Convexity, Currents, and Lelong numbers

Abstract: This thesis treats different aspects of convexity, both real and complex. On the real side, we study convexity in relation with certain currents. On the complex side, we study the singularities of plurisubharmonic functions. More precisely, we introduce a generalization of the Lelong number, which is a measurement of the strength of the singularities of a plurisubharmonic function. This generalization measures the singularity of a plurisubharmonic function $varphi$ at a point with respect to another function $psi$. For the special choice of $psi = log|z|$, we obtain the classical Lelong number. We study various properties which this generalization satisfies. In particular, we prove that our generalized Lelong number satisfies a certain analyticity property, namely that the upper level-sets of the Lelong number define analytic varieties of $C^n$. Another aspect of this thesis is the consideration of super currents. We study positivity properties of such super currents and show that this relates to convexity in $R^n$, in much the same way as plurisubharmonicity relates to positive currents on a complex manifold. We also consider how super currents can be used to study tropical geometry, proving in particular a natural correspondence between tropical hypersurfaces and certain super currents. We define an intersection theory for super currents, in the same spirit as for currents on complex manifolds, and show how this gives a natural intersection theory for tropical varieties. We also introduce the notion of an $R$-Kähler metric on $R^n$. In the setting of super forms on $R^n$ endowed with such an $R$-Kähler metric, we consider the $d-$equation, $$ d alpha = eta,$$ for a given $d$-closed super form $eta$. Using the ideas of Hörmander's $L^2$-estimates for the $dbar-$equation on a complex Kähler manifold, we prove existence theorems for the $d$-equation acting on super forms, and we find weighted $L^2$-estimates of the solutions in terms of the given data, $eta$.