Dmitri Panov

Hyperbolic geometry and non-Kahler manifolds with trivial canonical bundle

We use hyperbolic geometry to construct simply connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C 4 : the smoothing is a natural S 3 ‐bundle over H 3 , its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S 2 ‐bundle over H 4 with symplectic geometry determined by the metric. Using hyperbolic geometry, we find orbifold quotients with trivial canonical bundle; smooth examples are produced via crepant resolutions. In particular, we find the first example of a simply connected symplectic 6‐manifold with c1D 0 that does not admit a compatible Kahler structure. We also find infinitely many distinct complex structures on 2.S 3 S 3 / #.S 2 S 4 / with trivial canonical bundle. Finally, we explain how an analogous construction for hyperbolic manifolds in higher dimensions gives symplectic non-Kahler “Fano” manifolds of dimension 12 and higher. 53D35, 32Q55; 51M10, 57M25

The diversity of symplectic calabi-yau 6-manifolds

Given an integer b and a finitely presented group G, we produce a compact symplectic 6-manifold with c1 = 0, b2 > b, b3 > b and pi = G. In the simply connected case, we can also arrange for b3 = 0; in particular, these examples are not diffeomorphic to Kahler manifolds with c1 = 0. The construction begins with a certain orientable, four-dimensional, hyperbolic orbifold assembled from right-angled 120-cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi- Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the required properties.

Polyhedral Kähler manifolds

In this article we introduce the notion of Polyhedral Kahler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4-dimensional case, prove that such manifolds are smooth complex surfaces, and classify the singularities of the metric. The singularities form a divisor and the residues of the flat connection on the complement of the divisor give us a system of cohomological equations. Parabolic version of Kobayshi-Hitchin correspondence of T. Mochizuki permits us to characterize polyhedral Kahler metrics of non-negative curvature on CP^2 with singularities at complex line arrangements.

Counting meromorphic functions with critical points of large multiplicities

We study the number of meromorphic functions on a Riemann surface with given critical values and prescribed multiplicities of critical points and values. If the Riemann surface is ℂP1 and the function is a polynomial, we give an elementary way of finding this number. In the general case, we show that, as the multiplicities of critical points tend to infinity, the asymptotics for the number of meromorphic functions is given by the volume of some space of graphs glued from circles. We express this volume as a matrix integral. Bibliography: 7 titles.

Circle-invariant fat bundles and symplectic Fano 6-manifolds

We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S^4 and CP^2-bar are the only 4-manifolds admitting circle-invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang-Klieners theorem that only S^4 and CP^2 admit circle-invariant metrics of positive sectional curvature.

Enumeration of almost polynomial rational functions with given critical values

Enumerating ramified coverings of the sphere with fixed ramification types is a well-known problem first considered by Hurwitz [A. Hurwitz, Uber Riemannsche Flachen mit gegebenen Verzweigungspunkten, Mathematische Annalen 39 (1891) 1-61. [7]]. Up to now, explicit solutions have been obtained only for some families of ramified coverings, for instance, those realized by polynomials in one complex variable. In this paper we obtain an explicit answer for a large new family of coverings, namely, the coverings realized by simple almost polynomials, defined below. Unlike most other results in the field, our formula is obtained by elementary methods.

Real line arrangements with the hirzebruch property

A line arrangement of

Slope stability and exceptional divisors of high genus

3n

Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold

lines in

Spherical Metrics with Conical Singularities on a 2-Sphere: Angle Constraints

\mathbb CP^2