Local Fundamental Group Research Articles

A new family of isolated symplectic singularities with trivial local fundamental group

AbstractWe construct a new infinite family of four‐dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of by the dihedral group of order , (2) as singular points of Calogero–Moser spaces associated with dihedral groups of order at equal parameters, and (3) as singularities of a certain Slodowy slice in the ‐fold cover of the nilpotent cone in .

On the local fundamental group of the complement of a curve in a normal surface

We give a presentation of the fundamental group of the complement of a curve $C$ in its "tubular" neighbourhood in a normal surface $S$. The presentation is given in terms of the double weighted dual graph of the resolution of singularities of $C$ (and $S$). This result generalizes the presentation of the fundamental group of the complement of a normal singularity in its neighbourhood given by Mumford in the case, where the dual graph of the resolution is a tree and all exceptional curves of the resolution are rational curves.

Variation of local fundamental group and local first homology group

In this paper, we discuss various results about variation of the local fundamental group of normal complex spaces. It is proved that the finite Galois descent of upper semicontinuity of the local fundamental group holds at a factorial complex analytic germ. We also show by an example that finite Galois descent of upper semicontinuity of the local first homology group at a smooth germ is not true in general. We prove that the local fundamental groups of a normal complex algebraic variety are finite in number upto isomorphism.

The arithmetic local Nori fundamental group

In this paper we introduce the local Nori fundamental group scheme of a reduced scheme or algebraic stack over a perfect field k k . We give particular attention to the case of fields: to any field extension K / k K/k we attach a pro-local group scheme over k k . We show how this group has many analogies, but also some crucial differences, with the absolute Galois group. We propose two conjectures, analogous to the classical Neukirch-Uchida Theorem and Abhyankar Conjecture, providing some evidence in their favor. Finally we show that the local fundamental group of a normal variety is a quotient of the local fundamental group of an open, of its generic point (as it happens for the étale fundamental group) and even of any smooth neighborhood.

The local fundamental group of a Kawamata log terminal singularity is finite

We prove a conjecture of Kollár stating that the local fundamental group of a klt singularity x is finite. In fact, we prove a stronger statement, namely that the fundamental group of the smooth locus of a neighbourhood of x is finite. We call this the regional fundamental group. As the proof goes via a local-to-global induction, we simultaneously confirm finiteness of the orbifold fundamental group of the smooth locus of a weakly Fano pair.

Connected Fundamental Groups and Homotopy Contacts in Fibered Topological (C, R) Space

The algebraic as well as geometric topological constructions of manifold embeddings and homotopy offer interesting insights about spaces and symmetry. This paper proposes the construction of 2-quasinormed variants of locally dense p-normed 2-spheres within a non-uniformly scalable quasinormed topological (C, R) space. The fibered space is dense and the 2-spheres are equivalent to the category of 3-dimensional manifolds or three-manifolds with simply connected boundary surfaces. However, the disjoint and proper embeddings of covering three-manifolds within the convex subspaces generates separations of p-normed 2-spheres. The 2-quasinormed variants of p-normed 2-spheres are compact and path-connected varieties within the dense space. The path-connection is further extended by introducing the concept of bi-connectedness, preserving Urysohn separation of closed subspaces. The local fundamental groups are constructed from the discrete variety of path-homotopies, which are interior to the respective 2-spheres. The simple connected boundaries of p-normed 2-spheres generate finite and countable sets of homotopy contacts of the fundamental groups. Interestingly, a compact fibre can prepare a homotopy loop in the fundamental group within the fibered topological (C, R) space. It is shown that the holomorphic condition is a requirement in the topological (C, R) space to preserve a convex path-component. However, the topological projections of p-normed 2-spheres on the disjoint holomorphic complex subspaces retain the path-connection property irrespective of the projective points on real subspace. The local fundamental groups of discrete-loop variety support the formation of a homotopically Hausdorff (C, R) space.

Uniqueness of the minimizer of the normalized volume function

We confirm a conjecture of Chi Li which says that the minimizer of the normalized volume function for a klt singularity is unique up to rescaling. This is achieved by defining stability thresholds for valuations, and then showing that a valuation is a minimizer if and only if it is K-semistable, and that K-semistable valuation is unique up to rescaling. As applications, we prove a finite degree formula for volumes of klt singularities and an effective bound of the local fundamental group of a klt singularity.

