Divisorial Log Terminal Pair

A divisorial log terminal pair, abbreviated dlt pair, is a log canonical pair admitting a log resolution on which every exceptional prime divisor has strictly positive log discrepancy. The boundary is effective and has coefficients at most one. A log resolution is a proper birational morphism from a smooth variety on which the strict transform of the boundary together with the exceptional divisors has simple normal crossing support. For example, a smooth variety with a simple normal crossing boundary of coefficients in is divisorial log terminal (Kollár and Mori 1998).

OpenAI (2026) announced supported boundary lifting for a normal projective rationally factorial divisorial log terminal pair over an algebraically closed field of characteristic zero. Put , and suppose a nonzero effective Cartier divisor , for a sufficiently divisible positive integer , has support contained in the coefficient-one part of . If is semiample on its whole reduced support, the asserted conclusion is . The support inclusion may be strict, and generation on separate components without gluing is insufficient.

Cao and Păun (2026) prove a boundary extension theorem for a complex projective Q-factorial variety with divisorial log terminal boundary . Assume that is a nef divisor and is a pseudo-effective divisor for some . On a log resolution , write , where , is reduced, are rational, the coefficients of are less than 1, and is -exceptional. The divisors have no common components in pairs and their union has simple normal crossing support. For every sufficiently divisible integer , the restriction

is surjective. The supported-boundary application gives the restricted abundance result in the log abundance conjecture.