The elliptic genus of Calabi-Yau 3- and 4-folds, product formulae and generalized Kac-Moody algebras

In this paper the elliptic genus for a general Calabi-Yau fourfold is derived. The recent work of Kawai calculating N=2 heterotic string one-loop threshold corrections with a Wilson line turned on is extended to a similar computation where K3 is replaced by a general Calabi-Yau 3- or 4-fold. In all cases there seems to be a generalized Kac-Moody algebra involved, whose denominator formula appears in the result.Comment: 10 pages, latex, no figure

Complete intersection singularities of splice type as universal abelian covers

It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "splice type singularities", which generalize Brieskorn complete intersections. Second, these arise as universal abelian covers of a class of normal surface singularities with Q-homology sphere links, called "splice-quotient singularities". According to the Main Theorem, splice-quotients realize a large portion of the possible topologies of singularities with Q-homology sphere links. As quotients of complete intersections, they are necessarily Q-Gorenstein, and many Q-Gorenstein singularities with Q-homology sphere links are of this type. We conjecture that rational singularities and minimally elliptic singularities with Q-homology sphere links are splice-quotients. A recent preprint of T Okuma presents confirmation of this conjecture.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper17.abs.htm

Immersed and virtually embedded pi_1-injective surfaces in graph manifolds

We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed pi_1-injective surface of negative Euler characteristic. We also determine the class of closed graph manifolds which have no finite cover containing an embedded such surface. This is a larger class. Thus, manifolds M^3 exist which have immersed pi_1-injective surfaces of negative Euler characteristic, but no such surface is virtually embedded (finitely covered by an embedded surface in some finite cover of M^3).Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-20.abs.htm