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 Model. Anal. Inform. Sist., 2016, Volume 23, Number 4, Pages 440–465 (Mi mais514)

On algebraic cycles on fibre products of non-isotrivial families of regular surfaces with geometric genus 1

O. V. Nikol'skaya

A.G. and N.G. Stoletov Vladimir State University, Gorky str., 87, Vladimir, 600000, Russia

Abstract: Let $\pi_k:X_k\to C (k = 1, 2)$ be a projective family of surfaces (possibly with degenerations) over a smooth projective curve $C$. Assume that the discriminant loci $\Delta_k=\{\delta\in C \vert \operatorname{Sing}(X_{k\delta})\neq\varnothing\} \quad (k = 1, 2)$ are disjoint, $h^{2,0}(X_{ks})=1,\quad h^{1,0}(X_{ks}) = 0$ for any smooth fibre $X_{ks}$ and the period map associated with the variation of Hodge structures $R^2\pi'_{k\ast}\mathbb{Q}$ (where $\pi'_k:X'_k\to C\setminus\Delta_k$ is a smooth part of the morphism $\pi_k$), is non-constant. If for generic geometric fibres $X_{1s}$ and $X_{2s}$ the following conditions hold:
(i) $b_2(X_{1s})-\operatorname{rank} \operatorname{NS}(X_{1s})$ is an odd integer;
(ii) $b_2(X_{1s})-\operatorname{rank}\operatorname{NS}(X_{1s})\neq b_2(X_{2s})-\operatorname{rank} \operatorname{NS}(X_{2s})$,
then for any smooth projective model $X$ of the fibre product $X_1\times_CX_2$ the Hodge conjecture on algebraic cycles is true.
If, besides, the morphisms $\pi_k$ are smooth, $p_k=b_2(X_{ks}) -\operatorname{rank} \operatorname{NS}(X_{ks}) (k = 1,2)$ are odd prime numbers and $p_1\neq p_2$, then for $X_1\times_CX_2$ and for the fibre square $X_1\times_CX_1$ the Hodge conjecture and the Grothendieck standard conjecture on algebraicity of operators $\ast$ and $\Lambda$ of Hodge theory hold as well.
This result yields new examples of smooth projective 5-dimensional varieties satisfying the Hodge and the Grothendieck conjectures, because one can take as smooth fibres of the morphism $\pi_k:X_k\to C$ some $K3$ surfaces, minimal regular surfaces of general type (of Kodaira dimension $\varkappa=2$) with geometric genus $1$ belonging to one of the following types: (a) surfaces with $K^2\leq 2$; (b) surfaces with $3\leq K^2\leq 8$, whose moduli are in the same component of the space of moduli as Todorov surface; (c) surfaces with $K^2 = 3$ with torsion of the Picard group $\mathbb{Z}/3\mathbb{Z}$.

Keywords: Hodge conjecture, standard conjecture, fibre product, Hodge group, Poincaré cycle.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-31-00266_ìîë_à This work was supported by the Russian Foundation for basic research under the Grant No 16-31-00266.

DOI: https://doi.org/10.18255/1818-1015-2016-4-440-465

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UDC: 512.7

Citation: O. V. Nikol'skaya, “On algebraic cycles on fibre products of non-isotrivial families of regular surfaces with geometric genus 1”, Model. Anal. Inform. Sist., 23:4 (2016), 440–465

Citation in format AMSBIB
\Bibitem{Nik16} \by O.~V.~Nikol'skaya \paper On algebraic cycles on fibre products of non-isotrivial families of regular surfaces with geometric genus~1 \jour Model. Anal. Inform. Sist. \yr 2016 \vol 23 \issue 4 \pages 440--465 \mathnet{http://mi.mathnet.ru/mais514} \crossref{https://doi.org/10.18255/1818-1015-2016-4-440-465} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3549346} \elib{http://elibrary.ru/item.asp?id=26561563} 

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This publication is cited in the following articles:
1. O. V. Oreshkina, “O gipotezakh Khodzha, Teita i Mamforda–Teita dlya rassloennykh proizvedenii semeistv regulyarnykh poverkhnostei s geometricheskim rodom 1”, Model. i analiz inform. sistem, 25:3 (2018), 312–322
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