Projective klt pairs with nef anti-canonical divisor

Frédéric Campana, Junyan Cao, Shin Ichi Matsumura

Research output: Contribution to journalArticlepeer-review


In this paper, we study a projective klt pair (X,Δ) with the nef anti-log canonical divisor-(KX + Δ) and its maximally rationally connected fibration Ψ : X → Y . We prove that the numerical dimension of the anti-log canonical divisor -(KX+ Δ) on X coincides with that of the anti-log canonical divisor -(KXy+ ΔXy) on a general fiber Xyof Ψ X → Y , which is an analogue of Ejiri-Gongyo's result formulated for the Kodaira dimension. As a corollary, we reveal a relation between positivity of the anti-canonical divisor and the rationally connectedness, which gives a sharper estimate than the question posed by Hacon-McKernan. Moreover, in the case of X being smooth, we show that a maximally rationally connected fibration Ψ : X → Y can be chosen to be a morphism to a smooth projective variety Y with numerically trivial canonical divisor, and further that it is locally trivial with respect to the pair (X, Δ), which can be seen as a generalization of Cao-Höring's structure theorem to klt pair cases. Finally, we study the structure of the slope rationally connected quotient for a pair (X, Δ) with -(KX+Δ) nef, and obtain a structure theorem for projective orbifold surfaces.

MSC Codes Primary 14D06, Secondary 14E30, 32J25

Original languageEnglish
JournalUnknown Journal
Publication statusPublished - 2019 Oct 14


  • KLT pairs
  • Mrc fibrations
  • Nef anti-canonical divisors
  • Numerical dimension
  • Numerically flatness
  • Positivity of direct image sheaves
  • Rationally connectedness
  • Singular hermitian metrics
  • Slope rationally connected quotients

ASJC Scopus subject areas

  • General

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