Representations of braid groups and construction of projective surfaces
Francesco Polizzi
备注：
Note written for the Proceedings of the Conference "Group 32 - The 32nd International Colloquium on Group Theoretical Methods in Physics", held on Czech Technical University (Prague) on July 9-13, 2018
发表信息：
Journal of Physics: Conference Series, Volume 1194, Number 1 (2019)
Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will focus on their algebraic-geometric aspects, explaining how the representation theory of higher genus braid groups can be used to produce interesting examples of projective surfaces defined over the field of complex numbers.
Sometime ago, we showed that a pure Artin braid group is not Kähler, i.e. it is not the fundamental group of a compact Kähler manifold. This used a result of Bressler, Ramachandran and the author that Kähler groups cannot be too "big". The goal here is to study the problem of Kählerness for other braid groups. The main result is that, with some trivial exceptions, the pure braid group of a Riemann surface with at least 2 strands is never Kähler. In some cases the proof uses the previous strategy, for others it plays off some homological properties of braid groups established beforehand against…
We construct an explicit bundle with flat connection on the configuration space of n points of a complex curve. This enables one to recover the `formality' isomorphism between the Lie algebra of the prounipotent completion of the pure braid group of n points on a surface and an explicitly presented Lie algebra t_{g,n} (Bezrukavnikov), and to extend it to a morphism from the full braid group of the surface to the semidirect product exp(hat t_{g,n}) rtimes S_n.
We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $Γ$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\it topologically} distinct algebraic objects such as extremal elliptic surfaces, real trigonal curves, and real elliptic surfaces.
Moduli spaces of points on $n$-spheres carry natural actions of braid groups. For $n=0$, $1$, and $3$, we prove that these symmetries extend to actions of mapping class groups of positive genus surfaces, by establishing exceptional isomorphisms with certain moduli of local systems. This relies on the existence of group structure for spheres in these dimensions. We also use the connection to demonstrate that the space of rank 4 Stokes matrices with fixed Coxeter invariant of nonzero discriminant contains only finitely many integral braid group orbits.
We investigate the local contribution of the braid monodromy factorization in the context of the links obtained by the closure of these braids. We consider plane curves which are arrangements of lines and conics as well as some algebraic surfaces, where some of the former occur as local configurations in degenerated and regenerated surfaces in the latter. In particular we focus on degenerations which involve intersection points of multiplicity two and three. We demonstrate when the same links arise even when the local arrangements are different.
The 3-cuspidal quartic and braid monodromy of degree 4 coverings
研究了变形双重覆盖的4次扩张的判别式，通过方程z^2= u + a w，w^2= v + bz。我们首先证明了判别曲面是一个四次曲面，它在扭曲的三次曲面上是尖点的，即，是3次一般方程的判别式。然后取a（u，v），b（u，v），得到一条3-尖点仿射四次曲线，计算其辫子单值性。这种局部编织单值性的计算是朝向确定全局编织单值性的一个步骤，例如，对于作者先前考虑的（a，b，c）表面。在修订中，我们填补了一个空白（注意到裁判）的经典定理证明，任何四次曲面的扭三次尖点曲线是其切向可展的，我们改变了一个基点，以使图片正确。
This article is a revised version of the talk I gave at the conference ``Beauville Surfaces and groups'' held in Newcastle in June 2012. It presents some group theoretical methods to give bounds on the number of connected components of the moduli space of surfaces of general type, focusing on some families of regular surfaces isogenous to a product of curves.
We are interested in the 3-Calabi-Yau categories $\mathcal{D}$ arising from quivers with potential associated to a triangulated marked surface $\mathbf{S}$ (without punctures). We prove that the spherical twist group ST of $\mathcal{D}$ is isomorphic to a subgroup (generated by braid twists) of the mapping class group of the decorated marked surface $\mathbf{S}_{\bigtriangleup}$. Here $\mathbf{S}_{\bigtriangleup}$ is the surface obtained from $\mathbf{S}$ by decorating with a set of decorated points, where the number of points equals the number of triangles in any triangulations of $\mathbf{S}…
We study some special systems of generators on finite groups, introduced in previous work by the first author and called "diagonal double Kodaira structures", in order to investigate non-abelian, finite quotients of the pure braid group on two strands $\mathsf{P}_2(Σ_b)$, where $Σ_b$ is a closed Riemann surface of genus $b$. In particular, we prove that, if a finite group $G$ admits a diagonal double Kodaira structure, then $|G|\geq 32$, and equality holds if and only if $G$ is extra-special. In the last section, as a geometrical application of our algebraic results, we construct two $3$-dimen…
We consider a set of toric Calabi-Yau varieties which arise as deformations of the small resolutions of type A surface singularities. By careful analysis of the heuristics of B-brane transport in the associated GLSMs, we predict the existence of a mixed braid group action on the derived category of each variety, and then prove that this action does indeed exist. This generalizes the braid group action found by Seidel and Thomas for the undeformed resolutions. We also show that the actions for different deformations are related, in a way that is predicted by the physical heuristics.
We study algebraic isomonodromic deformations of flat logarithmic connections on the Riemann sphere with $n\geq 4$ poles, for arbitrary rank. We introduce a natural property of algebraizability for the germ of universal deformation of such a connection. We relate this property to a peculiarity of the corresponding monodromy representation: to yield a finite braid group orbit on the appropriate character variety. Under reasonable assumptions on the deformed connection, we may actually establish an equivalence between both properties. We apply this result in the rank two case to relate finite br…
