Special Session

Derivations occur as natural objects in many different mathematical contexts in which they provide power-ful bridges between algebra and geometry. During the last decades, the study of interactions between algebraicaspects of Lie theory and other non-commutative algebras with geometric questions related for instance to the structure of automorphism groups of affine varieties, to the rigidity and flexibility properties of complex Stein manifolds and to the classification of singularities has been a source of important progress in these areas.

The algebraic theory of certain classes of derivations called locally nilpotent and locally finite derivations has manydeep connections to famous challenging problems of more geometric nature in the context of affine algebraicgeometry, such as the Jacobian conjecture, the 14th Hilbert’s Problem, the Zariski Cancellation Problem and Linearization problems. Recent progress on the study of automorphism groups of affine varieties also shed anew light on possible interactions with infinite dimensional Lie theory. There is thus a rich and highly activefield of investigation at the interface between the algebraic theory of derivations and the geometric study ofalgebraic varieties.

The purpose of this session organized at the occasion of the First Joint Meeting Brazil-France in Mathematics is a wide dissemination of different types of problems and techniques related tothe Theory of Derivations and can culminate in future research partnerships between the involved researchers and its open public at the event.

• Simple derivations and foliations with one singularity

  Severino Collier Coutinho (UFRJ, Rio de Janeiro - Brazil)

  Abstract

  A number of results, in recent years, have been concerned with foliations with one singularity on the complex projective plane. Thus, [1] presents a classification of foliations of degree 2 with one singularity, while [2, Theorem 1, p. 192] introduces a family of foliations with one singularity of algebraic multiplicity one. In this talk I will present a new family of foliations of degree $d≥4$ whose unique sin-gularity has multiplicityd−1. The generic foliations in this familyhave trivial isotropy groups and a unique invariant algebraic curve,thus giving rise to a new family of simple derivations of the affineplane
  [1] Cerveau, D. and Deserti, J. and Garba Belko, D. and Meziani, Geometrie classique de certains feuilletages de degré deux, Bull. Braz. Math. Soc. (N.S.),41, (2010), 161–198
  [2] Alc´ntara, C. R., Foliations on $\mathbb{CP}^2$ of degree $d$ with a singular point with Milnor number $d^2+d+ 1$, Rev. Mat. Complut.,31, (2018), 187–199.

• Pro-locally nilpotent derivations

  Roberto Carlos Diaz Vivanco (Talca University, Chile and Université de Bourgogne, France)

  Abstract

  Let $V$ be an affine algebraic variety on $\mathbb{C}$ and $\mathcal{O}(V)$ its ring of regular functions, a known result is the correspondence between the actions of the additive group $\mathbb{G}_a$ on $V$ and the locally nilpotent derivations on $\mathcal{O}(V)$. For the case of an affine ind-variety $\mathcal{V}$ and its ring of regular functions $\mathcal{O}(\mathcal{V})$ we can also associate each action of the additive group $\mathbb{G}_a$ on $\mathcal{V}$, in the category of ind-varieties, a continuous derivation on its ring of regular functions $\mathcal{O}(\mathcal{V})$. However, this derivation is not necessarily locally nilpotent. In this talk I describe a particular type of derivations, the ``Pro-Locally Nilpotent Derivations", and I show some principles that make possible the correspondence between the actions of the additive group $\mathbb{G}_a$ on an affine ind-variety $\mathcal{V}$ and the pro-locally nilpotent derivations of its ring of regular functions $\mathcal{O}(\mathcal{V})$ .

• Locally Nilpotent Derivations

  Gene Freudenburg (Western Michigan University, USA)

  Abstract

  This talk will provide a brief introduction and overview for locally nilpotent derivations (LNDs) of integral domains over a field $k$ of characteristic zero. For rings that are finitely generated over $k$, there is a correspondence between LNDs and algebraic actions of the additive group $(k,+)$ on the corresponding algebraic $k$-varieties. This is one of the main reasons that LNDs are of interest. Indeed, LNDs can be seen to play a role in many fundamental problems of algebraic geometry, including Hilbert's Fourteenth Problem, the Embedding Problem and Cancelation Problem for Affine Spaces, the Jacobian Conjecture, and the Dolgachev-Weisfeiler Conjecture. We will survey known results and highlight recent developments in this area.

• Some results about the classification of algebraic group actions in characteristic $p$

  Kevin Langlois (Heinrich Heine Universität Düsseldorf, Germany)

  Abstract

  One of the great achievements of the Lie theory is to have created many beautiful and important relationships between algebraic and geometric objects dependent on continuous parameters (like Lie groups, Lie algebras, algebraic groups, group schemes ...) and discrete objects of combinatorial nature (such as lattices, root systems, polyhedra, graphs, finite reflection groups, ...). This fundamental picture is illustrated in the classification of Chevalley groups in terms of root systems. From this perspective, one may ask to describe not only the group object itself but its transformations on geometric spaces. In the case of reductive group actions, the Luna-Vust theory (1983) gives an answer in the situation that the equivariant birational type is known. A more mysterious issue to elucidate is thereby the case of \emph{actions of non-reductive groups}, where the first prototype to look at is the additive group $\mathbb{G}_{a} = (k, +)$ of the ground field $k$. While the abundancy of the $\mathbb{G}_{a}$-actions for a given finite type scheme plays a key role in affine geometry, in order to have a Lie type correspondence, one needs to rigidify those actions by adding some geometric constraints. A natural condition that appears many times in practice is to consider somehow the $\mathbb{G}_{a}$-actions that are 'homogeneous' with respect to the grading provided by the action of an algebraic torus. For instance, this recently intervened in the \emph{non-reductive geometric invariant theory} d ue to Bérczi, Doran, Hawes and Kirwan (2007-2017). A leading and pioneering work that reflects this idea is the one of M. Demazure (1970) who described the automorphism groups of smooth complete toric varieties in terms of the so called \emph{Demazure roots}. In 2003, H. Flenner and M. Zaidenberg classified all the $\mathbb{G}_{a}$-actions on complex normal affine $\mathbb{C}^{\star}$-surfaces that are normalized by the $\mathbb{C}^{\star}$-actions. This latter classification has been extended in higher dimension by A. Liendo (2010) for torus actions with general orbits of codimension one. In this talk, we will present the classification of such $\mathbb{G}_{a}$-actions over an arbitrary ground field by emphasizing the difference between characteristic $0$ and characteristic $p$ and the interaction with the objects of combinatorial nature. This is joint work with A. Liendo.

