TY - JOUR

T1 - Projective klt pairs with nef anti-canonical divisor

AU - Campana, Frédéric

AU - Cao, Junyan

AU - Matsumura, Shin Ichi

N1 - Publisher Copyright:
Copyright © 2019, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2019/10/14

Y1 - 2019/10/14

N2 - 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

AB - 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

KW - KLT pairs

KW - Mrc fibrations

KW - Nef anti-canonical divisors

KW - Numerical dimension

KW - Numerically flatness

KW - Positivity of direct image sheaves

KW - Rationally connectedness

KW - Singular hermitian metrics

KW - Slope rationally connected quotients

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