Characteristic classes of flat bundles, II

On a smooth varietyX defined over a fieldK of characteristic zero, one defines characteristic classes of bundles with an integrableK-connection in a group lifting the Chow group, which map, whenK is the field of complex numbers andX is proper, to Cheeger-Simons' secondary analytic invariants, compatibly with the cycle map in the Deligne cohomology.     

Characteristic classes for complex bundles with trivial real reduction

This note concerns itself with a theory of characteristic classes for those complex bundles whose real reductions are trivial.

     

Complexification of foliations and complex secondary classes

Some properties of complex secondary classes are discussed. It is shown that the Godbillon-Vey class and the Bott class are related via complexification.Supported by Ministry of Education, Culture, Sports, Science and Technology, Grant No. 13740042     

Divisibility of the stable Miller-Morita-Mumford classes

We determine the sublattice generated by the Miller-Morita- Mumford classes in the torsion free quotient of the integral cohomology ring of the stable mapping class group. We further decide when the mod reductions vanish.

     

A short proof of smooth implies flat

Proofs that a smooth morphism is flat available in the literature are long and di?cult. We give a short proof of this fact.     

On a characterization of finite vector bundles as vector bundles admitting a flat connection with finite monodromy group

We prove that a holomorphic vector bundle over a compact connected Kähler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials and , with nonnegative integral coefficients, such that the vector bundle is isomorphic to . An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group.

When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.

     

Vector bundles with holomorphic connection over a projective manifold with tangent bundle of nonnegative degree

For a projective manifold whose tangent bundle is of nonnegative degree, a vector bundle on it with a holomorphic connection actually admits a compatible flat holomorphic connection, if the manifold satisfies certain conditions. The conditions in question are on the Harder-Narasimhan filtration of the tangent bundle, and on the Neron-Severi group.

     

Morse theory and conjugacy classes of finite subgroups

We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type F _n inside mapping class groups, Aut(), and Out() which have infinitely many conjugacy classes of finite-order elements.      

Extremal Kähler metrics on projective bundles over a curve

Let M=P(E) be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle E→Σ over a compact complex curve Σ of genus ?2. Building on ideas of Fujiki (1992) [27], we prove that M admits a Kähler metric of constant scalar curvature if and only if E is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of E as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in Kähler classes sufficiently far from the boundary of the Kähler cone. The methods used to prove the above results apply to a wider class of manifolds, called rigid toric bundles over a semisimple base, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds.     

On the isomorphism classes of Legendre elliptic curves over finite fields

In this paper,the number of isomorphism classes of Legendre elliptic curves over finite field is enumerated.     

On the number of conjugacy classes of finite nilpotent groups

We establish the first super-logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group. In particular, we obtain that for any constant c there are only finitely many finite p-groups of order pm with at most c⋅m conjugacy classes. This answers a question of L. Pyber.     

Vector bundles on a product of real cubic curves

In this paper we give a characterization of then-tuples (C ₁,...,C _n ) of nonsingular projective real cubic curves such that every topological complex vector bundle onC ₁×...×C _n admits an algebraic structure. The results are very explicit and can be expressed in an especially simple form for cubies defined over the rationals.The second author was supported by an NSF grant.     

On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane

To each once-punctured-torus bundle, Tφ, over the circle with pseudo-Anosov monodromy φ, there are associated two tessellations of the complex plane: one, Δ(φ), is (the projection from ∞ of) the triangulation of a horosphere at ∞ induced by the canonical decomposition into ideal tetrahedra, and the other, CW(φ), is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through ∞. In this paper, we fully describe the relation between Δ(φ) and CW(φ).     

Flat holomorphic connections on principal bundles over a projective manifold

Let be a connected complex linear algebraic group and its unipotent radical. A principal -bundle over a projective manifold will be called polystable if the associated principal -bundle is so. A -bundle over is polystable with vanishing characteristic classes of degrees one and two if and only if admits a flat holomorphic connection with the property that the image in of the monodromy of the connection is contained in a maximal compact subgroup of .

     

On trivialities of Stiefel-Whitney classes of vector bundles over highly connected complexes

It is a classical theorem of Milnor that for every vector bundle over Sn, all the Stiefel-Whitney classes vanish if and only if n≠1,2,4,8. We describe a space B as W-trivial (except for one dimension) if for every vector bundle over B, all the Stiefel-Whitney classes vanish (except for a single fixed dimension). We establish theorems which state that certain high-connectivities of B imply these trivialities as well as a theorem which states that there are infinitely many “W-trivial except for one dimension” spaces.     

On multiply of sections of line bundles on compact Riemann surfaces

In this paper,we give some conditions on the surjective of multiply maps H~0(R,L)×H~0(R,K)→H~0(R,L(?)K).Here R is a compact Riemann surface,L a line bundle on R and K is the canonical line bundle.     

Generalization of a criterion for semistable vector bundles

It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that both H⁰(X,EF) and H¹(X,EF) vanishes. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined over a perfect field k, and let E be a vector bundle on X. We prove that E is semistable if and only if there is a vector bundle F on X such that Hⁱ(X,EF)=0 for all i. We also give an explicit bound for the rank of F.