This paper studies Batyrev's notion of primitive collection. We use primitive collections to characterize the nef cone of a quasi-projective toric variety whose fan has convex support, a result stated without proof by Batyrev in the smooth projective case. When the fan is non-simplicial, we modify the definition of primitive collection and explain how our definition relates to primitive collections of simplicial subdivisons. The paper ends with an open problem.

We present one property of the Riemannian metric which is derived from the positive power of potential functions. Then this property is applied to the study of the $\Gamma$-convergence of energy functionals which are associated with the Euler-Lagrange $p$-Laplacian equation.

We consider solvable Lie groups which are isomorphic to unimodularizations of products of affine groups. It is shown that a lattice of such a Lie group is determined, up to commensurability, by a totally real algebraic number field. We also show that the outer automorphism group of the lattice is represented faithfully in the automorphism group of the number field. As an application, we obtain a classification of codimension one, volume preserving, locally free actions of products of affine groups.

We consider a complete nonsingular complex algebraic variety having a normal crossing divisor such that the associated logarithmic tangent bundle is generated by its global sections. We obtain an optimal vanishing theorem for logarithmic Dolbeault cohomology of nef line bundles in that setting. This implies a vanishing theorem for ordinary Dolbeault cohomology which generalizes results of Broer for flag varieties, and of Mavlyutov for toric varieties.

An infinitesimal derivative of the Bott class is defined by generalizing Heitsch'es construction. We prove a formula relating the infinitesimal derivative to the Schwarzian derivatives, which gives a generalization of the Maszczyk formula for the Godbillon-Vey class of real codimension-one foliations. As an application, a residue of infinitesimal derivatives with respect to the Julia set in the sense of Ghys, Gomez-Mont and Saludes is introduced.

In a certain Banach space called an M-type 2 Banach space (including Hilbert spaces), we consider a set-valued stochastic differential equation with a set-valued drift term and a single valued diffusion term, under the Lipschitz continuity conditions, and we prove the existence and uniqueness of strong solutions which are continuous in the Hausdorff distance.