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 Odesskiis Poisson structures of type $q_{5,1}$ are the only log symplectic structures on projective four-space whose singular points are all elliptic.
We describe the possible noncommutative deformations of complex projective three-space by exhibiting the Calabi--Yau algebras that serve as their homogeneous coordinate rings. We prove that the space parametrizing such deformations has exactly six irreducible components, and we give explicit presentations for the generic members of each family in terms of generators and relations. The proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic Poisson structures in four dimensions, which we extract from Cerveau and Lins Netos classification of degree-two foliations on projective space. Corresponding to the ``exceptional component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical Schwarzenberger bundles.
We present an axisymmetric, equilibrium model for late-type galaxies which consists of an exponential disk, a Sersic bulge, and a cuspy dark halo. The model is specified by a phase space distribution function which, in turn, depends on the integrals of motion. Bayesian statistics and the Markov Chain Monte Carlo method are used to tailor the model to satisfy observational data and theoretical constraints. By way of example, we construct a chain of 10^5 models for the Milky Way designed to fit a wide range of photometric and kinematic observations. From this chain, we calculate the probability distribution function of important Galactic parameters such as the Sersic index of the bulge, the disk scale length, and the disk, bulge, and halo masses. We also calculate the probability distribution function of the local dark matter velocity dispersion and density, two quantities of paramount significance for terrestrial dark matter detection experiments. Though the Milky Way models in our chain all satisfy the prescribed observational constraints, they vary considerably in key structural parameters and therefore respond differently to non-axisymmetric perturbations. We simulate the evolution of twenty-five models which have different Toomre Q and Goldreich-Tremaine X parameters. Virtually all of these models form a bar, though some, more quickly than others. The bar pattern speeds are ~ 40 - 50 km/s/kpc at the time when they form and then decrease, presumably due to coupling of the bar with the halo. Since the Galactic bar has a pattern speed ~50 km/s/kpc we conclude that it must have formed recently.
