We show that a closed, connected and orientable Riemannian manifold
M of dimension d that admits a nonconstant
quasiregular mapping from ℝ
d
must have bounded
dimension of the cohomology independent ...of the distortion of the map. The
dimension of the degree l de Rham cohomology of
M is bounded above by
(
d
l
).
This is a sharp upper bound that proves the Bonk-Heinonen conjecture. A
corollary of this theorem answers an open problem posed by Gromov in 1981. He
asked whether there exists a d-dimensional, simply connected
manifold that does not admit a quasiregular mapping from
ℝ
d
. Our result gives an affirmative answer
to this question.
For a projective variety V⊂Pkn over a field of characteristic zero, with homogeneous ideal I in A=kx0,…,xn, we consider the local cohomology modules HIi(A). These have a structure of holonomic ...D-module over A, and we investigate their filtration by simple D-modules. In case V is nonsingular, we can describe completely the simple D-module components of HIi(A) for all i, in terms of the Betti numbers of V.
We prove a version of the Mayer–Vietoris sequence for De Rham differential forms in diffeological spaces. It is based on the notion of a generating family instead of that of a covering by open ...subsets.
We consider the (graded) Matlis dual D(M) of a graded D-module M over the polynomial ring R=kx1,…,xn (k is a field of characteristic zero), and show that it can be given a structure of D-module in ...such a way that, whenever dimkHdRi(M) is finite, then HdRi(M) is k-dual to HdRn−i(D(M)). As a consequence, we show that if M is a graded D-module such that HdRn(M) is a finite-dimensional k-space, then dimk(HdRn(M)) is the maximal integer s for which there exists a surjective D-linear homomorphism M→Es, where E is the top local cohomology module H(x1,…,xn)n(R). This extends a recent result of Hartshorne and Polini on formal power series rings to the case of polynomial rings; we also apply the same circle of ideas to provide an alternate proof of their result. When M is a finitely generated graded D-module such that dimkHdRi(M) is finite, we generalize the above result further, showing that HdRn−i(M) is k-dual to ExtDi(M,E).
Let be a proper closed subset in the projective n-space over a field k of characteristic zero and let C(Y) be its affine cone. Let be the homogeneous defining ideal of Y with homogeneous maximal ...ideal . In this paper, we find out a lower bound of de Rham depth of C(Y) in terms of .
We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic ...cohomology by saying that this period regulator is surjective. Showing that a suitable Betti–de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. These divisibility properties are treated in an appendix by B. Kahn (extending previous work of Bloch and Colliot-Thélène–Raskind).
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