THE NOETHER-LEFSCHETZ CONJECTURE AND GENERALIZATIONS NICOLAS BERGERON, ZHIYUAN LI, JOHN MILLSON, AND COLETTE MOEGLIN Abstract. We prove the Noether-Lefschetz conjecture on the moduli space of quasi-polarized K3 surfaces. This is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles, i.e. sub-arithmetic manifolds of the same type. For compact manifolds this was proved in [2], here we extend the results of [2] to non-compact manifolds. This allows us to apply our results to the moduli spaces of quasi-polarized K3 surfaces. Contents 1. Introduction 1 2. Notations and Conventions 8 3. Arthur’s classification theory 8 4. Cohomological unitary representations for orthogonal groups 13 5. Residual representations 15 6. Thetacorrespondencefororthogonalgroups 22 7. Cohomology of arithmetic manifolds 26 8. Special cycles on arithmetic manifolds of orthogonal type 29 References 35 1. Introduction 1.1. The Noether-Lefschetz conjecture. The study of Picard groups of moduli problems was started by Mumford [50] in the 1960’s. For the moduli space g M arXiv:1412.3774v2 [math.AG] 14 Apr 2015 of genus g curves, Mumford and Harer (cf. [51, 25]) showed that the Picard group 2 Pic g of g is isomorphic to its second cohomology group H g, Z , which is a(M finitely) generatedM abelian group of rank one for g 3. Moreover,(M the generator) 2 ≥ of H g, Q is the first Chern class of the Hodge bundle on g. In(M higher dimensional) moduli theory, a quasi-polarized K3M surface of genus g is a two dimensional analogue of the genus g smooth projective curve. Here, a quasi-polarized K3 surface of genus g 2 is defined by a pair S,L where S is a ≥ ( ) 2 K3 surface and L is line bundle on S with primitive Chern class c1 L H S, Z satisfying ( )∈ ( ) 2 L L = c1 L = 2g 2 and L C = c1 L ≥ 0 ⋅ SS ( ) − ⋅ SC ( ) N.B. is a member of the Institut Universitaire de France. J.M. was supported by NSF grant DMS -1206999. 1 2 NICOLAS BERGERON, ZHIYUAN LI, JOHN MILLSON, AND COLETTE MOEGLIN for every curve C ⊂ S. Let g be the coarse moduli space of quasi-polarized K3 K surfaces of genus g. Unlike the case of Pic g , O’Grady [52] has shown that (M ) the rank of Pic g can be arbitrarily large. Besides the Hodge line bundle, there (K ) are actually many other natural divisors on g coming from Noether-Lefschetz theory developed by Griffiths and Harris in [23]K (see also [43]). More precisely, the Noether-Lefschetz locus in g parametrizes K3 surfaces in g with Picard number greater than 2; it is a countableK union of divisors. EachK of them parametrizes the K3 surfaces whose Picard lattice contains a special curve class; these divisors are called Noether-Lefschetz (NL) divisors on g. Oguiso’s theorem [53, Main K Theorem] implies that any curve on g will meet some NL-divisor on g (see also K K [7, Theorem 1.1]). So it is natural to ask whether the Picard group PicQ g of (K ) g with rational coefficients is spanned by NL-divisors. This is conjectured to be trueK by Maulik and Pandharipande, see [43, Conjecture 3]. More generally, one can extend this question to higher NL-loci on g, which parametrize K3 surfaces in K g with higher Picard number, see [32]. Call the irreducible components of higher K NL-loci the NL-cycles on g. Each of them parametrizes K3 surfaces in g whose Picard lattice contains a specialK primitive lattice. K 2r Theorem 1.2. For all g ≥ 2 and all r ≤ 4, the cohomology group H g, Q is (K ) spanned by NL-cyles of codimension r. In particular (taking r = 1), PicQ g ≅ 2 (K ) H g, Q and the Noether-Lefschetz conjecture holds on g for all g ≥ 2. (K ) K Remark 1.3. There is a purely geometric approach (cf. [22]) for low genus case (g ≤ 12), but it can not be applied for large genera. It remains interesting to give a geometric proof for this conjecture. Combined with works of Borcherds and Bruinier in [6] and [13] (see also [38]), we get the following: Corollary 1.4. We have 31g 24 1 1 (1.1) rank Pic g = + αg βg ( (K )) 24 − 4 − 6 g−1 k2 k2 Q k ∈ Z, 0 ≤ k ≤ g 1 − 4g 4¡ − ♯ S 4g 4 − ¡ k=0 − − where g−1 ⎧0, if g is even, ⎧ 4g−5 1, if g ≡ 1 mod3, αg = ⎪ 2g−2 βg = ⎪ − ⎨ otherwise, ⎨ g−1 g−1 otherwise, ⎪ 2g−3 ⎪ 4g−5 3 ⎩⎪ ⎩⎪ + and a is the Jacobi symbol. b 1.5. From