef anticanonical class. It is fundamental for our work, and many of our proofs use it as a starting point.…”

Open problems on structure of positively curved projective varieties

“…Toward the above problem, it seems to be the first step to consider a surjective morphism φ : X → Y with the pseudo-effective relative anti-canonical −K X/Y . We in [EIM20] systematically studied a relation between the geometric structure of φ and positivity conditions on −K X/Y . As a result, we showed that the non-nef locus B − (−K X/Y ) is empty or dominant over Y (see [EIM20] for more details).…”

“…We in [EIM20] systematically studied a relation between the geometric structure of φ and positivity conditions on −K X/Y . As a result, we showed that the non-nef locus B − (−K X/Y ) is empty or dominant over Y (see [EIM20] for more details). This implies that the structure of φ : X → Y is restricted even when −K X/Y is pseudo-effective.…”