Soit R le corps des séries de Puiseux réelles. C’est un corps réel clos. On construit les premiers exemples d’intersections lisses de deux quadriques dans ℙ R 5 et d’hypersurfaces cubiques lisses dans ℙ R 4 qui ne sont pas stablement rationnelles mais pour lesquelles l’espace X ( R ) des R -points est semi-algébriquement connexe. La question de construire de tels exemples sur le corps des réels ℝ reste ouverte.
In [1, p. 1666], the proof of Theorem 6 had a misprint, which turned into a wrong argument that we correct here.
We prove that for generic parameters, the quantum radial parts map of Varagnolo and Vasserot gives an isomorphism between the spherical double affine Hecke algebra of GL n and a quantized multiplicative quiver variety, as defined by Jordan.
We continue our study of fixed loci of antisymplectic involutions on projective hyper-K & auml;hler manifolds of K3[n]-type induced by an ample class of square 2 in the Beauville-Bogomolov-Fujiki lattice. We prove that if the divisibility of the ample class is 2, then one connected component of the fixed locus is a Fano manifold of index 3, thus generalizing to higher dimensions the case of the LLSvS 8-fold associated to a cubic fourfold. We also show that, in the case of the LLSvS 8-fold associated to a cubic fourfold, the second component of the fixed locus is of general type, thus answering a question by Manfred Lehn. R & Eacute;SUM & Eacute; (La g & eacute;om & eacute;trie des involutions anti-symplectiques, II). Nous poursuivons notre & eacute;tude des lieux fixes des involutions anti-symplectiques sur les vari & eacute;t & eacute;s hyper-k & auml;hl & eacute;riennes projectives de type K3[n] induites par une classe ample de carr & eacute; 2 dans le r & eacute;seau de Beauville-Bogomolov-Fujiki. Nous prouvons que si la divisibilit & eacute; de la classe ample est 2, alors une composante connexe du lieu fixe est une vari & eacute;t & eacute; de Fano d'indice 3, g & eacute;n & eacute;ralisant ainsi aux dimensions sup & eacute;rieures le cas de la vari & eacute;t & eacute; de LLSvS de dimension 8 associ & eacute;e & agrave; une vari & eacute;t & eacute; cubique de dimension 4. Nous montrons & eacute;galement que, dans le cas de la vari & eacute;t & eacute; de dimension 8 de LLSvS associ & eacute;e & agrave; une vari & eacute;t & eacute; cubique de dimension 4, la deuxi & egrave;me composante du lieu fixe est de type g & eacute;n & eacute;ral, r & eacute;pondant ainsi & agrave; une question pos & eacute;e par Manfred Lehn.
We extend to higher dimensions the notion of Birkhoff attractor of a dissipative map. We prove that this notion coincides with the classical Birkhoff attractor. We prove that for the dissipative system associated to the discounted Hamilton-Jacobi equation, the graph of the solution is contained in the Birkhoff attractor. The appendix provides instructive counterexamples in the non-Tonelli case. We also study what happens when we perturb a Hamiltonian system to make it dissipative and let the perturbation go to zero. The paper contains two main results on gamma-supports and elements of the gamma-completion of the space of exact Lagrangians. Firstly the gamma-support of a Lagrangian in a cotangent bundle carries the cohomology of the base and secondly given an exact Lagrangian L, any Floer theoretic equivalent Lagrangian is the gamma-limit of Hamiltonian images of L.
