Constant scalar curvature Kähler metrics on fibred complex surfaces

By Joel Fine. 2004

98 citations recorded by the source corpus.

This article finds constant scalar curvature Kahler metrics on certain compact complex surfaces. The surfaces considered are those admitting a holomorphic submersion to a curve, with fibres of genus at least 2. The proof is via an adiabatic limit. An approximate solution is constructed out of the hyperbolic metrics on the fibres and a large multiple of a certain metric on the base. A parameter-dependent inverse function theorem is then used to perturb the approximate solution to a genuine solution in the same cohomology class. The arguments also apply to certain higher dimensional fibred Kahler manifolds.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/48befce949e3810e7e426a2a711091183895d51a

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In Applied Mathematics

1. Xiuxiong Chen
2. R. Dervan
3. Weiyong He
4. Chi Li
5. G'abor Sz'ekelyhidi
6. Song Sun
7. Robert J. Berman
8. Jingrui Cheng

In Astronomy and Astrophysics

1. Weiyong He
2. R. Dervan
3. Xiuxiong Chen
4. Jeffrey Streets
5. Jacob Sturm
6. Duong H. Phong
7. Robert J. Berman
8. Song Sun

In Geometry and Topology

1. R. Dervan
2. Xiuxiong Chen
3. Weiyong He
4. Chi Li
5. G'abor Sz'ekelyhidi
6. Robert J. Berman
7. Jingrui Cheng
8. Song Sun

Citations in the corpus

• Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, background): “The main result of Fine [16, 17] is that in this case X admits a cscK metric on rπ∗LB + LF in the adiabatic limit r ≫ 0.”
• On the Calabi Flow (2006, methodology): “More recently, J. Fine [30] has used adabiatic limit techniques to construct non-trivial CscK metrics on complex surfaces.”