Elliptic singularities on log symplectic manifolds and Feigin--Odesskii Poisson brackets
| Autor: | Brent Pym |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
Pure mathematics
Transversality FOS: Physical sciences 01 natural sciences Poisson bracket Mathematics - Algebraic Geometry 0103 physical sciences FOS: Mathematics Elliptic surface 0101 mathematics Algebraic Geometry (math.AG) Mathematics::Symplectic Geometry Mathematical Physics Symplectic manifold Mathematics Algebra and Number Theory 010102 general mathematics Mathematical Physics (math-ph) 53D17 32S25 32S65 14J45 Elliptic curve Hypersurface Mathematics - Symplectic Geometry Symplectic Geometry (math.SG) Computer Science::Programming Languages 010307 mathematical physics Complex manifold Symplectic geometry |
| Popis: | A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities $\tilde{E}_6,\tilde{E}_7$ and $\tilde{E}_8$. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskii's Poisson structures of type $q_{5,1}$ are the only log symplectic structures on projective four-space whose singular points are all elliptic. 33 pages, comments welcome |
| Databáze: | OpenAIRE |
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