On the Hodge, Tate and Mumford--Tate conjectures for fibre products of families of regular surfaces with geometric genus~1

O. V. Oreshkina (Nikol'skaya)

Modelirovanie i analiz informacionnyh sistem, Tome 25 (2018) no. 3, pp. 312-322.

Résumé

The Hodge, Tate and Mumford–Tate conjectures are proved for the fibre product of two non-isotrivial 1-parameter families of regular surfaces with geometric genus 1 under some conditions on degenerated fibres, the ranks of the Néron–Severi groups of generic geometric fibres and representations of Hodge groups in transcendental parts of rational cohomology. Let $\pi_i:X_i\to C\quad (i = 1, 2)$ be a projective non-isotrivial family (possibly with degeneracies) over a smooth projective curve $C$. Assume that the discriminant loci $\Delta_i=\{\delta\in C \vert \mathrm{Sing}(X_{i\delta})\neq\varnothing\} \quad (i = 1, 2)$ are disjoint, $h^{2,0}(X_{ks})=1,\quad h^{1,0}(X_{ks}) = 0$ for any smooth fibre $X_{ks}$, and the following conditions hold: $(i)$ for any point $\delta \in \Delta_i$ and the Picard–Lefschetz transformation $ \gamma \in \mathrm{GL}(H^2 (X_{is}, \mathbb{Q})) $, associated with a smooth part $\pi'_i: X'_i\to C\setminus\Delta_i$ of the morphism $\pi_i$ and with a loop around the point $\delta \in C$, we have $(\log(\gamma))^2\neq0$; $(ii)$ the variety $X_i (i = 1, 2)$, the curve $C$ and the structure morphisms $\pi_i:X_i\to C$ are defined over a finitely generated subfield $k \hookrightarrow \mathbb{C}$. If for generic geometric fibres $X_{1s}$ and $X_{2s}$ at least one of the following conditions holds: $(a)$ $b_2(X_{1s})-{\mathrm{rank}}\, {\mathrm{NS}}(X_{1s})$ is an odd prime number, $\quad $ $b_2(X_{1s})-{\mathrm{rank}}\,{\mathrm{NS}}(X_{1s})\neq b_2(X_{2s})-{\mathrm{rank}} \,{\mathrm{NS}}(X_{2s})$; $(b)$ the ring ${\mathrm{End}}_{\mathrm{Hg}(X_{1s})} {\mathrm{NS}}_{\mathbb{Q}}(X_{1s})^\perp$ is an imaginary quadratic field, $\quad b_2(X_{1s})-{\mathrm{rank}}\,{\mathrm{NS}}(X_{1s})\neq 4$, ${\mathrm{End}}_{\mathrm{Hg}(X_{2s})} {\mathrm{NS}}_{\mathbb{Q}}(X_{2s})^\perp$ is a totally real field or $b_2(X_{1s})-{\mathrm{rank}}\,{\mathrm{NS}}(X_{1s}) > b_2(X_{2s})-{\mathrm{rank}}\, {\mathrm{NS}}(X_{2s})$; $(c)$ $[b_2(X_{1s})-{\mathrm{rank}}\,{\mathrm{NS}}(X_{1s})\neq 4, {\mathrm{End}}_{\mathrm{Hg}(X_{1s})} {\mathrm{NS}}_{\mathbb{Q}}(X_{1s})^\perp= \mathbb{Q}$; $b_2(X_{1s})-{\mathrm{rank}}\,{\mathrm{NS}}(X_{1s})\neq b_2(X_{2s})-{\mathrm{rank}} \,{\mathrm{NS}}(X_{2s})$, then for the fibre product $X_1 \times_C X_2$ the Hodge conjecture is true, for any smooth projective $k$-variety $X_0$ with the condition $X_1 \times_C X_2$ $\widetilde{\rightarrow}$ $X_0 \otimes_k \mathbb{C}$ the Tate conjecture on algebraic cycles and the Mumford–Tate conjecture for cohomology of even degree are true.

