A note on negative isotropic curvature

We prove that any compact four-manifold admits a Riemannian metric with negative isotropic curvature in the sense of Micallef and Moore.Comment: 6 Pages. To appear in Mathematical Research Letter

Statistics of spatial averages and optimal averaging in the presence of missing data

We consider statistics of spatial averages estimated by weighting observations over an arbitrary spatial domain using identical and independent measuring devices, and derive an account of bias and variance in the presence of missing observations. We test the model relative to simulations, and the approximations for bias and variance with missing data are shown to compare well even when the probability of missing data is large. Previous authors have examined optimal averaging strategies for minimizing bias, variance and mean squared error of the spatial average, and we extend the analysis to the case of missing observations. Minimizing variance mainly requires higher weights where local variance and covariance is small, whereas minimizing bias requires higher weights where the field is closer to the true spatial average. Missing data increases variance and contributes to bias, and reducing both effects involves emphasizing locations with mean value nearer to the spatial average. The framework is applied to study spatially averaged rainfall over India. We use our model to estimate standard error in all-India rainfall as the combined effect of measurement uncertainty and bias, when weights are chosen so as to yield minimum mean squared error

Quot schemes and Ricci semipositivity

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${\mathcal Q}_X(r,d)$ be the quot scheme that parametrizes all the torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree $d$. This ${\mathcal Q}_X(r,d)$ is also a moduli space of vortices on $X$. Its geometric properties have been extensively studied. Here we prove that the anticanonical line bundle of ${\mathcal Q}_X(r,d)$ is not nef. Equivalently, ${\mathcal Q}_X(r,d)$ does not admit any K\"ahler metric whose Ricci curvature is semipositive