## Duke Mathematical Journal

### On the Tate and Mumford–Tate conjectures in codimension $1$ for varieties with $h^{2,0}=1$

Ben Moonen

#### Abstract

We prove the Tate conjecture for divisor classes and the Mumford–Tate conjecture for the cohomology in degree $2$ for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic $0$, under a mild assumption on their moduli. As an application of this general result, we prove the Tate and Mumford–Tate conjectures for several classes of algebraic surfaces with $p_{g}=1$.

#### Article information

Source
Duke Math. J., Volume 166, Number 4 (2017), 739-799.

Dates
Revised: 21 April 2016
First available in Project Euclid: 9 December 2016

https://projecteuclid.org/euclid.dmj/1481252670

Digital Object Identifier
doi:10.1215/00127094-3774386

Mathematical Reviews number (MathSciNet)
MR3619305

Zentralblatt MATH identifier
1372.14008

Subjects
Primary: 14C
Secondary: 14D 14J20: Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx]

#### Citation

Moonen, Ben. On the Tate and Mumford–Tate conjectures in codimension $1$ for varieties with $h^{2,0}=1$. Duke Math. J. 166 (2017), no. 4, 739--799. doi:10.1215/00127094-3774386. https://projecteuclid.org/euclid.dmj/1481252670

#### References

• [1] Y. André, Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part, Compos. Math. 82 (1992), 1–24.
• [2] Y. André, On the Shafarevich and Tate conjectures for hyper-Kähler varieties, Math. Ann. 305 (1996), 205–248.
• [3] Y. André, Pour une théorie inconditionnelle des motifs, Publ. Math. Inst. Hautes Études Sci. 83 (1996), 5–49.
• [4] Y. André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panor. Synthèses 17, Soc. Math. France, Paris, 2004.
• [5] A. Borel, Linear Algebraic Groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York, 1991.
• [6] A. Borel and J. Tits, Groupes réductifs, Publ. Math. Inst. Hautes Études Sci. 27 (1965), 55–152.
• [7] G. Borrelli, Surfaces with $p_{g}=q=1$, $K^{2}=8$ and nonbirational bicanonical map, Manuscripta Math. 130 (2009), 523–531.
• [8] F. Catanese, The moduli and the global period mapping of surfaces with $K^{2}=p_{g}=1$: A counterexample to the global Torelli problem, Compos. Math. 41 (1980), 401–414.
• [9] F. Catanese, “On the period map of surfaces with $K^{2}=\chi=2$” in Classification of Algebraic and Analytic Manifolds (Katata, 1982), Progr. Math. 39, Birkhäuser, Boston, 1983, 27–43.
• [10] F. Catanese, “On a class of surfaces of general type” in Algebraic Surfaces (Cortona, 1977), Springer, Heidelberg, 2010, 267–284.
• [11] F. Catanese and C. Ciliberto, “Surfaces with $p_{g}=q=1$” in Problems in the Theory of Surfaces and their Classification (Cortona, 1988), Sympos. Math. XXXII, Academic Press, London, 1991, 49–79.
• [12] F. Catanese and C. Ciliberto, Symmetric products of elliptic curves and surfaces of general type with $p_{g}=q=1$, J. Algebraic Geom. 2 (1993), 389–411.
• [13] F. Catanese and O. Debarre, Surfaces with $K^{2}=2$, $p_{g}=1$, $q=0$, J. Reine Angew. Math. 395 (1989), 1–55.
• [14] F. Catanese and R. Pignatelli, Fibrations of low genus, I, Ann. Sci. École Norm. Supér. (4) 39 (2006), 1011–1049.
• [15] P. Deligne, La conjecture de Weil pour les surfaces K3, Invent. Math. 15 (1972), 206–226.
• [16] P. Deligne, Cohomologie étale, Séminaire de Géométrie Algebrique du Bois-Marie (SGA $4\frac{1}{2}$), Lecture Notes in Math. 569, Springer, Berlin, 1977.
• [17] P. Deligne, “Hodge cycles on abelian varieties” in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math. 900, Springer, Berlin, 1982.
