300 research outputs found
Asymptotic cohomological functions on projective varieties
In this paper we define certain analogues of the volume of a divisor - called
asymptotic cohomological functions - and investigate their behaviour on the
Neron--Severi space. We establish that asymptotic cohomological functions are
invariant with respect to the numerical equivalence of divisors, and that they
give rise to continuous functions on the real Neron--Severi space. To
illustrate the theory, we work out these invariants for abelian varieties,
smooth surfaces, and certain homogeneous spaces.Comment: 32 pages, 3 figure
Triangulated categories of mixed motives
This book discusses the construction of triangulated categories of mixed
motives over a noetherian scheme of finite dimension, extending Voevodsky's
definition of motives over a field. In particular, it is shown that motives
with rational coefficients satisfy the formalism of the six operations of
Grothendieck. This is achieved by studying descent properties of motives, as
well as by comparing different presentations of these categories, following and
extending insights and constructions of Deligne, Beilinson, Bloch, Thomason,
Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, R\"ondigs, {\O}stv{\ae}r,
and others. In particular, the relation of motives with -theory is addressed
in full, and we prove the absolute purity theorem with rational coefficients,
using Quillen's localization theorem in algebraic -theory together with a
variation on the Grothendieck-Riemann-Roch theorem. Using resolution of
singularities via alterations of de Jong-Gabber, this leads to a version of
Grothendieck-Verdier duality for constructible motivic sheaves with rational
coefficients over rather general base schemes. We also study versions with
integral coefficients, constructed via sheaves with transfers, for which we
obtain partial results. Finally, we associate to any mixed Weil cohomology a
system of categories of coefficients and well behaved realization functors,
establishing a correspondence between mixed Weil cohomologies and suitable
systems of coefficients. The results of this book have already served as ground
reference in many subsequent works on motivic sheaves and their realizations,
and pointers to the most recent developments of the theory are given in the
introduction.Comment: This is the final version. To appear in the series Springer
Monographs in Mathematic
On the Kottwitz conjecture for local Shimura varieties
Kottwitz’s conjecture describes the contribution of a supercuspidal represention to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. Using a Lefschetz-Verdier fixedpoint formula, we prove a weakened generalized version of Kottwitz’s conjecture. The weakening comes from ignoring the action of the Weil group and only considering the actions of the groups G and Jb up to non-elliptic representations. The generalization is that we allow arbitrary connected reductive groups G and non-minuscule coweights µ
An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization
In this paper we develop an axiomatic setup for algorithmic homological
algebra of Abelian categories. This is done by exhibiting all existential
quantifiers entering the definition of an Abelian category, which for the sake
of computability need to be turned into constructive ones. We do this
explicitly for the often-studied example Abelian category of finitely presented
modules over a so-called computable ring , i.e., a ring with an explicit
algorithm to solve one-sided (in)homogeneous linear systems over . For a
finitely generated maximal ideal in a commutative ring we
show how solving (in)homogeneous linear systems over can be
reduced to solving associated systems over . Hence, the computability of
implies that of . As a corollary we obtain the computability
of the category of finitely presented -modules as an Abelian
category, without the need of a Mora-like algorithm. The reduction also yields,
as a by-product, a complexity estimation for the ideal membership problem over
local polynomial rings. Finally, in the case of localized polynomial rings we
demonstrate the computational advantage of our homologically motivated
alternative approach in comparison to an existing implementation of Mora's
algorithm.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu
The present moment in quantum cosmology: challenges to the arguments for the elimination of time
Barbour, Hawking, Misner and others have argued that time cannot play an
essential role in the formulation of a quantum theory of cosmology. Here we
present three challenges to their arguments, taken from works and remarks by
Kauffman, Markopoulou and Newman. These can be seen to be based on two
principles: that every observable in a theory of cosmology should be measurable
by some observer inside the universe, and all mathematical constructions
necessary to the formulation of the theory should be realizable in a finite
time by a computer that fits inside the universe. We also briefly discuss how a
cosmological theory could be formulated so it is in agreement with these
principles.Comment: This is a slightly revised version of an essay published in Time and
the Instant, Robin Durie (ed.) Manchester: Clinamen Press, 200
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