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Izv. Akad. Nauk SSSR Ser. Mat., 1991, Volume 55, Issue 4, Pages 716–746 (Mi izv986)  

This article is cited in 20 scientific papers (total in 20 papers)

Characteristic classes of vector bundles on a real algebraic variety

V. A. Krasnov


Abstract: For a vector bundle $E$ on a real algebraic variety $X$, the author studies the connections between the characteristic classes
$$ c_k(E(\mathbf C))\in H^{2k}(X(\mathbf C),\mathbf Z),\quad w_k(E(\mathbf R))\in H^k(X(\mathbf R),\mathbf F_2). $$
It is proved that for $M$-varieties the equality $w_k(E(\mathbf R))=0$ implies the congruence $c_k(E(\mathbf C))\equiv 0 \operatorname{mod}2$. Sufficient conditions are found also for the converse to hold; this requires the construction of new characteristic classes $cw_k(E(\mathbf C))\in H^{2k}(X(\mathbf C);G,\mathbf z(k))$. The results are applied to study the topology of $X(\mathbf R)$.

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English version:
Mathematics of the USSR-Izvestiya, 1992, 39:1, 703–730

Bibliographic databases:

UDC: 513.6+517.6
MSC: Primary 14P25; Secondary 14F05, 14J60, 55R40
Received: 19.03.1990

Citation: V. A. Krasnov, “Characteristic classes of vector bundles on a real algebraic variety”, Izv. Akad. Nauk SSSR Ser. Mat., 55:4 (1991), 716–746; Math. USSR-Izv., 39:1 (1992), 703–730

Citation in format AMSBIB
\Bibitem{Kra91}
\by V.~A.~Krasnov
\paper Characteristic classes of vector bundles on a real algebraic variety
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1991
\vol 55
\issue 4
\pages 716--746
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1137584}
\zmath{https://zbmath.org/?q=an:0783.14037|0746.14024}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1992IzMat..39..703K}
\transl
\jour Math. USSR-Izv.
\yr 1992
\vol 39
\issue 1
\pages 703--730
\crossref{https://doi.org/10.1070/IM1992v039n01ABEH002223}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. A. Krasnov, “Algebraic cycles on a real algebraic GM-manifold and their applications”, Russian Acad. Sci. Izv. Math., 43:1 (1994), 141–160  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. V. A. Krasnov, “On equivariant Grothendieck cohomology of a real algebraic variety, and its applications”, Russian Acad. Sci. Izv. Math., 44:3 (1995), 461–477  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. V. A. Krasnov, “The cohomological Brauer group of a real algebraic variety”, Izv. Math., 60:5 (1996), 933–962  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. V. A. Krasnov, “Real algebraic GM$\mathbb Z$-surfaces”, Izv. Math., 62:4 (1998), 695–721  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. V. A. Krasnov, “Real algebraic GM-varieties”, Izv. Math., 62:3 (1998), 465–491  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. V. A. Krasnov, “The etale and equivariant cohomology of a real algebraic variety”, Izv. Math., 62:5 (1998), 1013–1034  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. V. A. Krasnov, “Picard and Lefschetz numbers of real algebraic surfaces”, Math. Notes, 63:6 (1998), 747–751  mathnet  crossref  crossref  mathscinet  zmath  isi
    8. V. A. Krasnov, “Analogues of the Harnack–Thom inequality for a real algebraic surface”, Izv. Math., 64:5 (2000), 915–937  mathnet  crossref  crossref  mathscinet  zmath  isi
    9. V. A. Krasnov, “On the Picard group and the Brauer group of a real algebraic surface”, Math. Notes, 67:2 (2000), 168–175  mathnet  crossref  crossref  mathscinet  zmath  isi
    10. V. A. Krasnov, “The Brauer group of an noncomplete real algebraic surface”, Math. Notes, 67:3 (2000), 296–300  mathnet  crossref  crossref  mathscinet  zmath  isi
    11. V. A. Krasnov, “The Brauer and Witt Groups of Real Ruled Surfaces”, Math. Notes, 72:5 (2002), 652–659  mathnet  crossref  crossref  mathscinet  zmath  isi
    12. V. A. Krasnov, “The Nikulin Congruence for Four-Dimensional $M$-Varieties”, Math. Notes, 76:2 (2004), 191–199  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. V. A. Krasnov, “On the Algebraic Cohomology of Real Algebraic $M$-Varieties”, Math. Notes, 76:6 (2004), 796–809  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. V. A. Krasnov, “On the Fano Surface of a Real Cubic $M$-Threefold”, Math. Notes, 78:5 (2005), 662–668  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    15. V. A. Krasnov, “Real algebraic varieties and cobordism”, Izv. Math., 71:3 (2007), 573–601  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    16. V. A. Krasnov, “On Bordisms of Real Algebraic $M$-Varieties”, Math. Notes, 81:5 (2007), 649–655  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    17. V. A. Krasnov, “Maximal intersections of three real quadrics”, Izv. Math., 75:3 (2011), 569–587  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    18. V. A. Krasnov, “Real Four-Dimensional $\mathit{GM}$-Triquadrics”, Math. Notes, 93:6 (2013), 830–836  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    19. Giuseppe De Nittis, Kiyonori Gomi, “Classification of “Real” Bloch-bundles: Topological Quantum Systems of type AI”, Journal of Geometry and Physics, 2014  crossref
    20. V. A. Krasnov, “Real Kummer surfaces”, Izv. Math., 83:1 (2019), 65–103  mathnet  crossref  crossref  adsnasa  isi  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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