Obstructions to the Existence of Sasaki-Einstein Metrics

We describe two simple obstructions to the existence of Ricci-flat Kahler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kahler-Einstein metrics on Fano orbifolds. We present several families of hypersurface singularities that are obstructed, including 3-fold and 4-fold singularities of ADE type that have been studied previously in the physics literature. We show that the AdS/CFT dual of one obstruction is that the R-charge of a gauge invariant chiral primary operator violates the unitarity bound.Comment: 35 pages, 1 figure; references and a footnote adde

Weakly--exceptional quotient singularities

A singularity is said to be weakly--exceptional if it has a unique purely log terminal blow up. In dimension $2$, V. Shokurov proved that weakly--exceptional quotient singularities are exactly those of types $D_{n}$, $E_{6}$, $E_{7}$, $E_{8}$. This paper classifies the weakly--exceptional quotient singularities in dimensions $3$ and $4$

A global invariant for three dimensional CR-manifolds

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46621/1/222_2005_Article_BF01404456.pd

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

Szeg\"o kernel asymptotics and Morse inequalities on CR manifolds

We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szeg\"o kernel on (0, q)-forms with values in the high tensor powers of the line bundle. This gives after integration weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities which we apply to the embedding of some convex-concave manifolds.Comment: 40 pages, the constants in Theorems 1.1-1.8 have been modified by a multiplicative constant 1/2 ; v.2 is a final updat

Low-field magnetoresistance in GaAs 2D holes

We report low-field magnetotransport data in two-dimensional hole systems in GaAs/AlGaAs heterostructures and quantum wells, in a large density range, $2.5 \times 10^{10} \leq p \leq 4.0 \times 10^{11}$ cm$^{-2}$, with primary focus on samples grown on (311)A GaAs substrates. At high densities, $p \gtrsim 1 \times 10^{11}$ cm$^{-2}$, we observe a remarkably strong positive magnetoresistance. It appears in samples with an anisotropic in-plane mobility and predominantly along the low-mobility direction, and is strongly dependent on the perpendicular electric field and the resulting spin-orbit interaction induced spin-subband population difference. A careful examination of the data reveals that the magnetoresistance must result from a combination of factors including the presence of two spin-subbands, a corrugated quantum well interface which leads to the mobility anisotropy, and possibly weak anti-localization. None of these factors can alone account for the observed positive magnetoresistance. We also present the evolution of the data with density: the magnitude of the positive magnetoresistance decreases with decreasing density until, at the lowest density studied ($p = 2.5 \times 10^{10}$ cm$^{-2}$), it vanishes and is replaced by a weak negative magnetoresistance.Comment: 8 pages, 8 figure

Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

We give an expression for the {\L}ojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ in \O_n using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of {\L}ojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions $(\mathbb{C}^n,0)\to (\mathbb{C},0)$ for which the {\L}ojasiewicz of its gradient map $\nabla f$ attains the maximum possible value.Comment: 25 pages. Updated with minor change

Holographic Uniformization

We derive and study supergravity BPS flow equations for M5 or D3 branes wrapping a Riemann surface. They take the form of novel geometric flows intrinsically defined on the surface. Their dual field-theoretic interpretation suggests the existence of solutions interpolating between an arbitrary metric in the UV and the constant-curvature metric in the IR. We confirm this conjecture with a rigorous global existence proof.Comment: 52 pages, 3 figure