Einstein Metrics on Spheres

We prove the existence of an abundance of new Einstein metrics on odd dimensional spheres including exotic spheres, many of them depending on continuous parameters. The number of families as well as the number of parameter grows double exponentially with the dimension. Our method of proof uses Brieskorn-Pham singularities to realize spheres (and exotic spheres) as circle orbi-bundles over complex algebraic orbifolds, and lift a Kaehler-Einstein metric from the orbifold to a Sasakian-Einstein metric on the sphere.Comment: 19 pages, some references added and clarifications made. to appear in Annals of Mathematic

On Sasaki-Einstein manifolds in dimension five

We prove the existence of Sasaki-Einstein metrics on certain simply connected 5-manifolds where until now existence was unknown. All of these manifolds have non-trivial torsion classes. On several of these we show that there are a countable infinity of deformation classes of Sasaki-Einstein structures.Comment: 18 pages, Exposition was expanded and a reference adde

Effect of Resonance in Soil-Structure Interaction for Finite Soil Layers

In case of seismic design the deformability of the soil should be considered, which can be performed in several ways. Most of the methods do not take into account the finite dimensions of the soil, which results significantly different behavior than the spring-dashpot systems. For an infinite medium, which is used in many cases, there are no eigenmodes, however in practical applications the soft soil is always bounded by rocks. For these cases the soil has eigenmodes and the resonance may influence considerably the response of the system. This question was investigated numerically by FE calculations, and it was found that in certain cases the resonance, which is neglected in the common design process, may significantly enhance the earthquake loads. In this paper this phenomenon is investigated and the parameter range is defined when this effect must be taken into account

Model of Soil-structure Interaction of Objects Resting on Finite Depth Soil Layers for Seismic Design

In case of seismic design of structures the deformability and damping of the soil should be considered, which can be performed in several ways. The infinite soil half space can be approximated with the cone model, which gives constant values for the spring stiffnesses and dashpot characteristics, and an additional mass element for rocking motion. To approximate the dynamic impedance function of a soil layer more complex models were also applied. Most of the methods do not take into account the finite dimensions of the soil, which results significantly different behavior than spring-dashpot systems. To consider the effect of a finite layer a new simple model based on a physical approach is given for the horizontal excitation of strip foundations. Numerical verification is presented, and the parameter range is determined, where the application of the new model is recommended, since applying a spring-dashpot model results in significant errors

Hodge metrics and positivity of direct images

Building on Fujita-Griffiths method of computing metrics on Hodge bundles, we show that the direct image of an adjoint semi-ample line bundle by a projective submersion has a continuous metric with Griffiths semi-positive curvature. This shows that for every holomorphic semi-ample vector bundle $E$ on a complex manifold, and every positive integer $k$, the vector bundle $S^kE\otimes\det E$ has a continuous metric with Griffiths semi-positive curvature. If $E$ is ample on a projective manifold, the metric can be made smooth and Griffiths positive.Comment: revised and expanded version of "A positivity property of ample vector bundles

Graph hypersurfaces and a dichotomy in the Grothendieck ring

The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of Z generated by the class of a point. Thus, the span of the graph hypersurfaces in the Grothendieck ring is nearly killed by setting the Lefschetz motive L to zero, while it is known that graph hypersurfaces generate the Grothendieck ring over a localization of Z[L] in which L becomes invertible. In particular, this shows that the graph hypersurfaces do not generate the Grothendieck ring prior to localization. The same result yields some information on the mixed Hodge structures of graph hypersurfaces, in the form of a constraint on the terms in their Deligne-Hodge polynomials.Comment: 8 pages, LaTe

Highly connected manifolds with positive Ricci curvature

We prove the existence of Sasakian metrics with positive Ricci curvature on certain highly connected odd dimensional manifolds. In particular, we show that manifolds homeomorphic to the 2k-fold connected sum of S^{2n-1} x S^{2n} admit Sasakian metrics with positive Ricci curvature for all k. Furthermore, a formula for computing the diffeomorphism types is given and tables are presented for dimensions 7 and 11.Comment: This is the version published by Geometry & Topology on 29 November 200