On a discriminant knot group problem of Brieskorn

Quite some time ago, at the singularity conference at Cargese 1972 Brieskorn asked the following question: Is the local fundamental group $\pi_1^{s}(S-D)$ of the discriminant complement inside the semi-universal unfolding $S$ of an isolated hypersurface singularity constant for $s$ in the $\mu$-constant stratum $\Sigma_E$? We review this question and give an affirmative answer in case of singular plane curve germs of multiplicity at most $3$.

Finiteness of fundamental groups

We show that the finiteness of the fundamental groups of the smooth locus of lower dimensional log Fano pairs would imply the finiteness of the local fundamental group of Kawamata log terminal (klt) singularities. As an application, we verify that the local fundamental group of a three-dimensional klt singularity and the fundamental group of the smooth locus of a three-dimensional Fano variety with canonical singularities are always finite.

Finiteness of algebraic fundamental groups

AbstractWe show that the algebraic local fundamental group of any Kawamata log terminal singularity as well as the algebraic fundamental group of the smooth locus of any log Fano variety are finite.

A New Property of Quotients Modulo Pseudo-Reflection Groups

In this paper we prove that the induced homomorphism between local fundamental groups of germs of normal complex analytic spaces $\pi_1^0(W)\rar\pi_1^0(W/G)$ is a surjection, where $G$ is a finite group of automorphisms generated by pseudo-reflections.

Affine normal surfaces with simply-connected smooth locus

Given a normal affine surface V defined over \({\mathbb{C}}\), we look for algebraic and topological conditions on V which imply that V is smooth or has at most rational singularities. The surfaces under consideration are algebraic quotients \({\mathbb{C}^n/G}\) with an algebraic group action of G and topologically contractible surfaces. Theorem 3.6 can be considered as a global version of the well-known result of Mumford giving a smoothness criterion for a germ of a normal surface in terms of the local fundamental group.

Singularities appearing on generic fibers of morphisms between smooth schemes

I give various criteria for singularities to appear on geometric generic fibers of morphism between smooth schemes in positive characteristics. This involves local fundamental groups, jacobian ideals, projective dimension, tangent and cotangent sheaves, and the effect of Frobenius. As an application, I determine which rational double points do appear on geometric generic fibers.

Some remarks on the local fundamental group scheme

We define the local fundamental group scheme and study its properties under base change of the base field.

Reflexive modules on normal surface singularities and representations of the local fundamental group

We consider the Riemann–Hilbert correspondence on the complement of a normal surface singularity ( X , x ) . Through a closure operation we obtain a correspondence between the category of finite dimensional representations of the local fundamental group π 1 loc ( X , x ) and the category of left D X , x -modules that are reflexive as O X , x -modules. We show that under this correspondence profinite representations correspond to invariant modules and that these admit a canonical structure as left D X , x -modules. We prove that the fundamental module is an invariant module if and only if ( X , x ) is a quotient singularity. Finally we investigate some algebraisation aspects.

Surface classification and local and global fundamentals groups, I

Given a smooth complex surface S , and a compact connected global normal crossings divisor D = \bigcup_i D_i , we consider the local fundamental group \pi_1 (T \setminus D) , where T is a good tubular neighbourhood of D . One has an exact sequence 1 \to \mathcal K \to \Gamma : = \pi_1 (T - D) \rightarrow \Pi : = \pi_1 (D) \to 1 , and the kernel \mathcal K is normally generated by geometric loops \gamma_i around the curve D_i . Among the main results, which are strong generalizations of a well known theorem of Mumford, is the nontriviality of \gamma_i in \Gamma = \pi_1 (T - D) , provided all the curves D_i of genus zero have selfintersection D_i^2 \leq -2 (in particular this holds if the canonical divisor K_S is nef on D ), and under the technical assumption that the dual graph of D is a tree.

Perfectly Weighted Graphs

We present a classification of those graphs which arise as dual intersection graphs in the resolutions of complex surface singularities with perfect local fundamental group. Using computation intensive methods, we examine the possibilities for weights on the vertices corresponding to self-intersections of the exceptional curves.

Generic covers branched over [formula omitted

In this paper the authors study generic covers of C 2 branched over {x n+y m}=0 s.t. the total space is a normal analytic surface. They found a complete description of the monodromy of the cover in terms of the monodromy graphs and an almost complete description of the local fundamental groups in case (n,m)=1. For the general case, they give explicit descriptions of base changes in terms of monodromy graphs; they describe completely the embedded resolution graphs in the case n|m. Via these base changes every cover is a quotient of such a cover.

Local fundamental groups of algebraic varieties

Some progress is made in the calculation of local fundamental groups.

Local fundamental groups of surface singularities in characteristicp

Local fundamental groups of surface singularities in characteristicp