Projective klt pairs with nef anti-canonical divisor

Frédéric Campana, Junyan Cao, Shin Ichi Matsumura

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

In this paper, we study projective klt pairs (X, ∆) with nef anti-log canonical divisor −(KX + ∆) and their maximal rationally connected fibration ψ: X −−• Y. We prove that the numerical dimension of −(KX +∆) on X coincides with that of −(KXy +∆Xy ) on a general fiber Xy of ψ: X −−• Y, which is an analogue of Ejiri–Gongyo’s result formulated for the Kodaira dimension. As a corollary, we obtain a relation between the positivity of the anti-canonical divisor and the rational connectedness, which provides a sharper estimate than that in Hacon–McKernan’s question. Moreover, in the case of X being smooth, we show that X admits a “holomorphic” maximal rationally connected fibration to a smooth projective variety Y with numerically trivial canonical divisor, and also that this is locally trivial with respect to the pair (X, ∆), which generalizes Cao–Höring’s structure theorem to the case of klt pairs. Finally, we consider slope rationally connected quotients of (X, ∆) and obtain a structure theorem for projective orbifold surfaces.

Original languageEnglish
Pages (from-to)430-464
Number of pages35
JournalAlgebraic Geometry
Volume8
Issue number4
DOIs
Publication statusPublished - 2021

Keywords

  • Beauville{Bogomolov decomposition
  • klt pairs
  • MRC fibrations
  • nef anti-canonical divisors
  • numerical atness
  • numerical dimension
  • positivity of direct image sheaves
  • rational connectedness
  • singular Hermitian metrics
  • slope rationally connected quotients

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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