Grothendieck and vanishing cycles

Vanishing Vanishing Cycles

If A• is a bounded, constructible complex of sheaves on a complex analytic space X, and f : X → C and g : X → C are complex analytic functions, then the iterated vanishing cycles φg[−1](φf [−1]A •) are important for a number of reasons. We give a formula for the stalk cohomology H∗(φg[−1]φf [−1]A •)x in terms of relative polar curves, algebra, and the normal Morse data and micro-support of A•.

Vanishing Cycles and Mutation

Using Floer cohomology, we establish a connection between Picard-Lefschetz theory and the notion of mutation of exceptional collections in homological algebra.

Vanishing Cycles and Hermitian Duality

We show the compatibility between the moderate or nearby cycle functor for regular holonomic D-modules, as defined by Beilinson, Kashiwara and Malgrange, and the Hermitian duality functor, as defined by Kashiwara. Introduction The Hermitian dual of a D-module was introduced by M. Kashiwara in [9], who showed that the Hermitian dual of a regular holonomic D-module is also regular holonomic (henc...

BPS States, Weight Spaces and Vanishing Cycles

We review some simple group theoretical properties of BPS states, in relation with the singular homology of level surfaces. Primary focus is on classical and quantum N = 2 supersymmetric Yang-Mills theory, though the considerations can be applied to string theory as well. There has been dramatic recent progress in understanding non-perturbative properties of certain supersymmetric field and str...

Vanishing Cycles , the Generalized Hodgeconjecture and Gr