Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userReal Torus Actions on Real Affine Algebraic Varieties
110
0
0.0
(
0
)
Authors
Pierre-Alexandre Gillard
Ask ChatGPT about the research
No Arabic abstract
We extend the Altmann-Hausen presentation of normal affine algebraic C-varieties endowed with effective torus actions to the real setting. In particular, we focus on actions of quasi-split real tori, in which case we obtain a simpler presentation.
rate research
Read More
Using the work of Guillen and Navarro Aznar we associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on Borel-Moore homology with Z/2 coefficients an analog of the weight filtration for complex algebraic varieties.
We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of real degree less than $d$, and for the size of the sum of $mod 2$ Betti numbers for the real form of complex manifolds of complex degree less than $d$.
We give an algebro-geometric classification of smooth real affine algebraic surfaces endowed with an effective action of the real algebraic circle group $mathbb{S}^1$ up to equivariant isomorphisms. As an application, we show that every compact differentiable surface endowed with an action of the circle $S^1$ admits a unique smooth rational real quasi-projective model up to $mathbb{S}^1$-equivariant birational diffeomorphism.
In this article we review the question of constructing geometric quotients of actions of linear algebraic groups on irreducible varieties over algebraically closed fields of characteristic zero, in the spirit of Mumfords geometric invariant theory (GIT). The article surveys some recent work on geometric invariant theory and quotients of varieties by linear algebraic group actions, as well as background material on linear algebraic groups, Mumfords GIT and some of the challenges that the non-reductive setting presents. The earlier work of two of the authors in the setting of unipotent group actions is extended to deal with actions of any linear algebraic group. Given the data of a linearisation for an action of a linear algebraic group H on an irreducible variety $X$, an open subset of stable points $X^s$ is defined which admits a geometric quotient variety $X^s/H$. We construct projective completions of the quotient $X^s/H$ by considering a suitable extension of the group action to an action of a reductive group on a reductive envelope and using Mumfords GIT. In good cases one can also compute the stable locus $X^s$ in terms of stability (in the sense of Mumford for reductive groups) for the reductive envelope.
Let $X$ be a smooth projective real algebraic variety. We give new positive and negative results on the problem of approximating a submanifold of the real locus of $X$ by real loci of subvarieties of $X$, as well as on the problem of determining the subgroups of the Chow groups of $X$ generated by subvarieties with nonsingular real loci, or with empty real loci.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
