Geodesic-Einstein metrics and nonlinear stabilities

Han, Feng; Liu, Kefeng; Wan, Xueyuan

doi:10.1090/tran/7658

Cited by 7 publications

(8 citation statements)

References 27 publications

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“…[24, page 82]). Combining with [19,Theorem 0.3], it shows that the existence of geodesic-Einstein metrics is equivalent to the existence of Hermitian-Einstein metrics, which is also equivalent to polystability of the holomorphic vector bundle by the Donaldson-Uhlenbeck-Yau Theorem. In the sense of this, we may identify a Hermitian-Einstein vector bundle E with a projective bundle pair (P (E), O P (E) (1)) of which admits a geodesic-Einstein metric.…”

Section: Introductionmentioning

confidence: 88%

“…In this section, we will recall some basic definitions and facts on geodesic-Einstein metrics of a relative ample line bundle over holomorphic fibration, and also recall Berndtsson's curvature formula of direct image bundles. For more details one may refer to [3,4,5,19,37].…”

Section: Preliminariesmentioning

confidence: 99%

“…For any smooth function φ on X , we denote where N k α = φ αj φj k . By a routine computation, one can show that { δ δz α } 1≤α≤m spans a well-defined horizontal subbundle of T X (see [19,Section 1]).…”

Section: Preliminariesmentioning

confidence: 99%

“…4.4. For the case of holomorphic vector bundle E → M , the geodesic-Einstein metric is equivalent to Finsler-Einstein metric (see[19, Lemma 3.6]). With the assumption of existence…”

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confidence: 99%

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Remarks on the geodesic-Einstein metrics of a relative ample line bundle

Wan¹,

Wang

2020

Math. Z.

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Let π : X → M be a holomorphic fibration over a compact complex manifold M with compact fibers, L be a relative ample line bundle over X . In this paper, we prove that the pair (X , L) is nonlinear semistable if the Donaldson type functional L(·, ψ) is bounded from below and the long time existence of the flow (0.1). For the case of trωc(φ) ≥ 0, we give a sufficient condition deg ω π * (L + K X /M ) = 0 for the existence of geodesic-Einstein metric. We also introduce the definitions of S-class and C-class inspired by the Segre class and Chern class, and find two inequalities in terms of S-forms and C-forms if there exists a geodesic-Einstein metric. If L is ample, the positivity of S-forms is proved. If moreover, the line bundle L admits a geodesic-Einstein metric and dim M = 2, the second C-class C2(L) is proved to be numerical positive. We also prove the equivalence between the geodesic-Einstein metrics on a relative ample line bundle and the Hermitian-Einstein metrics on a quasi-vector bundle A 0,0 . Lastly, we discuss some examples on geodesic-Einstein metrics.

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Section: Introductionmentioning

confidence: 88%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…4.4. For the case of holomorphic vector bundle E → M , the geodesic-Einstein metric is equivalent to Finsler-Einstein metric (see[19, Lemma 3.6]). With the assumption of existence…”

mentioning

confidence: 99%

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Remarks on the geodesic-Einstein metrics of a relative ample line bundle

Wan¹,

Wang

2020

Math. Z.

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“…Remark 2: The Poisson-Kähler condition is in general stronger than the geodesic-Einstein condition in [21,44]. It is known that (see [4,41]) every relative Kähler fibration is Poisson-Kähler locally in the following sense: for every t ∈ B, there exists a small open neighborhood U of t such that the fibration from p −1 (U) to U possesses a Poisson-Kähler structure.…”

Section: Introductionmentioning

confidence: 99%

Poisson--Kähler fibration I: curvature of the base manifold

Wan¹,

Wang²

2019

Preprint

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We start from a finite dimensional Higgs bundle description of a result of Burns on negative curvature property of the space of complex structures, then we apply the corresponding infinite dimensional Higgs bundle picture and obtain a precise curvature formula of a Weil-Petersson type metric for general relative Kähler fibrations. In particular, our curvature formula implies a Burns type negative curvature property of the base manifold for a special class of maximal variation Kähler fibrations (named Poisson-Kähler fibrations).

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The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle

Wan

Zhang

2021

Geom Dedicata

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Let $$\pi :\mathcal {X}\rightarrow M$$ π : X → M be a holomorphic fibration with compact fibers and L a relatively ample line bundle over $$\mathcal {X}$$ X . We obtain the asymptotic of the curvature of $$L^2$$ L 2 -metric and Qullien metric on the direct image bundle $$\pi _*(L^k\otimes K_{\mathcal {X}/M})$$ π ∗ ( L k ⊗ K X / M ) up to the lower order terms than $$k^{n-1}$$ k n - 1 , for large k. As an application we prove that the analytic torsion $$\tau _k(\bar{\partial })$$ τ k ( ∂ ¯ ) satisfies $$\partial \bar{\partial }\log (\tau _k(\bar{\partial }))^2=o(k^{n-1})$$ ∂ ∂ ¯ log ( τ k ( ∂ ¯ ) ) 2 = o ( k n - 1 ) , where n is the dimension of fibers.

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Geodesic-Einstein metrics and nonlinear stabilities

Cited by 7 publications

References 27 publications

Remarks on the geodesic-Einstein metrics of a relative ample line bundle

Remarks on the geodesic-Einstein metrics of a relative ample line bundle

Poisson--Kähler fibration I: curvature of the base manifold

The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle

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