real algebraic varieties
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Author(s):  
Adrien Dubouloz ◽  
Gene Freudenburg ◽  
Lucy Moser-Jauslin

AbstractWe construct smooth rational real algebraic varieties of every dimension {\geq 4} which admit infinitely many pairwise non-isomorphic real forms.



2020 ◽  
Author(s):  
Frédéric Mangolte




2019 ◽  
Vol 30 (09) ◽  
pp. 1950042
Author(s):  
Tomasz Kowalczyk

We introduce a concept of blown-up Čech cohomology for coherent sheaves of homological dimension [Formula: see text] and some quasi-coherent sheaves on a nonsingular real affine variety. Its construction involves a directed set of multi-blowups. We establish, in particular, long exact cohomology sequence and Cartan’s Theorem B. Finally, some applications are provided, including universal solution to the first Cousin problem (after blowing up).



2018 ◽  
Vol 70 (5) ◽  
pp. 1008-1037 ◽  
Author(s):  
Boaz Elazar ◽  
Ary Shaviv

AbstractWe define Schwartz functions, tempered functions, and tempered distributions on (possibly singular) real algebraic varieties. We prove that all classical properties of these spaces, defined previously on affine spaces and on Nash manifolds, also hold in the case of affine real algebraic varieties, and give partial results for the non-affine case.



2018 ◽  
Vol 2020 (7) ◽  
pp. 2054-2113
Author(s):  
Saugata Basu ◽  
Cordian Riener

Abstract We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees bounded by a fixed constant d. We prove that if a Specht module, $\mathbb{S}^{\lambda }$, appears with positive multiplicity in the isotypic decomposition of the cohomology modules of such sets, then the rank of the partition $\lambda$ is bounded by O(d). This implies a polynomial (in the dimension of the ambient space) bound on the number of such modules. Furthermore, we prove a polynomial bound on the multiplicities of those that do appear with positive multiplicity in the isotypic decomposition of the abovementioned cohomology modules. We give some applications of our methods in proving lower bounds on the degrees of defining polynomials of certain symmetric semi-algebraic sets, as well as improved bounds on the Betti numbers of the images under projections of (not necessarily symmetric) bounded real algebraic sets, improving in certain situations prior results of Gabrielov, Vorobjov, and Zell.



2018 ◽  
Vol 2019 (23) ◽  
pp. 7160-7192 ◽  
Author(s):  
Dmitry Gourevitch ◽  
Siddhartha Sahi ◽  
Eitan Sayag

Abstract We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein’s theory of analytic continuation of holonomic distributions. We use this to construct H-equivariant functionals on principal series representations of G, where G is a real reductive group and H is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the p-adic case using a recent result of Hong and Sun.



2016 ◽  
Vol 9 (4) ◽  
pp. 1257-1302 ◽  
Author(s):  
Max Karoubi ◽  
Marco Schlichting ◽  
Charles Weibel


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