<i>C</i><sup>3</sup>-Actions and algebraic threefolds with ample tangent bundle

Mabuchi, Toshiki

doi:10.1017/s0027763000017931

Cited by 14 publications

(3 citation statements)

References 21 publications

(29 reference statements)

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“…That such a characterization exists in the case of compact Kähler manifold is the famous Frankel conjecture which says that a compact Kähler manifold of positive holomorphic bisectional curvature is biholomorphic to a complex projective space. This conjecture was solved by Andreotti-Frankel [10] and Mabuchi [18] in complex dimensions two and three respectively and the general case was then solved by Mori [24], and Siu-Yau [34] independently.…”

Section: §1 Introductionmentioning

confidence: 97%

A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature

Chen¹,

Tang²,

Zhu³

2004

J. Differential Geom.

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In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional Kähler manifold M of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have Euclidean volume growth and its scalar curvature decays to zero at infinity in the average sense, then M is biholomorphic to C 2 . During the proof, we also discover an interesting gap phenomenon which says that a Kähler manifold as above automatically has quadratic curvature decay at infinity in the average sense.

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Section: §1 Introductionmentioning

confidence: 97%

A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature

Chen¹,

Tang²,

Zhu³

2004

J. Differential Geom.

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“…Prior to the complete resolution by Mori and Siu-Yau, Theorem 2 was proved by Hartshorne [10] in dimension 2, whereas Theorem 3 was proved by AndreottiFrankel [7] in dimension 2, and then in dimension 3 by Mabuchi [16] using the result of Kobayashi-Ochiai [14]. Also, it should be remarked that, since the positivity of the holomorphic bisectional curvature of a Kahler metric, or more generally a Hermitian metric, implies the positivity of the holomorphic tangent bundle, Theorem 2 proves Theorem 3 as its corollary.…”

Section: Frankel Conjecture Versus Hartshorne Conjecturementioning

confidence: 99%

Harmonic Maps in Complex Finsler Geometry

Nishikawa

2004

Variational Problems in Riemannian Geometry

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Abstract. Given a smooth map from a compact Riemann surface to a complex manifold M equipped with a strongly pseudoconvex Finsler metric F, we define a natural notion of the a-energy of the map. A harmonic map is then defined to be a critical point of the a-energy functional. Under the condition that F is weakly Kahler, we obtain the second variation formula of the functional, and prove that any a-energy minimizing harmonic map from a Riemann sphere to a weakly Kahler Finsler manifold M of positive curvature is either holomorphic or antiholomorphic. Significance of complex Finsler metrics in the study of holomorphic vector bundles, in particular its relation with the Hartshorne conjecture in complex algebraic geometry, is represented. A brief overview of complex Finsler geometry is also provided.

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“…On the other hand, we may consider Mori's theorem (originally called Hartshorne's conjecture) characterizing the projective spaces among smooth projective varieties by the ampleness of the tangent sheaf [Mor79], cf. [Har70,Mab78,MS78]. Concretely, a d-dimensional smooth projective variety X/k has ample tangent sheaf T X/k := Ω 1,∨ X/k if and only if X ∼ = P d k := Proj k[x 1 , .…”

Section: Introductionmentioning

confidence: 99%

Varieties with ample Frobenius-trace kernel

Carvajal-Rojas¹,

Patakfalvi²

2021

Preprint

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In the search of a projective analog of the Kunz's theorem and a Frobeniustheoretic analog of the Hartshorne-Mori's theorem, we investigate the positivity of the kernel of the Frobenius trace (equivalently, the negativity of the cokernel of the Frobenius endomorphism) on a smooth projective variety over an algebraically closed field of positive characteristic. For instance, such kernel is ample for projective spaces. Conversely, we show that for curves, surfaces, and threefolds the Frobenius trace kernel is ample only for Fano varieties of Picard rank 1. F = F X : X − → X denotes the (absolute) Frobenius endomorphism of X. Equivalently, let us consider the exact sequence 0 − → O X F #

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C³-Actions and algebraic threefolds with ample tangent bundle

Cited by 14 publications

References 21 publications

A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature

A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature

Harmonic Maps in Complex Finsler Geometry

Varieties with ample Frobenius-trace kernel

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C3-Actions and algebraic threefolds with ample tangent bundle

Cited by 14 publications

References 21 publications

A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature

A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature

Harmonic Maps in Complex Finsler Geometry

Varieties with ample Frobenius-trace kernel

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C³-Actions and algebraic threefolds with ample tangent bundle