• Integration of rational derivations on algebraic varieties

  Alvaro Liendo (Talca University, Chile)

  Abstract

  In this talk we show a classification of rational actions of the additive and the multiplicative group on algebraic varieties that generalizes the usual description of regular actions of the additive and the multiplicative group on affine varieties in terms of derivations. As a corollary, we provide a characterization of regular actions of the additive and the multiplicative group on the class of varieties that are proper over the spectrum of its ring of global regular functions. This class of varieties include affine varieties, complete varieties and their blowups. Different parts of this talk are issued from joint works with L. Cid and A. Dubouloz, respectively.

• Examples of $\mathbb{G}_a$-actions on cylinders over Danielewski hypersurfaces

  Lucy Moser-Jauslin (Université de Bourgogne, Dijon, France)

  Abstract

  Given a polynomial of the form $P_n=x^nz-y^2+1$, consider the affine hypersurface $S_n$ in affine complex three-space defined by $P_n=0$. It is well-known that the cylinders of these hypersurfaces are all isomorphic. In particular they have many $\mathbb{G}_a$-actions. Consider the cylinder on the hypersurface $S$ defined by the equation $x^2z-y^2+x=0$. We will use explicit examples of $\mathbb{G}_a$-actions on the cylinder over $S_2$ to describe several $\mathbb{G}_a$-actions on $S$ with particular properties. We will also discuss the question of extensions of $\mathbb{G}_a$-actions on a hypersurface to $\mathbb{G}_a$-actions on the ambient space.

• Automorphism groups of Koras-Russell threefolds

  Charlie Petitjean (Université de Bourgogne, Dijon, France)

  Abstract

  The Koras–Russel threefolds are smooth contractible rational affine varieties, they can be described as hypersurfaces in $\mathbb{A}_{\mathbb{C}}^4$. Although close to $\mathbb{A}_{\mathbb{C}}^3$ these varieties are not isomorphic to $\mathbb{A}_{\mathbb{C}}^3$, this has been demonstrated using the Makar-Limanov invariant. \\ A first part will focus on the construction of these varieties, a second on the study of the automorphism groups using their Makar-Limanov invariants, that is, considering additive group actions on them. The third part will be about rational properties related to torus actions on these threefolds.

• $\mathbb{A}^1$-homotopy theory and affine algebraic geometry

  Sabrina Pauli (University of Oslo, Norway)

  Abstract

  In $\mathbb{A}^1$-homotopy theory we apply techniques from algebraic topology to algebraic varieties. The affine line $\mathbb{A}^1$ plays the role of the unit interval. $\mathbb{A}^1$-homotopy theory provides many new applications, also in affine algebraic geometry. Nontrivial affine $\mathbb{A}^1$-contractible varieties serve as potential counter examples to the Zaraski cancellation problem which is still unsolved in dimension greater or equal to 3 and characteristic 0. In my talk I will give an introduction to $\mathbb{A}^1$-homotopy theory. In particular, I will try to provide tools that can be useful in affine algebraic geometry.

• Embedding some open Riemann surfaces into the complex plane

  Pierre-Marie Poloni (Universität Bern, Switzerland)

  Abstract

  It is a long-standing open problem whether every open Riemann surface (i.e. every one-dimensional Stein manifold) admits a proper holomorphic embedding into the complex plane. In this talk, we enlarge the class of examples for which a positive answer is known. More precisely, we will show that the Riemann sphere, with a non-empty countable closed subset containing at most two accumulation points removed, as well as any compact Riemann surface of genus one, with a non-empty countable closed subset containing at most one accumulation point removed, are all embeddable into the plane. Our construction is inspired by a result of Sathaye stating that every smooth affine algebraic curve of genus one is a plane algebraic curve. This is joint work with Frank Kutzschebauch.

• Rings of constants of linear derivations on Fermat rings

  Marcelo Oliveira Veloso (UFSJ, Ouro Branco - Brazil)

  Abstract

  It is well known how hard is to describe the ring of constants of an arbitrary derivation as well to decide if the ring of constants of a derivation is trivial. In this talk we deal with $\mathbb{C}$-derivations of a Fermat ring $$B_n^{m} =\frac{\mathbb{C}[X_1,\ldots, X_n]}{(X_1^{m_1}+\cdots +X_n^{m_n})},$$ where $\mathbb{C}[X_1,\ldots, X_n]$ is the polynomial ring in $n$ variables over the complex numbers $\mathbb{C}$, $n\geq3$, $m=(m_1,\ldots,m_n)\in \mathbb{Z}^n $, and $m_i \geq 2$ for $i=1,\ldots,n$. Specifically, we study the ring of constants of linear derivations on Fermat rings and its locally nilpotent derivations. We present a description of all the linear $\mathbb{C}$-derivations and provide examples of linear derivations with trival ring constants for certain Fermat rings.