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We study when the period and the index of a class in the Brauer group of the function field of a real algebraic surface coincide. We prove that it is always the case if the surface has no real points (more generally, if the class vanishes in restriction to the real points of the locus where it is well-defined), and give a necessary and sufficient condition for unramified classes. As an application, we show that the -invariant of the function field of a real algebraic surface is equal to 4, answering questions of Lang and Pfister. Our strategy relies on a new Hodge-theoretic approach to de Jong’s period-index theorem on complex surfaces.
Benoist, Olivier 1
@article{PMIHES_2019__130__63_0,
author = {Benoist, Olivier},
title = {The period-index problem for real surfaces},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {63--110},
publisher = {Springer Berlin Heidelberg},
address = {Berlin/Heidelberg},
volume = {130},
year = {2019},
doi = {10.1007/s10240-019-00108-7},
mrnumber = {4028514},
zbl = {1442.14073},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-019-00108-7/}
}
TY - JOUR AU - Benoist, Olivier TI - The period-index problem for real surfaces JO - Publications Mathématiques de l'IHÉS PY - 2019 SP - 63 EP - 110 VL - 130 PB - Springer Berlin Heidelberg PP - Berlin/Heidelberg UR - http://geodesic.mathdoc.fr/articles/10.1007/s10240-019-00108-7/ DO - 10.1007/s10240-019-00108-7 LA - en ID - PMIHES_2019__130__63_0 ER -
%0 Journal Article %A Benoist, Olivier %T The period-index problem for real surfaces %J Publications Mathématiques de l'IHÉS %D 2019 %P 63-110 %V 130 %I Springer Berlin Heidelberg %C Berlin/Heidelberg %U http://geodesic.mathdoc.fr/articles/10.1007/s10240-019-00108-7/ %R 10.1007/s10240-019-00108-7 %G en %F PMIHES_2019__130__63_0
Benoist, Olivier. The period-index problem for real surfaces. Publications Mathématiques de l'IHÉS, Tome 130 (2019), pp. 63-110. doi: 10.1007/s10240-019-00108-7
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