# Vanishing cycle sheaves of one-parameter smoothings and quasi-semistable degenerations - Mathematics > Algebraic Geometry

Vanishing cycle sheaves of one-parameter smoothings and quasi-semistable degenerations - Mathematics > Algebraic Geometry - Download this document for free, or read online. Document in PDF available to download.

Abstract: We study the vanishing cycles of a one-parameter smoothing of a complexanalytic space and show that the weight filtration on its perverse cohomologysheaf of the highest degree is quite close to the monodromy filtration so thatits graded pieces have a modified Lefschetz decomposition. We describe itsprimitive part using the weight filtration on the perverse cohomology sheavesof the constant sheaves. As a corollary we show in the local completeintersection case that 1 is not an eigenvalue of the monodromy on the reducedMilnor cohomology at any points if and only if the total space and the singularfiber are both rational homology manifolds. Also we introduce quasi-semistabledegenerations and calculate the limit mixed Hodge structure by constructing theweight spectral sequence. As a corollary we show non-triviality of the space ofvanishing cycles of the Lefschetz pencil associated with a tensor product ofany two very ample line bundles except for the case of even-dimensionalprojective space where two has to be replaced by three.

Author: ** Alexandru Dimca, Morihiko Saito**

Source: https://arxiv.org/