Infinitesimal deformation quantization of complex analytic spaces

Global constructions of quantization deformation and obstructions are discussed for an arbitrary complex analytic space in terms of adapted (analytic) Hochschild cohomology. For K3-surfaces an explicit global construction of a Poisson bracket is given. It is shown that the analytic Hochschild (co)homology on a complex space has structure of coherent analytic sheaf in each degree

Global Geometric Deformations of the Virasoro algebra, current and affine algebras by Krichever-Novikov type algebra

In two earlier articles we constructed algebraic-geometric families of genus one (i.e. elliptic) Lie algebras of Krichever-Novikov type. The considered algebras are vector fields, current and affine Lie algebras. These families deform the Witt algebra, the Virasoro algebra, the classical current, and the affine Kac-Moody Lie algebras respectively. The constructed families are not equivalent (not even locally) to the trivial families, despite the fact that the classical algebras are formally rigid. This effect is due to the fact that the algebras are infinite dimensional. In this article the results are reviewed and developed further. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics. The algebras are of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory.Comment: 17 page

A simply connected surface of general type with p_g=0 and K^2=2

In this paper we construct a simply connected, minimal, complex surface of general type with p_g=0 and K^2=2 using a rational blow-down surgery and Q-Gorenstein smoothing theory.Comment: 19 pages, 6 figures. To appear in Inventiones Mathematica

Holomorphic synthesis of monogenic functions of several quaternionic variables

The system of differential equations for polymonogenic functions of several quaternionic variables is an analog of anti #partial deriv#-equation in complex analysis. We give a representation of polymonogenic functions by means of integration of a family of #sigma#-holomorphic functions of #sigma# runs over the variety #SIGMA# of all complex structures H#approx =#C"2, which are consistent with the metric and an orientation in H. The variety #SIGMA# is isomorphic to the manifold of all proper right ideals in the complexified quaternionic algebra and has a natural complex analytic structure. We construct a anti #partial deriv#-complex on #SIGMA# that provides a resolvent for the sheaf of polynomogenic functions. (orig.)Available from TIB Hannover: RO 7057(1998,26) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman