Notes on the weak positivity theorems

Abstract

We discuss the (twisted) weak positivity theorem. We also treat some applications.Date: 2015/6/30, version 0.54.

Nef anti-canonical divisors and rationally connected fibrations

“…so we find this sheaf is weakly positive by [5] or [13]. Take m 0 ∈ Z >0 such that O Y (m 0 L) ⊗ (f * O X (A)) * is globally generated.…”

“…The main ingredient of the proof is the so-called weak positivity theorem developed by several people including Fujita, Kawamata and Viehweg [Fuj78, Kaw81, Vie83]. We employ the log version of this theorem established in [Cam04, Eji17, Fuj17, Pat14], which concerns the positivity of the direct image sheaves of log pluricanonical bundles . Using this, we first show that given a -divisor on , if is nef, then is pseudo-effective (Theorem 3.1).…”

Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension

“…Further note that our proof of Theorem 1.2 is primarily algebraic. That is, we obtain our positivity results, from which Theorem 1.2 is deduced, algebraically, starting from the semi-positivity results of Fujino [Fuj12,Fuj14a]. Hence, our approach has a good chance to be portable to positive characteristic when the appropriate semi-positivity results (and other ingredients such as the mmp) become available in that setting.…”

Potentially Du Bois spaces

“…For instance, they form a natural class of singularities where Kodaira type vanishing theorems hold [Ste85,KSS10,Kov11,Kov13b,Pat13]. Other recent applications include extension theorems [GKKP11], positivity theorems [Sch12], categorical resolutions [Lun12], log canonical compactifications [HX13], semipositivity [FFS13], and injectivity theorems [Fuj13]. Besides applications in the minimal model program, Du Bois singularities play an important role in moduli theory as well [KK10,Kov13b].…”