Rational transformations of systems of commuting nonselfadjoint operators

具有有理系数的2秩交换微分算子

Commuting differential operators of rank 2 with rational coefficients

该研究发现了新的具有有理系数的2秩自伴交换算子，证明亏格2的椭圆、超椭圆曲线均为其谱曲线，并探讨了亏格3曲线作为其谱曲线的情况。

On the generalized commuting varieties of a reductive Lie algebra

The generalized commuting and isospectral commuting varieties of a reductive Lie algebra have been introduced in a preceding article. In this note, it is proved that their normalizations are Gorenstein with rational singularities. Moreover, their canonical modules are free of rank 1. In particular, the usual commuting variety is Gorenstein with rational singularities and its canonical module is free of rank 1.

Landen transforms as families of (commuting) rational self-maps of projective space

The classical (m,k)-Landen transform F_{m,k} is a self-map of the field of rational functions C(z) obtained by forming a weighted average of a rational function over twists by m'th roots of unity. Identifying the set of rational maps of degree d with an affine open subset of P^{2d+1}, we prove that F_{m,0} induces a dominant rational self-map R_{d,m,0} of P^{2d+1} of algebraic degree m, and for 0 < k < m, the transform F_{m,k} induces a dominant rational self-map R_{d,m,k} of algebraic degree m of a certain hyperplane in P^{2d+1}. We show in all cases that R_{d,m,k} extends nicely to a map of …

Commuting elements in reductive groups and Higgs bundles on abelian varieties

Let G be a connected real reductive algebraic group, and let K be a maximal compact subgroup of G. We prove that the conjugation orbit space Hom(Z^{2d},K)/K is a strong deformation retract of the space Hom(Z^{2d},G)/G of equivalence classes of representations of Z^{2d} into G. This is proved by showing that the homotopy type of the moduli space of principal G-Higgs bundles of vanishing rational characteristic classes on a complex abelian variety of dimension d depends only on K.

Several Applications of Bezout Matrices

The notion of Bezout matrix is an essential tool in studying broad variety of subjects: zeroes of polynomials, stability of differential equations, rational transformations of algebraic curves, systems of commuting nonselfadjoint operators, boundaries of quadrature domains etc. We present a survey of several properties of Bezout matrices and their applications in all mentioned topics. We use the framework of Vandermonde vectors because such approach allows us to give new proofs of both classical and modern results and in many cases to obtain new explicit formulas. These explicit formulas can s…

Hilbert schemes, commuting matrices, and hyperkähler geometry

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable twisted rational curves of degree $d$ in ${\Bbb P}^3\backslash {\Bbb P}^1$ is isomorphic to a complexified hyperkähler quotient of an open subset of a vector space by a non-reductive Lie group.