Détail

Keywords: Hodge, Tate and Mumford–Tate conjectures, Mumford–Tate group, $l$-adic representation.
Mots-clés : fibre product

Bibliographie

[1] Hodge W. V. D., “The topological invariants of algebraic varieties”, Proceedings of International Congress of Mathematicians, v. 1, 1952, 182–192 | MR | Zbl

[2] Tankeev S. G., “Cycles on simple abelian varieties of prime dimension”, Mathematics of the USSR–Izvestiya, 20:1 (1983), 157–171 | DOI | MR | MR | Zbl

[3] Gordon B. B., “A survey of the Hodge conjecture for Abelian varieties”, Appendix in: Lewis J.D.,, A survey of the Hodge conjecture, CRM Monograph Series, 10, 2 ed., American Mathematical Society, Providence, RI, 1999, 297–356 | MR

[4] Nikolskaya O. V., “On algebraic cycles on a fibre product of families of K3 surfaces”, Izv. Math., 77:1 (2013), 143–162 | DOI | DOI | MR | Zbl

[5] Nikolskaya O. V., “On the geometry of a smooth model of a fibre product of families of K3 surfaces”, Sbornik: Mathematics, 205:2 (2014), 269–276 | DOI | DOI | MR | Zbl

[6] Nikolskaya O. V. , “On algebraic cohomology classes on a smooth model of a fiber product of families of K3 surfaces”, Math. Notes, 96:5 (2014), 745–752 | DOI | DOI | MR | MR | Zbl

[7] Mamford D., “Families of abelian varieties”, Proc. of Symposium in Pure Math., IX, AMS, 1966, 347–351 | DOI | MR

[8] Serre J.-P. , “Representations l-adiques”, Algebraic number theory, International Symposium (Kyoto, 1976), ed. S. Iyanaga, Japan Society for the Promotion of Science, Tokyo, 1977, 177–193 | MR

[9] Tankeev S.G., “Surfaces of $K3$ type over number fields and the Mumford-Tate conjecture. II”, Russion Akad. Sci. Izv. Math., 59:3 (1995), 619–646 | DOI | MR | Zbl

[10] Tankeev S. G., “Cycles on simple abelian varieties of prime dimension over number fields”, Mathematics of the USSR-Izvestiya, 31:3 (1988), 527–540 | DOI | MR | Zbl

[11] Pink R., “l-adic algebraic monodromy group cocharacters, and the Mumford-Tate conjecture”, J. reine. angew. Math., 495 (1998), 187–237 | MR | Zbl

[12] Tate J., “Algebraic cycles and poles of zeta functions”, Arithmetical Algebraic Geometry, Proc. Conf. (Purdue Univ., 1963), Harper and Row, New York, 1965, 93–110 | MR

[13] Tate J., “Endomorphisms of abelian varieties over finite fields”, Invent. Math., 2 (1966), 134–144 | DOI | MR | Zbl

[14] Faltings G., “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Invent. Math., 73:3 (1983), 349–366 | DOI | MR | Zbl

[15] Tankeev S. G., “On cycles on abelian varieties of prime dimension over finite and number fields”, Mathematics of the USSR–Izvestiya, 22:2 (1984), 329–337 | DOI | MR | Zbl

[16] Moonen B., “On the Tate and Mumford-Tate conjectures in codimension one for varieties with $h^{2,0}=1$”, Duke Math. J., 166:4 (2017), 739–799 | DOI | MR | Zbl

[17] Pignatelli R., “Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g=q=1$ and $K^2=4$”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei $(9)$ Mat. Appl., 20:3 (2009), 207–226 | DOI | MR | Zbl

[18] Zarhin Yu. G., “Hodge groups of $K3$ surfaces”, Journal für die reine und angewandte Mathematik, 341 (1983), 193–220 | MR

[19] Nikol'skaya O.V., “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 (in Russian) | MR

[20] Mustafin G.A., “Families of algebraic varieties and invariant cycles”, Mathematics of the USSR–Izvestiya, 27:2 (1986), 251–278 | DOI | MR | Zbl

[21] Deligne P., “Théorie de Hodge. III”, Inst. Hautes Études Sci. Publ. Math., 44 (1974), 5–77 | DOI | MR | Zbl

[22] Kempf G. et al., Toroidal embeddings, v. I, Lecture Notes in Mathematics, 339, Springer-Verlag, Berlin–New York, 1973 | DOI | MR | Zbl

[23] Kulikov Vic. S., “Degenerations of $K3$ surfaces and Enriques surfaces”, Math. USSR–Izv., 11:5 (1977), 957–988 | DOI | MR | Zbl

[24] Tankeev S. G., “The arithmetic and geometry of a generic hypersurface section”, Izv. Math., 66:2 (2002), 393–424 | DOI | DOI | MR | Zbl

[25] Kawamata Y., “Kodaira dimension of algebraic fiber spaces over curves”, Invent. Math., 66:1 (1982), 57–71 | DOI | MR | Zbl

[26] Birkar C., “Iitaka conjecture $C_{n,m}$ in dimension six”, Compositio Mathematica, 2009:6 (1979), 1442–1446 | DOI | MR