相关论文: Vanishing cycles in complex symplectic geometry

The paper is on the vanishing topology of singular Milnor fibres of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices with sufficiently many parameters. We define vanishing cycles on such fibres, prove an…

The variation operator associated with an isolated hypersurface singularity is a classical topological invariant that relates relative and absolute homologies of the Milnor fiber via a non trivial isomorphism. Here we work with a…

We employ the perverse vanishing cycles to show that each reduced cohomology group of the Milnor fiber, except the top two, can be computed from the restriction of the vanishing cycle complex to only singular strata with a certain lower…

In this work, we analyze vanishing cycles of Feynman loop integrals by means of the Mayer-Vietoris spectral sequence. A complete classification of possible vanishing geometries are obtained. We employ this result for establishing an…

We study Milnor fibers and symplectic fillings of links of sandwiched singularities, with the goal of contrasting their algebro-geometric deformation theory and symplectic topology. In the algebro-geometric setting, smoothings of sandwiched…

We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We then describe the topology of the regular and singular fibres, in particular calculating their middle Betti numbers. For the…

We study the monodromy of vanishing cycles for map-germs $f:(C^{2n},0) \to (\CM^k,0)$ whose components are in involution. Although the singular fibres of such maps have non-isolated singularities, it is shown that the regular fibres are…

We study vanishing cycles naturally attached to a meromorphic function with isolated singularities, in both local and global settings.

We prove an extended Lefschetz principle for a large class of pencils of hypersurfaces having isolated singularities, possibly in the axis, and show that the module of vanishing cycles is generated by the images of certain variation maps.

Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive…

Let f be an isolated plane curve singularity with Milnor fiber of genus at least 5. For all such f, we give (a) an intrinsic description of the geometric monodromy group that does not invoke the notion of the versal deformation space, and…

The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analogue for an isolated singularity. We define the monodromy Lagrangian Floer…

The paper continues the discussion of symplectic aspects of Picard-Lefschetz theory begun in "Vanishing cycles and mutation" (this archive). There we explained how to associate to a suitable fibration over a two-dimensional disc a…

In this paper we introduce a new homology theory devoted to the study of families such as semi-algebraic or subanalytic families and in general to any family definable in an o-minimal structure (such as Denjoy-Carleman definable or $ln-exp$…

We describe the local and Steenbrink vanishing problems for singular symplectic varieties with isolated singularities. We do this by constructing a morphism $$\mathbb D_X(\underline \Omega_X^{n+p}) \to \underline \Omega_X^{n+p}$$ for a…

Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds, i.e., manifolds equipped with a closed 2-form which is symplectic outside a union of embedded 1-dimensional submanifolds, and…

We study the vanishing neighbourhood of non-isolated singularities of functions on singular spaces by associating a general linear function. We use the carrousel monodromy in order to show how to get a better control over the attaching of…

A classical way to construct a Lagrangian in a symplectic manifold $\Sigma$ is to let $\Sigma$ appear as a smooth fiber in a Lefschetz fibration. If this is possible the singularities of the fibration induce Lagrangian spheres in $\Sigma$…

By using our previous results on L\^e modules and an upper-bound on the betti numbers which we proved with L\^e, we investigate the cohomology of Milnor fibers and the internal local systems given by the vanishing cycles of hypersurfaces…

We give a survey on some aspects of the topological investigation of isolated singularities of complex hypersurfaces by means of Picard-Lefschetz theory. We focus on the concept of distinguished bases of vanishing cycles and the concept of…