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 Bridgeland stability. For instance stability of coherent sheaves in the case of slope of coherent sheaves?
The Stokes phenomenon in the study of asymptotics of differential equations may be regarded 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?.
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 numbersâ€¦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.
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 integralsâ€¦
L. Katzarkov, M. Kontsevich, T. Pantev, Hodge theoretic aspects of mirror symmetry, arxiv/0806.0107
Maxim Kontsevich, Yan Soibelman, Motivic Donaldson-Thomas invariants: summary of results, 0910.4315
Last revised on December 11, 2018 at 13:08:26. See the history of this page for a list of all contributions to it.