2019/10/15 by Frédéric Campana, Campana, Frédéric, Junyan Cao +4 · 8 citations
Mathematics · #32J25 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Primary 14D06 #Secondary 14E30 #math.AG #math.CV #math.DG #msc:14D06 #msc:14E30 #msc:32J25
paper · pdf · doi:10.48550/arxiv.1910.06471
36 pages, comments are welcome
arxiv created 2019/10/15 · openalex publication_date 2019/10/15 · arxiv updated 2019/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \dashrightarrow 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 Xy of ψ: X \dashrightarrow 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 rational connectedness, which gives a sharper estimate than the question posed by Hacon-\mathrmMcKernan. 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.