Fundamental Groups Of Smooth Varieties Research Articles

Milnor–Wood inequality for klt varieties of general type and uniformization

AbstractWe generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to consider are those of smooth loci. Assuming that the rank of the target Lie group is not greater than two, we show that the Toledo invariant satisfies a Milnor–Wood‐type inequality and we characterize the corresponding maximal representations.

On the Semi-absoluteness of Isomorphisms between the Pro-$p$ Arithmetic Fundamental Groups of Smooth Varieties

On the Semi-absoluteness of Isomorphisms between the Pro-$p$ Arithmetic Fundamental Groups of Smooth Varieties

Anabelian group-theoretic properties of the absolute Galois groups of discrete valuation fields

Let K be a field. Write GK for the absolute Galois group of K. In the present paper, we discuss the slimness [i.e., the property that every open subgroup is center-free] and the elasticity [i.e., the property that every nontrivial topologically finitely generated normal closed subgroup of an open subgroup is open] of GK. These two group-theoretic properties are closely related to [various versions of] the Grothendieck Conjecture in anabelian geometry. For instance, with regard to the slimness, Mochizuki proved that GK is slim if K is a subfield of a finitely generated extension of the field of fractions of the Witt ring W(F‾p) as a consequence of a [highly nontrivial] Grothendieck Conjecture-type result. In the present paper, we generalize this result to the case where K is a subfield of the field of fractions of an arbitrary mixed characteristic Noetherian local domain. Our proof is based on elementary field theories such as Kummer theory. On the other hand, with regard to the elasticity, Mochizuki proved that GK is elastic if K is a finite extension of the field of p-adic numbers. In the present paper, we generalize this result to the case where K is an arbitrary mixed characteristic Henselian discrete valuation field. As a corollary of this generalization, we prove the semi-absoluteness of isomorphisms between the étale fundamental groups of smooth varieties over mixed characteristic Henselian discrete valuation fields. Moreover, we also prove the weak version of the Grothendieck Conjecture for hyperbolic curves of genus 0 over subfields of finitely generated extensions of mixed characteristic higher local fields.

On the fundamental groups of normal varieties

We show that the fundamental groups of normal complex algebraic varieties share many properties of the fundamental groups of smooth varieties. The jump loci of rank one local systems on a normal variety are related to the jump loci of a resolution and of a smoothing of this variety.

The Toledo invariant on smooth varieties of general type

We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: assuming that the rank of the target Lie group is not greater than two, we prove that the Toledo invariant satisfies a Milnor-Wood type inequality and we characterize the corresponding maximal representations.

Weight decompositions on étale fundamental groups

Let $X_0$ be a smooth or proper variety defined over a finite field $k$, and let $X$ be the base extension of $X_0$ to the algebraic closure $\bar{k}$. The geometric \'etale fundamental group $\pi_1(X,\bar{x})$ of $X$ is a normal subgroup of the Weil group, so conjugation gives it a Weil action. For $l$ not dividing the characteristic of $k$, we consider the pro-${\Bbb Q}_l$-algebraic completion of $\pi_1(X,\bar{x})$ as a nonabelian Weil representation. Lafforgue's Theorem and Deligne's Weil II theorems imply that this affine group scheme is mixed, in the sense that its structure sheaf is a mixed Weil representation. When $X$ is smooth, weight restrictions apply, affecting the possibilities for the structure of this group. This gives new examples of groups which cannot arise as \'etale fundamental groups of smooth varieties.