Reflection functors for quiver varieties and Weyl group actions

We introduce reflectionfunctors on quiver varieties. They are hyper-Kähler isometries between quiver varieties with different parameters, related by elements in the Weyl group. The definition is motivated by the origial reflection functor given by Bernstein-Gelfand-Ponomarev [1], but they behave much nicely. They are isomorphisms and satisfy the Weyl group relations. As an application, we define Weyl group representations of homology groups of quiver varieties. They are analogues of Slodowys construction of Springer representations of the Weyl group. Mathematics Subject Classification (2000):Primary 53C26; Secondary 14D21, 16G20, 20F55, 33D80Supported by the Grant-in-aid for Scientific Research (No.11740011), the Ministry of Education, Japan.     

Abelian group actions on algebraic varieties with one fixed point

Dedicated to the memory of Ed Floyd with admiration and friendship     

The automorphism group of certain factorial threefolds and a cancellation problem

A new class of counterexamples to a generalized cancellation problem for affine varieties is presented. Each member of the class is an affine factorial complex threefold admitting a locally trivial action of the additive group, hence the total space for a principal G _a bundle over a quasiaffine base. The automorphism groups for these varieties are also determined.     

On the cancellation problem

Let k be an algebraically closed field. For every n ≥ 8 we give examples of Zariski open, dense, affine subsets of the affine space A ⁿ (k) which do not have the cancellation property. Dedicated to Professor Mikhail Zaidenberg. The author was partially supported by the grant of Polish Ministry of Science, 2006–2009.     

Polarizations on abelian subvarieties of principally polarized abelian varieties with dihedral group actions

For any $n\ge 2$ we study the group algebra decomposition of an $([\frac{n}{2}]+1)$ -dimensional family of principally polarized abelian varieties of dimension $n$ with an action of the dihedral group of order $2n$ . For any odd prime $p, n=p$ and $n=2p$ we compute the induced polarization on the isotypical components of these varieties and some other distinguished subvarieties. In the case of $n=p$ the family contains a one-dimensional family of Jacobians. We use this to compute a period matrix for Klein’s icosahedral curve of genus 5.     

Ergodicity of mapping class group actions on representation varieties, I. Closed surfaces

We prove that the mapping class group of a closed surface acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group. Received: March 21, 2001     

Affine pseudo-planes and cancellation problem

We define affine pseudo-planes as one class of -homology planes. It is shown that there exists an infinite-dimensional family of non-isomorphic affine pseudo-planes which become isomorphic to each other by taking products with the affine line . Moreover, we show that there exists an infinite-dimensional family of the universal coverings of affine pseudo-planes with a cyclic group acting as the Galois group, which have the equivariant non-cancellation property. Our family contains the surfaces without the cancellation property, due to Danielewski-Fieseler and tom Dieck.

     

A survey on Zariski cancellation problem

In this survey article we describe known results and open questions on the Zariski cancellation problem, highlighting recent developments on the problem. We also discuss its close relationship with some of the other central problems on polynomial rings.     

On the cancellation problem for surfaces

In this Note we prove a conjecture formulated by Danielewski. As a consequence we obtain an infinite dimensional family of non-isomorphic surfaces X_λ with the property that all X_λ × ℂ are isomorphic.     

Rationality problem of three-dimensional purely monomial group actions: the last case

A -automorphism of the rational function field is called purely monomial if sends every variable to a monic Laurent monomial in the variables . Let be a finite subgroup of purely monomial -automorphisms of . The rationality problem of the -action is the problem of whether the -fixed field is -rational, i.e., purely transcendental over , or not. In 1994, M. Hajja and M. Kang gave a positive answer for the rationality problem of the three-dimensional purely monomial group actions except one case. We show that the remaining case is also affirmative.

     

Ergodicity of mapping class group actions on representation varieties,II. Surfaces with boundary

The mapping class group of a compact oriented surface of genus greater than one with boundary acts ergodically on connected components of the representation moduli corresponding to a connected compact Lie group, for every choice of conjugacy class boundary condition.     

The multiplicative group action on singular varieties and Chow varieties

We answer two questions of Carrell on a singular complex projective variety admitting the multiplicative group action, one positively and the other negatively. The results are applied to Chow varieties and we obtain Chow groups of 0-cycles and Lawson homology groups of 1-cycles for Chow varieties. A brief survey on the structure of Chow varieties is included for comparison and completeness. Moreover, we give counterexamples to Shafarevich's problem on the rationality of the irreducible components of Chow varieties.     

On the torsion of group extensions and group actions

In this note, we study the torsion of extensions of finitely generated abelian by elementary abelian groups. When the action is trivial , we make a specific choice of a 1-cochain for a vanishing multiple of the cohomology class defining the extension and use it to completely describe the torsion of central extensions. As an application, one gets that, under the assumption of trivial action on homology, Z_p^r may act freely on (S¹)^k if and only if r?k, providing an alternative proof of the main theorem in [Trans. Amer. Math. Soc. 352 (6) (2000) 2689-2700] for central extensions.     

Hamiltonian torus actions on symplectic orbifolds and toric varieties

In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly, we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each open facet and that all such orbifolds are algebraic toric varieties.

     

Automorphism group actions on trees

We study the situation when the automorphism group of a recursively saturated structure acts on an ?‐tree. The cases of (?, <) and models of Peano Arithmetic are central in the paper. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)