Singularities of functions: a global point of view

Abstract

This text surveys cohomological properties of pairs $(U,f)$ consisting of a smooth complex quasi-projective variety $U$ together with a regular function on~it. On the one hand, one tries to mimic the case of a germ of holomorphic function in its Milnor ball and, on the other hand, one takes advantage of the algebraicity of~$U$ and $f$ to apply technique of algebraic geometry, in particular Hodge theory. The monodromy properties are expressed by means of tools provided by the theory of linear differential equations, by mimicking the Stokes phenomenon. In the case of tame functions on smooth affine varieties, which is an algebraic analogue of that of a holomorphic function with an isolated critical point, the theory simplifies much and the formulation of the results are nicer. Examples of such tame functions are exhibited.

59 pages, 3 figures. To appear as a chapter in the book series "Handbook of Geometry and Topology of Singularities". V2: A few typos corrected and a reference added. V3: Final version, 66 pages

References in corpus (13)

• The orbifold Chow ring of toric Deligne-Mumford stacks
• Introduction to Stokes structures
• E_1-degeneration of the irregular Hodge filtration (with an appendix by Morihiko Saito)
• Examples of non-commutative Hodge structures
• On the irregular Hodge filtration of exponentially twisted mixed Hodge modules
• Hodge theory of Kloosterman connections
• Quadratic relations between Bessel moments
• Hodge-Tate conditions for Landau-Ginzburg models
• Irregular Hodge theory
• Topological computation of some Stokes phenomena on the affine line
• Quadratic relations between periods of connections
• A canonical Frobenius structure
• Hodge numbers of motives attached to Kloosterman and Airy moments