Narrow Results

An affine algebraic variety $X$ is rigid if the algebra of regular functions $𝕂[X]$ admits no nonzero locally nilpotent derivation. We prove that a factorial trinomial hypersurface is rigid if and only if every exponent in the trinomial is at least $2$.

Let M be a closed even n-manifold of positive sectional curvature on which a torus T^k acts isometrically. We show that if (respectively, k > 1) for n ≠ 12 (respectively, n = 12), then the Euler characteristic of each T^k-fixed point component is positive. This implies that the Euler characteristic of M is positive. We also extend this result to an isometric elementary p-group -action on a closed manifold of positive sectional curvature.

In this paper, we obtain a classification for the fundamental groups of positively curved n-manifolds which admit isometric torus T^k-actions with and n ≥ 25.

This paper takes a detailed look at a subject that occurs in various contexts in mathematics, the fixed-point sets of torus actions on flag manifolds, and considers it from the (perhaps nontraditional) perspective of moment maps and length functions on Weyl groups. The approach comes from earlier work of the author where it is shown that certain singular flows in the Hamiltonian system known as the Toda lattice generate the action of a group A on a flag manifold, where A is a direct product of a non-maximal torus and unipotent group. As a first step in understanding the orbits of A in connection with the Toda lattice, this paper seeks to understand the fixed points of the non-maximal tori in this setting.

We provide an explicit description of homogeneous locally nilpotent derivations of the algebra of regular functions on affine trinomial hypersurfaces. As an application, we describe the set of roots of trinomial algebras.

We show that every irreducible, simply connected curve on a toric affine surface X over ℂ is an orbit closure of a 𝔾_m-action on X. It follows that up to the action of the automorphism group Aut(X) there are only finitely many nonequivalent embeddings of the affine line 𝔸¹ in X. A similar description is given for simply connected curves in the quotients of the affine plane by small finite linear groups. We provide also an analog of the Jung-van der Kulk theorem for affine toric surfaces, and apply this to study actions of algebraic groups on such surfaces.