25,124 research outputs found
Algebraic computation of some intersection D-modules
Let be a complex analytic manifold, a locally
quasi-homogeneous free divisor, an integrable logarithmic connection with
respect to and the local system of the horizontal sections of on
. In this paper we give an algebraic description in terms of of the
regular holonomic D-module whose de Rham complex is the intersection complex
associated with . As an application, we perform some effective computations
in the case of quasi-homogeneous plane curves.Comment: 18 page
An algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation
We give an algorithm to compute the following cohomology groups on U = \C^n
\setminus V(f) for any non-zero polynomial f \in \Q[x_1, ..., x_n]; 1.
H^k(U, \C_U), \C_U is the constant sheaf on with stalk \C. 2. H^k(U,
\Vsc), \Vsc is a locally constant sheaf of rank 1 on . We also give
partial results on computation of cohomology groups on for a locally
constant sheaf of general rank and on computation of H^k(\C^n \setminus Z,
\C) where is a general algebraic set. Our algorithm is based on
computations of Gr\"obner bases in the ring of differential operators with
polynomial coefficients.Comment: 38 page
Gauge Backgrounds and Zero-Mode Counting in F-Theory
Computing the exact spectrum of charged massless matter is a crucial step
towards understanding the effective field theory describing F-theory vacua in
four dimensions. In this work we further develop a coherent framework to
determine the charged massless matter in F-theory compactified on elliptic
fourfolds, and demonstrate its application in a concrete example. The gauge
background is represented, via duality with M-theory, by algebraic cycles
modulo rational equivalence. Intersection theory within the Chow ring allows us
to extract coherent sheaves on the base of the elliptic fibration whose
cohomology groups encode the charged zero-mode spectrum. The dimensions of
these cohomology groups are computed with the help of modern techniques from
algebraic geometry, which we implement in the software gap. We exemplify this
approach in models with an Abelian and non-Abelian gauge group and observe
jumps in the exact massless spectrum as the complex structure moduli are
varied. An extended mathematical appendix gives a self-contained introduction
to the algebro-geometric concepts underlying our framework.Comment: 41 pages + extended appendice
Computing Invariants of Simplicial Manifolds
This is a survey of known algorithms in algebraic topology with a focus on
finite simplicial complexes and, in particular, simplicial manifolds. Wherever
possible an elementary approach is chosen. This way the text may also serve as
a condensed but very basic introduction to the algebraic topology of simplicial
manifolds.
This text will appear as a chapter in the forthcoming book "Triangulated
Manifolds with Few Vertices" by Frank H. Lutz.Comment: 13 pages, 3 figure
Projective schemes: What is Computable in low degree?
This article first presents two examples of algorithms that extracts
information on scheme out of its defining equations. We also give a review on
the notion of Castelnuovo-Mumford regularity, its main properties (in
particular its relation to computational issues) and different ways that were
used to estimate it
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