moduli theory to Shimura varieties of orthogonal type. Theo- rem 1.2 is deduced from a general theorem on arithmetic manifolds. Indeed: let 2 S,L be a K3 surface in g, then the middle cohomology H S, Z is an even unimodular( ) lattice of signatureK 3, 19 under the intersection form( ), which is isometric to the K3 lattice ( ) ⟨ ⟩ ⊕3 ⊕ − ⊕2 LK3 = U E8 , ( ) where U is the hyperbolic lattice of rank two and E8 is the positive definite lattice associated to the Lie group of the same name; see [43]. In fact a fundamental THE NOETHER-LEFSCHETZ CONJECTURE 3 theorem of M. Friedman states that the homeomorphism type of a compact, ori- entable, simply connected (real) 4-manifold M is determined by the isometry type of H2 M, Z , , . One may then define a K3 surface to be a simply connected algebraic( ( surface) ⟨ ⟩)S whose middle cohomology lattice H2 S, Z , , is isometric to 3 ( ( ) ⟨ ⟩) the K3 lattice LK3. Any nonsingular quartic in P is a K3 surface. 2 A marking on a K3 surface S is a choice of an isometry u ∶ H S, Z → LK3. ( ) If S,L is a quasi-polarized K3 surface, the first Chern class c1 L is a primitive ( ) 2 2 ( ) 2 vector in H S, Z . Define the primitive sublattice H S, Z prim of H S, Z by ( ) ( ) ( ) 2 2 H S, Z prim = η ∈ H S, Z ∶ η ∧ c1 L = 0 . ( ) { ( ) ( ) } Then we have an orthogonal (for the intersection form) splitting 2 2 (1.2) H S, Z = Zc1 L + H S, Z prim. ( ) ( ) ( ) 2 There is a Hodge structure on H S, Z prim given by the Hodge decomposition induced by the Hodge structure on H( 2 S,) Z : ( ) 2 2,0 1,1 0,2 H S, Z prim ⊗Z C = H S, C ⊕ H S, C prim ⊕ H S, C ( ) ( ) ( ) ( ) with Hodge number 1, 19, 1 . We will now describe( the) moduli space of such polarized Hodge structures. Fix 2 − a primitive element v ∈ LK3 such that v is positive (and therefore equal to 2 g 1 ( ) for some g ≥ 2). Write ⊕⊥ LK3 = v Λ. ⟨ ⟩ The lattice Λ is then isometric to the even lattice ⊕2 ⊕2 Zw ⊕ U ⊕ −E8 , ( ) where w, w = 2−2g. A marked v-quasi-polarized K3 surface is a collection S,L,u ⟨ ⟩ ( ) where S,L is a quasi-polarized K3 surface, u is a marking and u c1 L = v. Note ( ) ( ( )) 2,0 that this forces S,L to be of genus g. The period point of S,L,u is uC H S , 2 ( ) ( ) ( ( )) where uC ∶ H S, Z ⊗Z C → LK3 C is the complex linear extension of u. It is a ( ) ( ) complex line C ω ∈ LK3 C satisfying ⋅ ( ) ω,ω = 0 and ω, ω > 0. ⟨ ⟩ ⟨ ⟩ It is moreover orthogonal to v = u c1 L ∈ LK3. Let V = Λ Z Q be the linear space orthogonal to v. We conclude( that( )) the period point belongs⊗ to × D = D V = ω ∈ V Q C ω,ω = 0, ω, ω > 0 C ( ) { ⊗ S ⟨ ⟩ ⟨ ⟩ }~ ≅ oriented positive 2-planes in VR = V Q R { ⊗ } ≅ SO VR KR, ( )~ where KR ≅ SO 2 SO 19 is the stabilizer of an oriented positive 2-plane in ( ) × ( ) SO VR ≅ SO 2, 19 . Here, by positive 2-plane P we mean a two dimensional ( ) ( ) subspace P ⊂ VR such that the restriction of the form , to P is positive definite. Now the global Torelli theorem for K3 surfaces (cf.⟨ [54,⟩ 18]) says that the period map is onto and that if S,L and S′,L′ are two quasi-polarized K3-surfaces and if there exists an isometry( of) lattices( ψ H) 2 S′, Z → H2 S, Z such that ∶ ( ) ( ) ′ 2,0 ′ 2,0 ψ c1 L = ψ c1 L and ψC H S = H S , ( ( )) ( ( )) ( ( )) ( ) then there exists a unique isomorphism of algebraic varieties f S → S′ such that ∗ ∶ f = ψ. Forgetting the marking, we conclude that the period map identifies the 4 NICOLAS BERGERON, ZHIYUAN LI, JOHN MILLSON, AND COLETTE MOEGLIN complex points of the moduli space g with the quotient Γ D where K ∨ Γ = γ ∈ SO Λ γ acts trivially on Λ Λ , { ( ) S ~ } is the natural monodromy group acting properly discontinuously on D. Note that the quotient YΓ = Γ D is a connected component of a Shimura variety associated to the group SO 2, 19 . ( ) We now interpret NL-cycles on g as special cycles on YΓ. Fix a vector x in Λ. Then the set of marked v-quasi-polarizedK K3 surfaces S,L,u for which x is the projection in Λ of an additional element in ( ) 1,1 2 Pic S ≅ H S, C H S, Z ( ) ( ) ∩ ( ) 2,0 corresponds to the subset of S,L,u ∈ g for which the period point uC H S belongs to ( ) K ( ( )) ⊥ Dx = ω ∈ D x, ω = 0 = D V x { S ⟨ ⟩ } ( ∩ ) = oriented positive 2-planes in VR that are orthogonal to x .