International Journal of Mathematics

Consider an effective Hamiltonian circle action on a compact symplectic 2n-dimensional manifold (M, ω). Assume that the fixed set M^S1 is minimal, in two senses: It has exactly two components, X and Y, and dim(X) + dim(Y) = dim(M) - 2.

We prove that the integral cohomology ring and Chern classes of M are isomorphic to either those of ℂℙⁿ or (if n ≠ 1 is odd) to those of , the Grassmannian of oriented two-planes in ℝⁿ⁺². In particular, Hⁱ(M;ℤ) = Hⁱ(ℂℙⁿ; ℤ) for all i, and the Chern classes of M are determined by the integral cohomology ring. We also prove that the fixed set data of M agrees exactly with the fixed set data for one of the standard circle actions on one of these two manifolds. In particular, we show that there are no points with stabilizer ℤ_k for any k > 2.

The same conclusions hold when M^S1 has exactly two components and the even Betti numbers of M are minimal, that is, b_2i(M) = 1 for all i ∈ {0, …, ½dim(M)}. This provides additional evidence that very few symplectic manifolds with minimal even Betti numbers admit Hamiltonian actions.

We generalize and refine the hyperkähler Arnold conjecture, which was originally established, in the non-degenerate case, for three-dimensional time by Hohloch, Noetzel and Salamon by means of hyperkähler Floer theory. In particular, we prove the conjecture in the case where the time manifold is a multidimensional torus and also establish the degenerate version of the conjecture. Our method relies on Morse theory for generating functions and a finite-dimensional reduction along the lines of the Conley–Zehnder proof of the Arnold conjecture for the torus.

The smallest primitive ideal spaces for which there exist counterexamples to the classification of non-simple, purely infinite, nuclear, separable C*-algebras using filtrated K-theory, are four-point spaces. In this article, we therefore restrict to real rank zero C*-algebras with four-point primitive ideal spaces. Up to homeomorphism, there are ten different connected T₀-spaces with exactly four points. We show that filtrated K-theory classifies real rank zero, tight, stable, purely infinite, nuclear, separable C*-algebras that satisfy that all simple subquotients are in the bootstrap class for eight out of ten of these spaces.

We consider a semidirect product of the sheaf of vector fields on a manifold ℂ* × X with a central extension of the sheaf of Lie algebras of maps from ℂ* × X into a finite-dimensional simple Lie algebra, viewed as sheaves on X. Using vertex algebra methods we construct sheaves of modules for this sheaf of Lie algebras. Our results extend the work of Malikov–Schechtman–Vaintrob on the chiral de Rham complex.

We will prove that given a genus-2 fibration f : X → C on a smooth projective surface X such that b₁(X) = b₁(C) + 2, the fundamental group of X is almost isomorphic to π₁(C) × π₁(E), where E is an elliptic curve. We will also verify the Shafarevich Conjecture on holomorphic convexity of the universal cover of surfaces X with genus-2 fibration X → C such that b₁(X) > b₁(C).

In this paper we generalize the notion of Cuntz–Pimsner algebras of C*-correspondences to the setting of subproduct systems. The construction is justified in several ways, including the Morita equivalence of the operator algebras under suitable conditions and examples are provided to illustrate its naturality. We also demonstrate why some features of the Cuntz–Pimsner algebras of C*-correspondences fail to generalize to our setting, and discuss what we have instead.

Let G be a reductive affine algebraic group and let X be an affine algebraic G-variety. We establish a (poly)stability criterion for points x ∈ X in terms of intrinsically defined closed subgroups H_x of G and relate it with the numerical criterion of Mumford and with Richardson and Bate–Martin–Röhrle criteria, in the case X = G^N. Our criterion builds on a close analogue of a theorem of Mundet and Schmitt on polystability and allows the generalization to the algebraic group setting of results of Johnson–Millson and Sikora about complex representation varieties of finitely presented groups. By well established results, it also provides a restatement of the non-abelian Hodge theorem in terms of stability notions.

The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of generators) on nonsingular cubic surfaces X ⊂ ℙ³. We give necessary and sufficient conditions on the first Chern class D for the existence of stable Ulrich bundles on X of rank r and c₁ = D. When such bundles exist, we prove that the corresponding moduli space of stable bundles is smooth and irreducible of dimension D² - 2r² + 1 and consists entirely of stable Ulrich bundles (see Theorem 1.1). We are also able to prove the existence of stable Ulrich bundles of any rank on nonsingular cubic threefolds in ℙ⁴, and we show that the restriction map from bundles on the threefold to bundles on the surface is generically étale and dominant.

In this paper, we prove a structure theorem for projectively flat Finsler metrics of negative constant flag curvature. We show that for such a Finsler metric if the orthogonal group acts as isometries, then the Finsler metric is a slight generalization of Chern–Shen's construction Riemann–Finsler geometry, Nankai Tracts in Mathematics, Vol. 6 (World Scientific Publishing, Hackensack, NJ, 2005), x+192 pp.

For a smooth projective curve C with genus g ≥ 2 and a degree 1 line bundle on C, let be the moduli space of stable vector bundles of rank r over C with the fixed determinant . In this paper, we study the small rational curves on M and estimate the codimension of the locus of the small rational curves. In particular, we determine all small rational curves when r = 3.

We give a criterion for the Kostant–Kirillov form on an adjoint orbit in a real semisimple Lie group to be exact. We explicitly compute the second cohomology of all the nilpotent adjoint orbits in every complex simple Lie algebra.