International Journal of Mathematics

Let $E_G$ be a $\mathrm{\Gamma }$-equivariant algebraic principal $G$-bundle over a normal complex affine variety $X$ equipped with an action of $\mathrm{\Gamma }$, where $G$ and $\mathrm{\Gamma }$ are complex linear algebraic groups. Suppose $X$ is contractible as a topological $\mathrm{\Gamma }$-space with a dense orbit, and $x_0\in X$ is a $\mathrm{\Gamma }$-fixed point. We show that if $\mathrm{\Gamma }$ is reductive, then $E_G$ admits a $\mathrm{\Gamma }$-equivariant isomorphism with the product principal $G$-bundle $X{\times }_{\rho }{E}_G(x_0)$, where $\rho :\mathrm{\Gamma }\to G$ is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal $G$-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal $G$-bundles over any complex toric variety, generalizing the main result of [A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math. 27(14) (2016)].

Let ${ℍ}_n$ be the $(2n+1)$-dimensional Heisenberg group and $K$ a closed subgroup of $U(n)$ acting on ${ℍ}_n$ by automorphisms such that $(K,{ℍ}_n)$ is a Gelfand pair. Let $G=K⋉{ℍ}_n$ be the semidirect product of $K$ and ${ℍ}_n$. Let $𝔤\supset 𝔨$ be the respective Lie algebras of $G$ and $K$, and $pr:{𝔤}^{\ast }\to {𝔨}^{\ast }$ the natural projection. For coadjoint orbits ${{𝒪}}^G\subset {𝔤}^{\ast }$ and ${{𝒪}}^K\subset {𝔨}^{\ast }$, we denote by $n({{𝒪}}^G,{{𝒪}}^K)$ the number of $K$-orbits in ${{𝒪}}^G\cap {pr}^{-1}({{𝒪}}^K)$, which is called the Corwin–Greenleaf multiplicity function. In this paper, we give two sufficient conditions on ${{𝒪}}^G$ in order that

In this paper, we establish the interior $C^2$ estimate for equation $det{D}^{2}u=f(x,\nabla u)$ with an auxiliary function in dimension $n=2$.

We determine the $\mathrm{\text{SL}}(2,ℂ)$-character variety for each odd classical pretzel knot $P(2k_1+1,2k_2+1,2k_3+1)$, and present a method for computing its A-polynomial.

Let $X$ be a compact connected strongly pseudoconvex Cauchy–Riemann (CR) manifold of real dimension $2n+1,n\ge 1$ with a transversal CR $S^1$ action on $X$. We establish an asymptotic expansion for the $m$th Fourier component of the Szegö kernel function as $m\to \infty$, where the expansion involves a contribution in terms of a distance function from lower dimensional strata of the $S^1$ action. We also obtain explicit formulas for the first three coefficients of the expansion.

Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid is the quotient of a finite-dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an “Almeida–Molino” integrable lift when the base manifold is simply connected. On the other hand, we notice that the classical de Rham isomorphism provides a universal integrable algebroid. Using it we construct a “de Rham” integrable lift for any given transitive Abelian Lie algebroid.