Bihermitian metrics on Del Pezzo surfaces
By N. Hitchin. 2006
62 citations recorded by the source corpus.
Recall that a Del Pezzo surface is defined as an algebraic surface with ample anticanonical bundle K. Ampleness means that the line bundle K has a hermitian metric whose curvature form F is positive, defining a Kahler metric g0. This is the data for our construction – we take a holomorphic section σ of K, and the function f = log ‖σ‖. This function is singular on the zero set of σ (which is an elliptic curve). The section σ can also be seen as a holomorphic Poisson structure on M and its real part is a real Poisson structure. We use this Poisson structure to define a Hamiltonian vector field for f , which turns out to be well-defined and smooth on the whole of M . Integrating it for a time t we get a Poisson diffeomorphism φt and take I + = I, the original complex structure on the Del Pezzo, and I = φ∗t I. For small enough t we show how to define a bihermitian metric g canonically from the imaginary part of σ.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/3b950f0ef41be94968b5cf1351c4d8c83c632e21
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- On the J-anti-invariant cohomology of almost complex 4-manifolds (2011, background): “The study of such structures has been active recently, (see, for instance, [18] and the references therein), especially due to the link with generalized Kähler geometry ([16]).”
- Generalized Kähler Manifolds, Commuting Complex Structures, and Split Tangent Bundles (2007, background): “Despite a growing number of explicit constructions [2, 8, 20, 21 , 24, 27], the general existence problem for non-trivial generalized Kahler structures remains open.”
- Generalized Kähler Geometry (2010, methodology): “This equivalence, obtained in the author’s thesis [23], was followed by a number of results, such as those contained in [8,10,17,19,22,24,29,30,32,35], which are not easily accessed purely from the bi-Hermitian point of view.”
- Branes on Poisson varieties (2007, methodology): “We remark finally upon the relation of our construction to Hitchin’s result for Del Pezzo surfaces [3].”
- Deformations of holomorphic Poisson manifolds (2011, result): “The viewpoint we get here of a fixed C manifold with different complex structures is close to that in [9] for Del Pezzo surfaces.”
- Deformations of generalized complex and generalized Kahler structures (2007, methodology): “Hitchin gave an explicit construction of generalized K¨ ahler structures on Del Pezzo surfaces by using holomorphic Poisson structures and suggested that generalized K¨ ahler structures are related to holomorphic Poisson structur es [12], [13].”
- Lectures on generalized geometry (2010, result): “This is consistent with some concrete constructions of one-parameter families of bihermitian metrics on Fano manifolds in [14] and [7], where I+(t) and I−(t) are all equivalent under a diffeomorphism.”