As on arXiv
Abstract
We use the Kähler-Ricci flow to give alternate proofs of several known Miyaoka-Yau inequalities and slope semi-stabilities, in particular, for the case of compact Kähler manifolds with semi-ample canonical line bundles and K-semistable Fano manifolds. Moreover, when the canonical line bundle is semi-ample, and the Kodaira dimension is , we prove the Miyaoka-Yau quantity is equal to the intersection of the Weil-Petersson metric and the Fubini-Study metric on the canonical model. Consequently, equality holds in the Miyaoka-Yau inequality if and only if the pluricanonical map is a holomorphic fibre bundle.
Comments: 24 pages
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Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Fano Fibrations and Twisted Kähler-Einstein Metrics I
Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Same author Alexander Bednarek · Shared terms “Weil Petersson”, “line bundle”, “canonical line”
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The Miyaoka-Yau inequality and the delta invariant for Fano varieties
Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “Miyaoka Yau”, “Yau inequality”, “Fano”
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On the Ricci curvature of Kahler-Ricci Flow
Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “semi ample”, “canonical line”, “ample canonical”
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Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow
Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Same author Alexander Bednarek · Shared terms “Fano”, “compact Kähler”
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Curvature Estimates for the Continuity Method
Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “semi ample”, “canonical line”, “ample canonical”
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The Miyaoka-Yau inequality and the delta invariant for Fano varieties II -soliton case-
Topics Kähler–Einstein & K-stability, Kähler geometry · Shared terms “Miyaoka Yau”, “Yau inequality”, “Fano manifold”
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On almost nonpositive -Ricci curvature
Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “canonical line”, “line bundle”, “compact Kähler”
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Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class
Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “semi ample”, “ample canonical”, “canonical model”
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 5A.6Where This Track Leads
The Kähler–Ricci flow, and when to come back to it.
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
- 10A.12Curve Shortening and Minimal Discs
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.