2,941 research outputs found
Shapes of polyhedra and triangulations of the sphere
The space of shapes of a polyhedron with given total angles less than 2\pi at
each of its n vertices has a Kaehler metric, locally isometric to complex
hyperbolic space CH^{n-3}. The metric is not complete: collisions between
vertices take place a finite distance from a nonsingular point. The metric
completion is a complex hyperbolic cone-manifold. In some interesting special
cases, the metric completion is an orbifold. The concrete description of these
spaces of shapes gives information about the combinatorial classification of
triangulations of the sphere with no more than 6 triangles at a vertex.Comment: 39 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTMon1/paper25.abs.htm
Germs of de Rham cohomology classes which vanish at the generic point
We show that hypergeometric differential equations, unitary and Gauss-Manin
connections give rise to de Rham cohomology sheaves which do not admit a
Bloch-Ogus resolution. The latter is in contrast to Panin's theorem asserting
that corresponding \'etale cohomology sheaves do fulfill Bloch-Ogus theory.Comment: latex 2e, 6 page
On the Deligne--Beilinson cohomology sheaves
We are showing that the Deligne--Beilinson cohomology sheaves are torsion free by assuming Kato's conjectures
hold true for function fields. This result is `effective' for ; in this
case, by dealing with `arithmetic properties' of the presheaves of mixed Hodge
structures defined by singular cohomology, we are able to give a cohomological
characterization of the Albanese kernel for surfaces with .Comment: 12 pages, LaTeX 2.0
Graded and Filtered Fiber Functors on Tannakian Categories
We study fiber functors on Tannakian categories which are equipped with a
grading or a filtration. Our goal is to give a comprehensive set of
foundational results about such functors. A main result is that each filtration
on a fiber functor can be split by a grading fpqc-locally on the base scheme
Chow rings and decomposition theorems for families of K3 surfaces and Calabi-Yau hypersurfaces
The decomposition theorem for smooth projective morphisms
says that decomposes as
. We describe simple examples where it is not
possible to have such a decomposition compatible with cup-product, even after
restriction to Zariski dense open sets of . We prove however that this is
always possible for families of surfaces (after shrinking the base), and
show how this result relates to a result by Beauville and the author on the
Chow ring of surfaces . We give two proofs of this result, the second
one involving a certain decomposition of the small diagonal in also
proved by Beauville and the author}. We prove an analogue of such a
decomposition of the small diagonal in for Calabi-Yau hypersurfaces
in , which in turn provides strong restrictions on their Chow
ring.Comment: Final version, to appear in Geometry \& Topolog
2-Gerbes bound by complexes of gr-stacks, and cohomology
We define 2-gerbes bound by complexes of braided group-like stacks. We prove
a classification result in terms of hypercohomology groups with values in
abelian crossed squares and cones of morphisms of complexes of length 3. We
give an application to the geometric construction of certain elements in
Hermitian Deligne cohomology groups.Comment: 70 pages, latex+amsmath+xypi
Integration of simplicial forms and Deligne cohomology
We present two approaches to constructing an integration map for smooth
Deligne cohomology. The first is defined in the simplicial model, where a class
in Deligne cohomology is represented by a simplicial form, and the second in a
related but more combinatorial model.Comment: 28 pages, section on products adde
Supersolutions
We develop classical globally supersymmetric theories. As much as possible,
we treat various dimensions and various amounts of supersymmetry in a uniform
manner. We discuss theories both in components and in superspace. Throughout we
emphasize geometric aspects. The beginning chapters give a general discussion
about supersymmetric field theories; then we move on to detailed computations
of lagrangians, etc. in specific theories. An appendix details our sign
conventions. This text will appear in a two-volume work "Quantum Fields and
Strings: A Course for Mathematicians" to be published soon by the American
Mathematical Society. Some of the cross-references may be found at
http://www.math.ias.edu/~drm/QFT/Comment: 130 pages, AMSTe
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