Let X be a smooth complex projective variety of dimension g. A Hodge class of degree 2d on X is, by definition, an element of H(X,Q)∩H(X). The cohomology class of an algebraic subvariety of codimension d of X is a Hodge class of degree 2d. The original Hodge conjecture states that any Hodge class on X is algebraic, i.e., a Q-linear combination of classes of algebraic subvarieties of X. Lefschet...

Let M be a Kobayashi-hyperbolic 2-dimensional complex manifold and Aut(M) the group of holomorphic automorphisms of M . We showed earlier that if dimAut(M) = 3, then Aut(M)-orbits are closed submanifolds in M of (real) codimension 1 or 2. In this paper we classify all connected Kobayashi-hyperbolic 2-dimensional manifolds with 3-dimensional automorphism groups in the case when every orbit has c...

A Birkhoff system is an algebra that has two binary operations · and +, with each being commutative, associative, and idempotent, and together satisfying x · (x + y) = x + (x · y). Examples of Birkhoff systems include lattices, and quasilattices, with the latter being the regularization of the variety of lattices. A number of papers have explored the bottom part of the lattice of subvarieties o...

Let X be a smooth complex projective variety of dimension g. A Hodge class of degree 2d on X is, by definition, an element of H(X,Q)∩H(X). The cohomology class of an algebraic subvariety of codimension d of X is a Hodge class of degree 2d. The classical Hodge conjecture states that any Hodge class on X is algebraic, i.e., a Q-linear combination of classes of algebraic subvarieties of X. Lefsche...

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U ⊂ X be an open set whose complement has codimension at least 2. We extend the theory of perverse coherent sheaves, due to Deligne and disseminated by Bezrukavnikov, by showing that a coherent middle extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheave...

A specific singularity of a vector field on R is considered, of codimension 2 in the dissipative case and of codimension 1 in the conservative case. In both contexts in generic unfoldings the existence is proved of subordinate Sil'nikov bifurcations, which have codimension 1. Special attention is paid to the C°°-flatness of this subordinate phenomenon.

In this paper we construct an additive basis for the cohomology ring of a regular nilpotent Hessenberg variety which is obtained by extending all Poincaré duals smaller subvarieties. particular, subvarieties are linearly independent.

Projective resolutions of modules over a ring R are constructed starting from appropriate projective resolutions over a ring Q mapping to R. It is shown that such resolutions may be chosen to be minimal in codimension ≤ 2, but not in codimension 3. This is used to obtain minimal resolutions for essentially all modules over local (or graded) rings R with codimension ≤ 2. Explicit resolutions are...