Families of conic Kähler–Einstein metrics

Guenancia, Henri

doi:10.1007/s00208-018-1769-6

Math. Ann.

2018

DOI: 10.1007/s00208-018-1769-6

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Families of conic Kähler–Einstein metrics

Henri Guenancia

Abstract: Let p : X → Y be an holomorphic surjective map between compact Kähler manifolds and let D be an effective divisor on X with generically simple normal crossings support and coefficients in (0, 1). Provided that the adjoint canonical bundle KX y + Dy of the generic fiber is ample, we show that the current obtained by glueing the fiberwise conic Kähler-Einstein metrics on the regular locus of the fibration is positive. Moreover, we prove that this current is bounded outside the divisor and that it extends to a po… Show more

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Cited by 5 publications

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“…Let F be the general fiber of f . Let α X be a Kähler class on X and let D be a Gue16] confirm the above question by studying the variation of Kähler-Einstein metrics (based on [Sch12]). In our article, we confirm Question 1.7 in two special cases: Theorem 3.4 and Theorem 5.2 by using the positivity of the fibrewise Bergman kernel which is established in [BP08,BP10].…”

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confidence: 99%

Rational curves on compact Kähler manifolds

Cao

Höring

2020

J. Differential Geom.

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Mori's theorem yields the existence of rational curves on projective manifolds such that the canonical bundle is not nef. In this paper we study compact Kähler manifolds such that the canonical bundle is pseudoeffective, but not nef. We present an inductive argument for the existence of rational curves that uses neither deformation theory nor reduction to positive characteristic. The main tool for this inductive strategy is a weak subadjunction formula for lc centres associated to certain big cohomology classes.

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confidence: 99%

Rational curves on compact Kähler manifolds

Cao

Höring

2020

J. Differential Geom.

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Positivity of direct images with a Poincaré type twist

Naumann

2022

Forum of Mathematics, Sigma

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We consider a holomorphic family $f:\mathscr {X} \to S$ of compact complex manifolds and a line bundle $\mathscr {L}\to \mathscr {X}$ . Given that $\mathscr {L}^{-1}$ carries a singular hermitian metric that has Poincaré type singularities along a relative snc divisor $\mathscr {D}$ , the direct image $f_*(K_{\mathscr {X}/S}\otimes \mathscr {D} \otimes \mathscr {L})$ carries a smooth hermitian metric. If $\mathscr {L}$ is relatively positive, we give an explicit formula for its curvature. The result applies to families of log-canonically polarized pairs. Moreover, we show that it improves the general positivity result of Berndtsson-Păun in a special situation of a big line bundle.

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Positivity of direct images of fiberwise Ricci-flat metrics on Calabi-Yau fibrations

Braun¹,

Choi²,

Schumacher³

2021

Trans. Amer. Math. Soc.

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Let X X be a Kähler manifold which is fibered over a complex manifold Y Y such that every fiber is a Calabi-Yau manifold. Let ω \omega be a fixed Kähler form on X X . By Yau’s theorem, there exists a unique Ricci-flat Kähler form ω K E , y \omega _{KE,y} on each fiber X y X_y for y ∈ Y y\in Y which is cohomologous to ω | X y \omega \vert _{X_y} . This family of Ricci-flat Kähler forms ω K E , y \omega _{KE,y} induces a smooth ( 1 , 1 ) (1,1) -form ρ \rho on X X under a normalization condition. In this paper, we prove that the direct image of ρ n + 1 \rho ^{n+1} is positive on the base Y Y . We also discuss several byproducts including the local triviality of families of Calabi-Yau manifolds.

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Families of conic Kähler–Einstein metrics

Cited by 5 publications

References 24 publications

Rational curves on compact Kähler manifolds

Rational curves on compact Kähler manifolds

Positivity of direct images with a Poincaré type twist

Positivity of direct images of fiberwise Ricci-flat metrics on Calabi-Yau fibrations

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