相关论文: Zariski density of monodromy groups via Picard-Lef…

Let $\mathcal{M}_{n,2n+2}$ be the coarse moduli space of CY manifolds arising from a crepant resolution of double covers of $\mathbb{P}^n$ branched along $2n+2$ hyperplanes in general position. We show that the monodromy group of a good…

For all $d$ belonging to a density-$1/8$ subset of the natural numbers, we give an example of a square-tiled surface conjecturally realizing the group $\mathrm{SO}^*(2d)$ in its standard representation as the Zariski-closure of a factor of…

We give conditions under which the monodromy group of an $A$-hypergeometric system is invariant under modifications of the collection of characters $A$. The key ingredient is a Zariski--Lefschetz type theorem for principal $A$-determinants.

We compute the Zariski closure of the Kontsevich-Zorich monodromy groups arising from certain square tiled surfaces that are geometrically motivated. Specifically we consider three surfaces that emerge as translation covers of platonic…

We give an example of a Teichm\"uller curve which contains, in a factor of its monodromy, a group which was not observed before. Namely, it has Zariski closure equal to the group $SO^*(6)$ in its standard representation; up to finite index,…

The fundamental group of a smooth projective variety is fibered if it maps onto the fundamental group of smooth curve of genus 2 or more. The goal of this paper is to establish some strong restrictions on these groups, and in particular on…

We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski dense, unless its Zariski closure acts transitively on a Hermitian symmetric space of tube type. This is a kind of converse to a…

The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space of dimension at least 3 and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of…

We give necessary and sufficient conditions for a linear reflection group in the sense of Vinberg to be Zariski-dense in the ambient projective general linear group. As an application, we show that every irreducible right-angled Coxeter…

We compute the mapping class group-valued monodromy of any sufficiently ample linear system on any smooth simply connected projective surface, identifying this with the r-spin mapping class group associated to a maximal root of the adjoint…

Symplectic four-manifolds give rise to Lefschetz fibrations, which are determined by monodromy representations of free groups in mapping class groups. We study the topology of Lefschetz fibrations by analysing the action of the monodromy on…

We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study Zariski equisingular families of surface, not necessarily isolated,…

We study the topology of the space of affine hyperplanes $L \subset \CC^n$ which are in general position with respect to a given generic quadratic hypersurface $A$, and calculate the monodromy action of the fundamental group of this space…

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…

Given an ample line bundle on a toric surface, a question of Donaldson asks which simple closed curves can be vanishing cycles for nodal degenerations of smooth curves in the complete linear system. This paper provides a complete answer.…

We classify the polycyclic totally ordered simple dimension groups, i.e. dimension groups given by a dense embedding of n-dimensional lattice into the real line. Our method is based on the geometry of simple geodesics on the hyperbolic…

We propose a conjecture that the monodromy group of a singular hyperbolic metric on a non-hyperbolic Riemann surface is {\it Zariski dense} in ${\rm PSL}(2,\,{\Bbb R})$. By using meromorphic differentials and affine connections, we obtain…

We prove a motivic enhancement of the classical Picard--Lefschetz formula. Our proof is completely motivic, and yields a description of the motivic nearby cycles at a quasi-homogeneous singularity, as well as its monodromy, in terms of an…

Consider the family of smooth cubic surfaces which can be realized as threefold-branched covers of $\mathbb{P}^{2}$, with branch locus equal to a smooth cubic curve. This family is parametrized by the space $\mathcal{U}_{3}$ of smooth cubic…

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the…