nLab
slope filtration

Overview

There are many examples that the objects of additive categories (abelian, abelian tensor categories, triangulated etc.) are equipped with slope filtrations, which are labelled by real numbers. Slope is here related to stability. Stokes phenomenon in the study of asymptotics of differential equations can be viewed as a similar structure on the D-modules (or by the Riemann-Hilbert correspondence, on the object in certain category of monodromy data), which correspond to the solutions of differential equations.

See also wall crossing, Stokes phenomenon, Newton polygon?.

Survey by Andre

For the slopes in algebraic and arithmetic geometry there is a recent survey

Here is an introductory passage:

Slope filtrations occur in algebraic and analytic geometry, in asymptotic analysis, in ramification theory, in p-adic theories, in geometry of numbers…Five basic examples are the Harder-Narasimhan filtration of vector bundles over a smooth projective curve, the Dieudonne-Manin filtration of F-isocrystals over the p-adic point, the Turrittin-Levelt filtration of formal differential modules, the Hasse-Arf filtration of finite Galois representations of local fields, and the Grayson-Stuhler filtration of euclidean lattices. Despite the variety of their origins, these filtrations share a lot of similar features.

Other aspects and references

The story of slope filtrations is important for the construction of various moduli spaces (esp. in algebraic geometry and in mathematical physics), for Hodge theory, theory of motives, study of motivic properties of Feynman integrals

  • L. Katzarkov, M. Kontsevich, T. Pantev, Hodge theoretic aspects of mirror symmetry, arxiv/0806.0107

  • M. Kontsevich, Y. Soibelman, Motivic Donaldson-Thomas invariants: summary of results, 0910.4315

Created on March 24, 2010 18:48:54 by Zoran Škoda (161.53.130.104)