Etude de certaines familles de variétés algébriques munies d'une action de groupe algébrique

An overview of the classification of spherical and complexity-one varieties

These notes, in which we give an overview of the combinatorial classification of spherical and complexity-one varieties, correspond to a 3h mini-course given by the author for the workshop "Arc schemes and algebraic group actions" held in Paris from December 2 to December 4, 2019.

Real structures on nilpotent orbit closures

We determine the equivariant real structures on nilpotent orbits and the normalizations of their closures for the adjoint action of a complex semisimple algebraic group on its Lie algebra.

Invariant deformation theory of affine schemes with reductive group action

We develop an invariant deformation theory, in a form accessible to practice, for affine schemes $W$ equipped with an action of a reductive algebraic group $G$. Given the defining equations of a $G$-invariant subscheme $X \subset W$, we device an algorithm to compute the universal deformation of $X$ in terms of generators and relations up to a given order. In many situations, our algorithm even computes an algebraization of the universal deformation. As an application, we determine new families of examples of the invariant Hilbert scheme of Alexeev and Brion, where $G$ is a classical group acting on a classical representation, and describe their singularities.

Stability conditions and related filtrations for $(G,h)$-constellations

Given an infinite reductive algebraic group $G$, we consider $G$-equivariant coherent sheaves with prescribed multiplicities, called $(G,h)$-constellations, for which two stability notions arise. The first one is analogous to the $\theta$-stability defined for quiver representations by King and for $G$-constellations by Craw and Ishii, but depending on infinitely many parameters. The second one comes from Geometric Invariant Theory in the construction of a moduli space for $(G,h)$-constellations, and depends on some finite subset $D$ of the isomorphy classes of irreducible representations of $G$. We show that these two stability notions do not coincide, answering negatively a question raise…