• [18] G. Faltings, “Complements to Mordell” in Rational Points, Aspects Math. E6, Friedr. Vieweg, Braunschweig, 1984.
• [19] D. Ferrand, Un foncteur norme, Bull. Soc. Math. France 126 (1998), 1–49.
• [20] G. Harder, R. Langlands, and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal-Flächen, J. Reine Angew. Math. 366 (1986), 53–120.
• [21] C. Klingenberg, Die Tate-Vermutungen für Hilbert-Blumenthal-Flächen, Invent. Math. 89 (1987), 291–317.
• [22] B. Kostant, A characterization of the classical groups, Duke Math. J. 25 (1958), 107–123.
• [23] V. Kumar Murty and D. Ramakrishnan, Period relations and the Tate conjecture for Hilbert modular surfaces, Invent. Math. 89 (1987), 319–345.
• [24] D. Lombardo, On the $\ell$-adic Galois representations attached to nonsimple abelian varieties, Ann. Inst. Fourier (Grenoble) 66 (2016), 1217–1245.
• [25] C. Lyons, The Tate conjecture for a family of surfaces of general type with $p_{g}=q=1$ and $K^{2}=3$, Amer. J. Math. 137 (2015), 281–311.
• [26] B. Moonen and Y. Zarhin, Hodge classes and Tate classes on simple abelian fourfolds, Duke Math. J. 77 (1995), 553–581.
• [27] D. Morrison, “On the moduli of Todorov surfaces” in Algebraic Geometry and Commutative Algebra, Vol. I, Kinokuniya, Tokyo, 1988, 313–355.
• [28] M. Murakami, Infinitesimal Torelli theorem for surfaces of general type with certain invariants, Manuscripta Math. 118 (2005), 151–160.
• [29] R. Pignatelli, 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 Rend. Lincei (9) Mat. Appl. 20 (2009), 207–226.
• [30] R. Pink, $\ell$-adic algebraic monodromy groups, cocharacters, and the Mumford-Tate conjecture, J. Reine Angew. Math. 495 (1998), 187–237.
• [31] F. Polizzi, On surfaces of general type with $p_{g}=q=1$, $K^{2}=3$, Collect. Math. 56 (2005), 181–234.
• [32] F. Polizzi, Surfaces of general type with $p_{g}=q=1$, $K^{2}=8$ and bicanonical map of degree $2$, Trans. Amer. Math. Soc. 358 (2006), no. 2, 759–798.
• [33] F. Polizzi, Standard isotrivial fibrations with $p_{g}=q=1$, J. Algebra 321 (2009), 1600–1631.
• [34] K. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), 751–804.
• [35] C. Riehm, The corestriction of algebraic structures, Invent. Math. 11 (1970), 73–98.
• [36] J-P. Serre, “Lettres à Ken Ribet du 1/1/1981 et du 29/1/1981” in Oeuvres: Collected Papers, IV, 1985–1998, Springer, Berlin, 2000.
• [37] J-P. Serre, Lectures on the Mordell-Weil Theorem, 3rd ed., Aspects Math. E15, Friedr. Vieweg, Braunschweig, 1997.
• [38] S. Tankeev, Surfaces of K3 type over number fields and the Mumford-Tate conjecture (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 4, 846–861; English translation in Math. USSR-Izv. 37 (1991), no. 1, 191–208.
• [39] S. Tankeev, Surfaces of K3 type over number fields and the Mumford-Tate conjecture, II (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 59 (1995), no. 3, 179–206; English translation in Izv. Math. 59 (1995), no. 3, 619–646.
• [40] A. Todorov, Surfaces of general type with $p_{g}=1$ and $(K,K)=1$, I, Ann. Sci. École Norm. Supér. (4) 13 (1980), 1–21.
• [41] A. Todorov, A construction of surfaces with $p_{g}=1$, $q=0$ and $2\leq(K^{2})\leq8$: Counterexamples of the global Torelli theorem, Invent. Math. 63 (1981), 287–304.
• [42] B. van Geemen, Half twists of Hodge structures of CM-type, J. Math. Soc. Japan 53 (2001), 813–833.
• [43] C. Voisin, “Hodge loci” in Handbook of Moduli, Vol. III, Adv. Lect. Math. 26, Int. Press, Somerville, Mass., 2013, 507–546.
• [44] Y. Zarhin, Hodge groups of K3 surfaces, J. Reine Angew. Math. 341 (1983), 193–220.