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.

     

Compact anti-self-dual orbifolds with torus actions

We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal structure. A compact anti-self-dual orbifold can also be constructed by adding a point at infinity to an asymptotically locally Euclidean (ALE) scalar-flat K?hler orbifold. We use this observation to classify ALE scalar-flat K?hler 4-orbifolds whose isometry group contain a 2-torus.     

On 4-manifolds with singular torus actions

Mathematische Annalen -     

Faithful tropicalisation and torus actions

For any affine variety equipped with coordinates, there is a surjective, continuous map from its Berkovich space to its tropicalisation. Exploiting torus actions, we develop techniques for finding an explicit, continuous section of this map. In particular, we prove that such a section exists for linear spaces, Grassmannians of planes (reproving a result due to Cueto, Häbich, and Werner), matrix varieties defined by the vanishing of 3 × 3-minors, and for the hypersurface defined by Cayley’s hyperdeterminant.     

Classification of torus manifolds with codimension one extended actions

The purpose of this paper is to classify torus manifolds (M ²ⁿ , T ⁿ ) with codimension one extended G-actions (M ²ⁿ , G) up to essential isomorphism, where G is a compact, connected Lie group whose maximal torus is T ⁿ . For technical reasons, we do not assume torus manifolds are orientable. We prove that there are seven types of such manifolds. As a corollary, if a nonsingular toric variety or a quasitoric manifold has a codimension one extended action then such manifold is a complex projective bundle over a product of complex projective spaces.     

Exact cohomology sequences with integral coefficients for torus actions

Using methods applied by Atiyah in equivariant K-theory, Bredon obtained exact sequences for the relative cohomologies (with rational coefficients) of the equivariant skeletons of (sufficiently nice) T-spaces X, with free equivariant cohomology over Here we characterise those finite T-CW complexes with connected isotropy groups for which an analogous result holds with integral coefficients.     

Centered complexity one Hamiltonian torus actions

We consider symplectic manifolds with Hamiltonian torus actions which are ``almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are ``centered" and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmannians and by two equal symplectic balls.

     

Toeplitz operators and Hamiltonian torus actions

This paper is devoted to semi-classical aspects of symplectic reduction. Consider a compact prequantizable Kähler manifold M with a Hamiltonian torus action. In the seminal paper [V. Guillemin, S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (3) (1982) 515-538], Guillemin and Sternberg introduced an isomorphism between the invariant part of the quantum space associated to M and the quantum space associated to the symplectic quotient of M, provided this quotient is non-singular. We prove that this isomorphism is a Fourier integral operator and that the Toeplitz operators of M descend to Toeplitz operators of the reduced phase space. We also extend these results to the case where the symplectic quotient is an orbifold and estimate the spectral density of a reduced Toeplitz operator, a result related to the Riemann-Roch-Kawasaki theorem.     

Equivariant Chow groups for torus actions

We study Edidin and Graham's equivariant Chow groups in the case of torus actions. Our main results are: (i) a presentation of equivariant Chow groups in terms of invariant cycles, which shows how to recover usual Chow groups from equivariant ones; (ii) a precise form of the localization theorem for torus actions on projective, nonsingular varieties; (iii) a construction of equivariant multiplicities, as functionals on equivariant Chow groups; (iv) a construction of the action of operators of divided differences on theT-equivariant Chow group of any scheme with an action of a reductive group with maximal torusT. We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations. In particular, we obtain a presentation of the Chow ring of any smooth, projective spherical variety.     

Affine actions of higher rank lattices

We consider actions of lattices in certain higher rank simple Lie groups by affine (i.e. connection-preserving) transformations of a compact Riemannian manifold. When the dimension of the manifold is not too large, such actions are partially described here in terms of affine actions on the flat torus and isometric actions. The main tools are Marguils' and Zimmer's rigidity theorems.     

Examples of non-Kähler Hamiltonian torus actions

An important question with a rich history is the extent to which the symplectic category is larger than the K?hler category. Many interesting examples of non-K?hler symplectic manifolds have been constructed [T] [M] [G]. However, sufficiently large symmetries can force a symplectic manifolds to be K?hler [D] [Kn]. In this paper, we solve several outstanding problems by constructing the first symplectic manifold with large non-trivial symmetries which does not admit an invariant K?hler structure. The proof that it is not K?hler is based on the Atiyah-Guillemin-Sternberg convexity theorem [At] [GS]. Using the ideas of this paper, C. Woodward shows that even the symplectic analogue of spherical varieties need not be K?hler [W]. Oblatum IX-1995 & 3-III-1997     

The biinvariant diagonal class for Hamiltonian torus actions

Suppose that an algebraic torus G acts algebraically on a projective manifold X with generically trivial stabilizers. Then the Zariski closure of the set of pairs {(x,y)∈X×X|y=gx for some g∈G} defines a nonzero equivariant cohomology class . We give an analogue of this construction in the case where X is a compact symplectic manifold endowed with a Hamiltonian action of a torus, whose complexification plays the role of G. We also prove that the Kirwan map sends the class [ΔG] to the class of the diagonal in each symplectic quotient. This allows to define a canonical right inverse of the Kirwan map.     

Local torus actions modeled on the standard representation

We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants called a characteristic pair and an Euler class of the orbit map, and prove that local torus actions are classified topologically by them. As a corollary, we obtain a topological classification of locally standard torus actions, which includes the topological classifications of quasi-toric manifolds by Davis and Januszkiewicz and of effective T²-actions on four-dimensional manifolds without nontrivial finite stabilizers by Orlik and Raymond. We discuss locally toric Lagrangian fibrations from the viewpoint of local torus actions. We also investigate the topology of a manifold equipped with a local torus action when the Euler class of the orbit map vanishes.     

Rational smoothness and fixed points of torus actions

We obtain a criterion for rational smoothness of an algebraic variety with a torus action, with applications to orbit closures in flag varieties, and to closures of double classes in regular group completions.Dedicated to the memory of Claude Chevalley