Iterated traces in 2–categories and Lefschetz theorems

Weak Lefschetz Theorems

A new approach to Nori's weak Lefschetz theorem is described. The new approach, which involves the ¯ ∂-method, avoids moving arguments and gives much stronger results. In particular, it is proved that if X and Y are connected smooth projective varieties of positive dimension and f : Y − → X is a holomorphic immersion with ample normal bundle, then the image of π 1 (Y) in π 1 (X) is of finite in...

Iterated combinatorial density theorems

Lefschetz Theorems for Torsion Algebraic Cycles in Codimension 2

Let Y be a smooth projective variety over C, and X be a smooth hypersurface in Y . We prove that the natural restriction map on Chow groups of codimension two cycles is an isomorphism when restricted to the torsion subgroups provided dimY ě 5. We prove an analogous statement for a very general hypersurface X Ă P of degree ě 5. In the more general setting of a very general hypersurface X of suff...

Generalized Lefschetz fixed point theorems in extension type spaces

Several Lefschetz fixed point theorems for compact type self maps in new classes of spaces are presented in this paper.

Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations

Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose severe restrictions on the cohomology algebra of a smooth compact Kähler manifold or on the intersection cohomology of a projective toric variety; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions o...