arXiv subjects
Publications and source records attributed to Stéphanie Cupit-Foutou.
The Gromov width of a uniruled projective Kähler manifold can be bounded from above by the symplectic area of its minimal curves. We apply this result to toric varieties and thus get in this case upper bounds expressed in toric combinatorial invariants.
We prove that the Gromov width of any Bott-Samelson variety birational to a Schubert variety and equipped with a rational Kähler form equals the symplectic area of a minimal curve. From this, we derive an estimate for the Seshadri constants of ample line bundles on Bott-Samelson varieties.
We identify the $G(\mathbb R)$-orbits of the real locus $X(\mathbb R)$ of any spherical complex variety $X$ defined over $\mathbb R$ and homogeneous under a split connected reductive group $G$ defined also over $\mathbb R$. This is done by introducing some reflection operators on the set of real Borel orbits of $X(\mathbb R)$. We thus investigate the existence problem for an action of the Weyl group of $G$ on the set of real Borel orbits of $X(\mathbb R)$. In particular, we determine the varieties $X$ for which these operators define an action of the very little Weyl group of $X$ on the set of open real Borel orbits of $X(\mathbb R)$. This enables us to give a parametrization of the $G(\mathbb R)$-orbits of $X(\mathbb R)$ in terms of the orbits of this new action.
We apply the combinatorial theory of spherical varieties to characterize the momentum polytopes of polarized projective spherical varieties. This enables us to derive a classification of these varieties, without specifying the open orbit, as well as a classification of all Fano spherical varieties. In the setting of multiplicity free compact and connected Hamiltonian manifolds, we obtain a necessary and sufficient condition involving momentum polytopes for such manifolds to be Kähler and classify the invariant compatible complex structures of a given Kähler multiplicity free compact and connected Hamiltonian manifold.
Given a connected reductive algebraic group $G$ and a finitely generated monoid $Γ$ of dominant weights of $G$, in 2005 Alexeev and Brion constructed a moduli scheme $\mathrm M_Γ$ for multiplicity-free affine $G$-varieties with weight monoid $Γ$. This scheme is equipped with an action of an `adjoint torus' $T_{\mathrm{ad}}$ and has a distinguished $T_{\mathrm{ad}}$-fixed point $X_0$. In this paper, we obtain a complete description of the $T_{\mathrm{ad}}$-module structure in the tangent space of $\mathrm M_Γ$ at $X_0$ for the case where $Γ$ is saturated. Using this description, we prove that the root monoid of any affine spherical $G$-variety is free. As another application, we obtain new proofs of uniqueness results for affine spherical varieties and spherical homogeneous spaces first proved by Losev in 2009. Furthermore, we obtain a new proof of Alexeev and Brion's finiteness result for multiplicity-free affine $G$-varieties with a prescribed weight monoid. At last, we prove that for saturated $Γ$ all the irreducible components of $\mathrm M_Γ$, equipped with their reduced subscheme structure, are affine spaces.
Let $G$ be a complex semisimple algebraic group and $X$ be a complex symmetric homogeneous $G$-variety. Assume that both $G$, $X$ as well as the $G$-action on $X$ are defined over real numbers. Then $G(\mathbb{R})$ acts on $X(\mathbb{R})$ with finitely many orbits. We describe these orbits in combinatorial terms using Galois cohomology, thus providing a patch to a result of A.Borel and L.Ji.
We give a combinatorial description of all affine spherical varieties with prescribed weight monoid $Γ$. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion's moduli scheme $\mathrm M_Γ$ for such varieties. Moreover, we find several sufficient conditions for $\mathrm M_Γ$ to be irreducible and exhibit several examples where $\mathrm M_Γ$ is reducible. Finally, we provide examples of non-reduced $\mathrm M_Γ$.
We investigate qualitative and quantitative properties of anti-holomorphic involutions on some spherical varieties.
In the setting of strict wonderful varieties we answer positively to Luna's conjecture, saying that wonderful varieties are classified by combinatorial objects, the so-called spherical systems. In particular, we prove that strict wonderful varieties are mostly obtained from symmetric spaces, spherical nilpotent orbits or model spaces. To make the paper self-contained as much as possible, we shall gather some known results on these families and more generally on wonderful varieties.
The invariant Hilbert schemes considered in \cite{BC1} were proved to be affine spaces. The proof relied on the classification of strict wonderful varieties. We obtain in the present article a classification-free proof of the affinity of these invariant Hilbert scheme by means of deformation theoretical arguments. As a consequence we recover in a shorter way the classification of strict wonderful varieties. This provides an alternative and new approach to answer Luna's conjecture.