The nondegenerate generalized Kähler Calabi–Yau problem
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Vestislav Apostolov
Abstract
We formulate a Calabi–Yau-type conjecture in generalized Kähler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized Kähler structures generalizing the notion of Kähler class, we conjecture unique solvability of Gualtieri’s Calabi–Yau equation within this class. We establish the uniqueness, and moreover show that all such solutions are actually hyper-Kähler metrics. We furthermore establish a GIT framework for this problem, interpreting solutions of this equation as zeroes of a moment map associated to a Hamiltonian action and finding a Kempf–Ness functional. Lastly we indicate the naturality of generalized Kähler–Ricci flow in this setting, showing that it evolves within the given Hamiltonian deformation class, and that the Kempf–Ness functional is monotone, so that the only possible fixed points for the flow are hyper-Kähler metrics. On a hyper-Kähler background, we establish global existence and weak convergence of the flow.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1454854
Funding statement: The first author was supported in par by an NSERC Discovery Grant and is grateful to the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences where a part of this project was realized. The second author gratefully acknowledges support from the NSF via DMS-1454854, and from an Alfred P. Sloan Fellowship.
Acknowledgements
The second author would like to thank Marco Gualtieri for helpful discussions on this topic. The authors thank the referee for useful comments.
References
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© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The nondegenerate generalized Kähler Calabi–Yau problem
- Parity sheaves and Smith theory
- Modular symbols for Teichmüller curves
- Eisenstein series and the top degree cohomology of arithmetic subgroups of SLn/ℚ
- Higher-page Bott–Chern and Aeppli cohomologies and applications
- Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation
- Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: An algebro-geometric approach
- Polynomially convex embeddings of odd-dimensional closed manifolds
Articles in the same Issue
- Frontmatter
- The nondegenerate generalized Kähler Calabi–Yau problem
- Parity sheaves and Smith theory
- Modular symbols for Teichmüller curves
- Eisenstein series and the top degree cohomology of arithmetic subgroups of SLn/ℚ
- Higher-page Bott–Chern and Aeppli cohomologies and applications
- Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation
- Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: An algebro-geometric approach
- Polynomially convex embeddings of odd-dimensional closed manifolds
