TY - JOUR

T1 - Projective klt pairs with nef anti-canonical divisor

AU - Campana, Frédéric

AU - Cao, Junyan

AU - Matsumura, Shin Ichi

N1 - Funding Information:
The third author would like to thank Professor Yoshinori Gongyo for fruitful discussions on the work reported in [EG19]. He would also like to express his thanks to the members of the Institut de Math?matiques de Jussieu - Paris Rive gauche for their hospitality during his stay there.
Funding Information:
The third author is supported by Grant-in-Aid for Young Scientists (A) ♯17H04821, Grant-in-Aid for Scientific Research (B) ♯21H00976, Fostering Joint International Research (A) ♯19KK0342 from JSPS, and the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers. This work is partially supported by the Agence Nationale de la Recherche through a “Convergence de Gromov–Hausdorff en géométrie kählérienne” (ANR-GRACK) grant.
Publisher Copyright:
© Foundation Compositio Mathematica 2021. This article is distributed with Open Access under the terms of the Creative Commons Attribution Non-Commercial License, which permits non-commercial reuse, distribution, and reproduction in any medium, provided that the original work is properly cited. For commercial re-use, please contact the Foundation Compositio Mathematica.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Beauville{Bogomolov decomposition

KW - klt pairs

KW - MRC fibrations

KW - nef anti-canonical divisors

KW - numerical atness

KW - numerical dimension

KW - positivity of direct image sheaves

KW - rational connectedness

KW - singular Hermitian metrics

KW - slope rationally connected quotients

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U2 - 10.14231/AG-2021-013

DO - 10.14231/AG-2021-013

M3 - Article

AN - SCOPUS:85100265045

VL - 8

SP - 430

EP - 464

JO - Algebraic Geometry

JF - Algebraic Geometry

SN - 2313-1691

IS - 4

ER -