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Positivity preserving along a flow over projective bundle
Differential Geometry · Submitted 30 Jan 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ is preserved along this flow under the null eigenvector assumption.
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The abstract, sentence by sentence
In this paper, we introduce a flow over the projective bundle $p:P(E^*)\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and K\"ahler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ is preserved along this flow under the null eigenvector assumption. As applications, we prove that the semipositivity is preserved along the this flow if the base manifold $M$ is a curve, which implies that the Griffiths semipositivity is preserved along the Hermitian-Yang-Mills flow over a curve. And we also reprove that the nonnegativity of holomorphic bisectional curvature is preserved under K\"ahler-Ricci flow.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we introduce a flow over the projective bundle $p:P(E^*)\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and K\"ahler-Ricci flow. | contribution 1.00 | 0.91 | 0.06 | asserted 1.00 |
| s1 | We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ is preserved along this flow under the null eigenvector assumption. | result 0.99 | 0.96 | 0.76 | asserted 1.00 |
| s2 | As applications, we prove that the semipositivity is preserved along the this flow if the base manifold $M$ is a curve, which implies that the Griffiths semipositivity is preserved along the Hermitian-Yang-Mills flow over a curve. | result 0.87 | 0.95 | 0.18 | asserted 1.00 |
| s3 | And we also reprove that the nonnegativity of holomorphic bisectional curvature is preserved under K\"ahler-Ricci flow. | result 0.92 | 0.88 | 0.08 | asserted 1.00 |
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The paper
Original abstract
In this paper, we introduce a flow over the projective bundle $p:P(E^*)\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and K\"ahler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ is preserved along this flow under the null eigenvector assumption. As applications, we prove that the semipositivity is preserved along the this flow if the base manifold $M$ is a curve, which implies that the Griffiths semipositivity is preserved along the Hermitian-Yang-Mills flow over a curve. And we also reprove that the nonnegativity of holomorphic bisectional curvature is preserved under K\"ahler-Ricci flow.
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How the literature uses it
3 arXiv papers cite this in 7 sentences
Papers citing it · 6 sentences without a confident label
Cites it as background 1 paper
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For more details we refer to [ref] .
Complex Finsler vector bundles with positive Kobayashi curvature · Complex Finsler vector bundle · 1.00
What it cites · 3 sentences without a confident label
Cites as background 1 paper
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For more details we refer to [ref] .
Geodesic-Einstein metrics and nonlinear stabilities · Preliminaries · 1.00
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