1,720,971 research outputs found
A remark on the generalized Hodge conjecture.
No full textIt is well known that for a smooth, projective variety X over C
the space of the coniveau filtration N^pH^i(X,Q) is contained in
the intersection of the space of the Hodge filtration F^pH^i(X,C)
with H^i(X,Q). In general this inclusion is strict. We introduce
a natural subspace S^{p,i} of F^pH^i(X,C) such that for any
integers i,p, N^pH^i(X,Q) is the intersection of S^{ p,i} with
H^i(X,Q). The main technical tool is the use of semi-algebraic
sets, which are available by the triangulation of complex projective
varieties
On the supports for cohomology classes of complex manifolds.
No full textLet c be a cohomology class of a compact connected complex manifold X. The paper deals with the possibility to construct a topological cycle S on X, whose class is the Poincare' dual of c, which is closely related in a precise sense to the complex structure of X. A topological obstruction for this is analyzed
On Grothendieck counterexample to the Generalized Hodge Conjecture
No full textFor a smooth complex projective variety X, let N^p the subspace of the cohomology space H^i(X, Q) of the classes supported by an algebraic subvariety of codimension at least p. Grothendieck showed that a conjectural description of this space given by Hodge is false, by an explicit example. Recently the point of view of Hodge was somewhat refined (Portelli, 2014), and we aimed to use
this refinement to revisit Grothendieck’s example. We explicitly compute the classes in this second space which are not in N^1H^3(X, Q). We also get a complete clarification that the representation of the homology customarily used for complex tori does not allow to apply the methods of (Portelli, 2014) to give a different proof that N^1H^3(X, Q) is different from the space conjectured by Hodge
On threefolds covered by lines
No full textA classification theorem is given of projective threefolds that are covered by the lines of a two-dimensional family, but not by a higher dimensional family. Precisely, if X is such a threefold, let S denote the Fano scheme of lines on X and m the number of lines contained in X and passing through a general point of X. Assume that S is generically reduced. Then m < 6. Moreover, X is birationally a scroll over a surface (m = 1), or X is a quadric bundle, or X belongs to a finite list of threefolds of degree at most 6. The smooth varieties of the third type are precisely the Fano threefolds with - K_X = 2H_X
On the Divisor Class Group of Localizations, Completions and Veronesean Subrings of Z-graded Krull Domains
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