RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Izv. RAN. Ser. Mat.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 6, Pages 199–231 (Mi izv8504)  

This article is cited in 1 scientific paper (total in 1 paper)

On an inductive approach to the standard conjecture for a fibred complex variety with strong semistable degeneracies

S. G. Tankeev

Vladimir State University

Abstract: We prove that Grothendieck's standard conjecture $B(X)$ of Lefschetz type on the algebraicity of the operators $\ast$ and $\Lambda$ of Hodge theory holds for a 4-dimensional smooth projective complex variety fibred over a smooth projective curve $C$ provided that every degenerate fibre is a union of smooth irreducible components of multiplicity 1 with normal crossings, the standard conjecture $B(X_{\overline\eta})$ holds for a generic geometric fibre $X_{\overline\eta}$, there is at least one degenerate fibre $X_\delta$ and the rational cohomology rings $H^\ast(V_i,\mathbb{Q})$ and $H^\ast(V_i\cap V_j,\mathbb{Q})$ of the irreducible components $V_i$ of every degenerate fibre $X_\delta=V_1+…+V_m$ are generated by classes of algebraic cycles. We obtain similar results for 3-dimensional fibred varieties with algebraic invariant cycles (defined by the smooth part $\pi'\colon X'\to C'$ of the structure morphism $\pi\colon X\to C$) or with a degenerate fibre all of whose irreducible components $E_i$ possess the property $H^2(E_i,\mathbb{Q})= \operatorname{NS}(E_i)\otimes_{\mathbb{Z}}\mathbb{Q}$.

Keywords: standard conjecture of Lefschetz type, Galois descent, algebraic cycle, Clemens–Schmid sequence.

DOI: https://doi.org/10.4213/im8504

Full text: PDF file (706 kB)
First page: PDF file
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2017, 81:6, 1253–1285

Bibliographic databases:

UDC: 512.7
MSC: 14C25, 14F25, 14J35
Received: 07.01.2016

Citation: S. G. Tankeev, “On an inductive approach to the standard conjecture for a fibred complex variety with strong semistable degeneracies”, Izv. RAN. Ser. Mat., 81:6 (2017), 199–231; Izv. Math., 81:6 (2017), 1253–1285

Citation in format AMSBIB
\Bibitem{Tan17}
\by S.~G.~Tankeev
\paper On an inductive approach to the standard conjecture for a~fibred
complex variety with strong semistable degeneracies
\jour Izv. RAN. Ser. Mat.
\yr 2017
\vol 81
\issue 6
\pages 199--231
\mathnet{http://mi.mathnet.ru/izv8504}
\crossref{https://doi.org/10.4213/im8504}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2017IzMat..81.1253T}
\elib{http://elibrary.ru/item.asp?id=30737854}
\transl
\jour Izv. Math.
\yr 2017
\vol 81
\issue 6
\pages 1253--1285
\crossref{https://doi.org/10.1070/IM8504}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000418891300009}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85040996040}


Linking options:
  • http://mi.mathnet.ru/eng/izv8504
  • https://doi.org/10.4213/im8504
  • http://mi.mathnet.ru/eng/izv/v81/i6/p199

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    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. S. G. Tankeev, “O standartnoi gipoteze dlya rassloennogo proizvedeniya trekh ellipticheskikh poverkhnostei s poparno neperesekayuschimisya diskriminantnymi lokusami”, Izv. RAN. Ser. matem., 83:3 (2019), 213–256  mathnet  crossref
  • Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya Izvestiya: Mathematics
    Number of views:
    This page:232
    References:20
    First page:20

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019