The notion of quantum root comes up in the study of the quantum cohomology of generalized flag varieties, quantum Bruhat graphs or in the study of the affinized Bruhat order on an affinized Weyl semi-group. In this paper, we provide a complete classification of quantum roots associated to general Kac–Moody root systems. In particular we prove that the set of quantum roots of a given root system is always finite. We then deduce several finiteness properties of the affinized Bruhat order, generalizing results of A. Welch. In particular, we prove that its intervals are finite. This affinized Bruhat order was introduced by A. Braverman, D. Kazhdan and M. Patnaik in their study of Kac-Moody groups over discretely valued fields, and of the associated Iwahori-Hecke algebras.
(Semi-algebraically connected real varieties that are not stably rational) Let R be the field of real Puiseux series. It is a real closed field. We construct the first examples of smooth intersections of two quadrics in IP'5 R and smooth cubic hypersurfaces in IP'4 which are not stably rational but for which the space X(R) of R-points is semi-algebraically connected. The question of constructing such examples over the field of real numbers I[8 remains open.
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.
