Narrow Results

We investigate the relative lie algebroid connections on a holomorphic vector bundle over a family of compact complex manifolds (or smooth projective varieties over $ℂ$). We provide a sufficient condition for the existence of a relative Lie algebroid connection on a holomorphic vector bundle over a complex analytic family of compact complex manifolds. We show that the relative Lie algebroid Chern classes of a holomorphic vector bundle admitting relative Lie algebroid connection vanish, if each of the fibers of the complex analytic family is compact and Kähler. Moreover, we consider the moduli space of relative Lie algebroid connections and we show that there exists a natural relative compactification of this moduli space.

We prove the existence of a parabolic curve for a germ of biholomorphic map tangent to the identity at an isolated singular point of a surface under some conditions. For this purpose, we present a Camacho–Sad type index theorem for fixed curves of biholomorphic maps of singular surfaces and develop a local intersection theory of curves in singular surfaces from an analytic approach by means of Grothendieck residues.

In 1981 W. Fulton and R. MacPherson introduced the notion of bivariant theory (BT), which is a sophisticated unification of covariant theories and contravariant theories. This is for the study of singular spaces. In 2001 M. Levine and F. Morel introduced the notion of algebraic cobordism, which is a universal oriented Borel–Moore functor with products (OBMF) of geometric type, in an attempt to understand better V. Voevodsky's (higher) algebraic cobordism. In this paper we introduce a notion of oriented bivariant theory (OBT), a special case of which is nothing but the oriented Borel–Moore functor with products. The present paper is a first one of the series to try to understand Levine–Morel's algebraic cobordism from a bivariant theoretical viewpoint, and its first step is to introduce OBT as a unification of BT and OBMF.

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.

Some cohomology elements, called $\nu$-classes, as a supergeneralization of universal Chern classes, are introduced for canonical super line bundles over $\nu$-projective spaces, a novel supergeometric generalization of projective spaces. It is shown that these classes may be described by analytic representatives of elements of generalized de Rham cohomology.

One provides a detailed construction of the Schwartz classes. They are characteristic classes associated to complex analytic singular varieties. In a first step, one gives the construction of Schwartz classes by obstruction theory. Then one relates these classes to Mather's and MacPherson's ones. The third part is devoted to the computation of examples. The last section deals with polar varieties and definitions of characteristic classes via polar varieties. These are old and new results, partly obtained jointly with M.-H. Schwartz, with G. Gonzalez-Sprinberg, with G. Barthel, K.-H. Fieseler and L. Kaup and with P. Aluffi.