Shokurov's boundary property

TL;DR

This paper proves Shokurov's BP Conjecture, showing that certain codimension one log structures on a base variety from fiber spaces uniquely glue together on a birational model, with applications to adjunction, FGA-algebras, and moduli parts.

Contribution

It establishes the uniqueness of a global log structure from fiber-induced local structures, confirming a key conjecture in birational geometry.

Findings

Proves the existence and uniqueness of the global log structure (Shokurov's BP Conjecture).

Demonstrates invariance of FGA-algebras under restriction to lc centers.

Provides insights on the moduli part of parabolic fiber spaces.

Abstract

For a birational analogue of minimal elliptic surfaces X/Y, the singularities of the fibers define a log structure in codimension one on Y. Via base change, we have a log structure in codimension one on Y', for any birational model Y' of Y. We show that these codimension one log structures glue to a unique log structure, defined on some birational model of Y (Shokurov's BP Conjecture). We have three applications: inverse of adjunction for the above mentioned fiber spaces, the invariance of Shokurov's FGA-algebras under restriction to exceptional lc centers, and a remark on the moduli part of parabolic fiber spaces.