On coupled constant scalar curvature Kähler metrics

Datar, Ved; Pingali, Vamsi Pritham

doi:10.48550/arxiv.1901.10454

Cited by 3 publications

(5 citation statements)

References 21 publications

(49 reference statements)

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“…In the particular case when f is a constant this comprises a complex Monge-Ampère equation coupled to a twisted cscK equation, and it is precisely of the type studied by Datar and Pingali [8].…”

Section: 3mentioning

confidence: 99%

Deformed Hermitian Yang-Mills connections, extended gauge group and scalar curvature

Schlitzer,

Stoppa

2019

Preprint

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The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background Kähler metric, to be constant. In this paper we introduce and study dHYM equations with variable Kähler metric. These are coupled equations involving both the Lagrangian phase and the radius function, at the same time. They are obtained by using the extended gauge group to couple the moment map interpretation of dHYM connections, due to Collins-Yau and mirror to Thomas' moment map for special Lagrangians, to the Donaldson-Fujiki picture of scalar curvature as a moment map. As a consequence one expects that solutions should satisfy a mixture of K-stability and Bridgeland-type stability. In special limits, or in special cases, we recover the Kähler-Yang-Mills system of Álvarez-Cónsul, Garcia-Fernandez and García-Prada, and the coupled Kähler-Einstein equations of Hultgren-Witt Nyström. After establishing several general results we focus on the equations and their large/small radius limits on abelian varieties, with a source term, following ideas of Feng and Székelyhidi.

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Section: 3mentioning

confidence: 99%

Deformed Hermitian Yang-Mills connections, extended gauge group and scalar curvature

Schlitzer,

Stoppa

2019

Preprint

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“…Sasakian analogues were also studied by Futaki-Zhang [FZ18]. Moreover, a moment map interpretation for CKE metrics was established very recently by Datar-Pingali [DP19].…”

Section: Introductionmentioning

confidence: 98%

“…One of the motivation for CKE metrics comes from stabilities. Indeed, Datar-Pingali [DP19] recently showed that CKE metrics have an infinite dimensional moment map/GIT interpretation which interpolates between polarized manifolds and holomorphic vector bundles over them. This result strongly indicates a potential application to construct their good moduli space.…”

Section: Introductionmentioning

confidence: 99%

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Ricci Iteration for Coupled Kähler–Einstein Metrics

Takahashi

2019

International Mathematics Research Notices

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We study the quantization of coupled Kähler-Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, so called the "balanced metrics". We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern class, we introduce some algebro-geometric obstruction which interpolates between the Donaldson-Futaki invariant and Chow weight. Then we show the existence and weak convergence of balanced metrics on CKE manifolds under the vanishing of this obstruction. Moreover, restricted to the case when the automorphism group is discrete, we also discuss approximate solutions and a gradient flow method towards the smooth convergence.

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“…Hultgren and Witt Nyström [16] proved the existence of the solution for λ = −1 under the condition c 1 (M ) < 0 extending [23] and [1], and there are many interesting results for λ = 1 under the condition c 1 (M ) > 0 including attempts to extend [3] and [22]. Further studies of coupled Kähler-Einstein metrics have been done in [19], [15], [13], [5], [20], [4], [21], [18].…”

Section: Introductionmentioning

confidence: 99%