Related papers: Vanishing Cycles and Thom's $a_f$ Condition

We show that Bloch's complex of relative zero-cycles can be used as a dualizing complex over perfect fields and number rings. This leads to duality theorems for torsion sheaves on arbitrary separated schemes of finite type over…

For a holomorphic function on a complex manifold, we show that the vanishing cohomology of lower degree at a point is determined by that for the points near it, using the perversity of the vanishing cycle complex. We calculate it explicitly…

We introduce a join construction as a way of completing the description of the relative conormal space of an analytic function on a complex analytic space that has a non-vanishing derivative at the origin. Then we show how to obtain a…

We use the geometry of the secant variety to an embedded smooth curve to prove some vanishing and regularity theorems for powers of ideal sheaves.

In this paper we study the cyclicity of sliding cycles for regularized piecewise smooth visible-invisible two-folds, in the presence of singularities of the Filippov sliding vector field located away from two-folds. We obtain a slow-fast…

In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. We explain the relation between nearby cycles for…

We prove that the shifted vanishing cycles and nearby cycles commute with Verdier dualizing up to a {\bf natural} isomorphism, even when the coefficients are not in a field.

We construct Morse homology groups associated with any regular function on a smooth complex algebraic variety, allowing singular and non-compact critical loci. These groups are generated by critical points of a certain large pertubation of…

In the first half of the paper, we translate in the geometric situation of Drinfeld varieties, the principal results of the Harris and Taylor's book. We give in particular the restriction to the open strata of the vanishing cycles sheaves…

Using the vanishing cycles of simple singularities, we study the eigenvectors of Cartan matrices of finite root systems, and of q-deformations of these matrices.

An introduction is given to the logic of sheaves of structures and to set theoretic forcing constructions based on this logic. Using these tools, it is presented an alternative proof of the independence of the Continuum Hypothesis; which…

We show that the deformation theory of a perfect complex and that of its determinant are related by the trace map, in a general setting of sheaves on a site. The key technical step, in passing from the setting of modules over a ring where…

We reprove and generalize the result that the intersection cohomology groups of a toric variety with coefficient in a nontrivial rank one local system vanish. We prove a similar vanishing result for a certain class of varieties on which a…

A correspondence between quasicoherent sheaves on toric schemes and graded modules over some homogeneous coordinate ring is presented, and the behaviour of several finiteness properties under this correspondence is investigated.

We show that under some conditions, two constructions of nearby cycles over general bases coincide. More specifically, we show that under the assumption of $\Psi$-factorizability, the constructions of unipotent nearby cycles over an affine…

We give a diagrammatic summary of the connections between various theorems and conjectures about the vanishing of the Euler characteristic.

We prove a Generic Vanishing Theorem for coherent sheaves on an abelian variety over an algebraically closed field $k$. When $k=\CC$ this implies a conjecture of Green and Lazarsfeld.

Remarks on the Hodge-Grothendieck class of the nearby cycles functor and a generalized local invariant cycles result.

We investigate the emptiness of adjoint linear systems associated to successive multiples of a given positive divisor with real coefficients

We develop a new method to construct the virtual fundamental classes for quasi-smooth derived schemes using the perverse sheaves of vanishing cycles on their $-1$-shifted contangent spaces. It is based on the author's previous work that can…