A1-homotopy groups,excision, and solvable quotients

We study some properties of ${\mathbb{A}}^1$-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of ${\mathbb{A}}^1$-homotopy theory. These concepts and results are well suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of solvable group quotients, we investigate ${\mathbb{A}}^1$-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree ${\mathbb{A}}^1$-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the “next” non-vanishing ${\mathbb{A}}^1$-homotopy group (beyond ${\pi }_1^{{\mathbb{A}}^1}$) of a smooth toric variety. From this point of view, ${\mathbb{A}}^1$-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost “as tractable” (in low degrees) as ordinary homotopy for large classes of interesting varieties.     

G-exceptional vector bundles on P2 and representations of quivers

It is known that the category of homogeneous bundles on ${\mathbb{P}}^2$ is equivalent to the category of representations of a quiver with relation. In this paper we make use of this equivalence to describe a family of G-exceptional bundles on ${\mathbb{P}}^2$ and to prove that they are stable. We also study the G-exceptionality of Fibonacci bundles on ${\mathbb{P}}^2$.     

Logarithmic potentials on Pn

We study the projective logarithmic potential ${\mathbb{G}}_{\mu }$ of a probability measure μ on the complex projective space ${\mathbb{P}}^n$. We prove that the range of the operator $\mu ?{\mathbb{G}}_{\mu }$ is contained in the (local) domain of definition of the complex Monge–Ampère operator acting on the class of quasi-plurisubharmonic functions on ${\mathbb{P}}^n$ with respect to the Fubini–Study metric. Moreover, when the measure μ has no atom, we show that the complex Monge–Ampère measure of its logarithmic potential is an absolutely continuous measure with respect to the Fubini–Study volume form on ${\mathbb{P}}^n$.     

Sharp bounds for Hardy operators on product spaces

In this article, we obtain the sharp bounds from L^P$\left({\mathbb{G}}^n\right)$ to the space wL^P$\left({\mathbb{G}}^n\right)$ for Hardy operators on product spaces. More generally, the precise norms of Hardy operators on product spaces from L^P$\left({\mathbb{G}}^n\right)$ to the space L^P_I$\left({\mathbb{G}}^n\right)$ are obtained.     

On two upper bounds for hypersurfaces involving a Thas’ invariant

Let $X^n$ be a hypersurface in ${\mathbb{P}}^{n+1}$ with $n\ge 1$ defined over a finite field ${\mathbb{F}}_q$ of $q$ elements. In this note, we classify, up to projective equivalence, hypersurfaces $X^n$ as above which reach two elementary upper bounds for the number of ${\mathbb{F}}_q$-points on $X^n$ which involve a Thas’ invariant.