Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: The canonical structure of the limit
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Benjamin Sibley
Abstract
In the following article we study the limiting properties of the Yang–Mills flow associated to a holomorphic vector bundle E over an arbitrary compact Kähler manifold (X,ω). In particular we show that the flow is determined at infinity by the holomorphic structure of E. Namely, if we fix an integrable unitary reference connection A0 defining the holomorphic structure, then the Yang–Mills flow with initial condition A0 converges (away from an appropriately defined singular set) in the sense of the Uhlenbeck compactness theorem to a holomorphic vector bundle E∞, which is isomorphic to the associated graded object of the Harder–Narasimhan–Seshadri filtration of (E,A0). Moreover, E∞ extends as a reflexive sheaf over the singular set as the double dual of the associated graded object. This is an extension of previous work in the cases of one and two complex dimensions and proves the general case of a conjecture of Bando and Siu.
Funding source: NSF
Award Identifier / Grant number: 1037094
The author would like to thank his thesis advisor, Richard Wentworth, who suggested this problem and gave him considerable help in solving it. He also owes a great debt to the anonymous referee, who has given comments of exceptional detail through three separate readings, pointed out errors, fixed numerous typos, and whose comments have in general improved the exposition considerably. Finally, the author would like to thank Xuwen Chen for occasional discussions about certain technical points in the paper.
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- Cosmetic surgeries on knots in S3
- Integral points in two-parameter orbits
- Arithmetic moduli and lifting of Enriques surfaces
- Analytic varieties with finite volume amoebas are algebraic
- Vanishing resonance and representations of Lie algebras
- A new notion of angle between three points in a metric space
- Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: The canonical structure of the limit
- An Oka principle for equivariant isomorphisms
- Birkhoff matrices, residues and rigidity for q-difference equations
Articles in the same Issue
- Frontmatter
- Cosmetic surgeries on knots in S3
- Integral points in two-parameter orbits
- Arithmetic moduli and lifting of Enriques surfaces
- Analytic varieties with finite volume amoebas are algebraic
- Vanishing resonance and representations of Lie algebras
- A new notion of angle between three points in a metric space
- Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: The canonical structure of the limit
- An Oka principle for equivariant isomorphisms
- Birkhoff matrices, residues and rigidity for q-difference equations
