DOCSLIB.ORG

The Noether-Lefschetz Conjecture and Generalizations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Noether-Lefschetz Conjecture and Generalizations 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 .
Load more

Recommended publications
  • Arthur's Multiplicity Formula for Certain Inner Forms of Special Orthogonal and Symplectic Groups
    Arthur's multiplicity formula for certain inner forms of special orthogonal and symplectic groups Olivier Taïbi ∗ Abstract Let G be a special orthogonal group or an inner form of a symplectic group over a number eld F such that there exists a non-empty set S of real places of F at which G has discrete series and outside of which G is quasi-split. We prove Arthur's multiplicity formula for automorphic representations of G having algebraic regular innitesimal character at all places in S. Contents 1 Introduction2 2 Review of Arthur's results4 2.1 Quasi-split symplectic and special orthogonal groups and their Langlands dual groups....................................4 2.2 Self-dual automorphic cuspidal representations of GLN ............5 2.3 Elliptic endoscopic data and embeddings of L-groups.............5 2.4 Arthur's substitute for global parameters....................7 2.5 Local Arthur packets in the quasi-split case..................8 2.6 Localisation of global parameters........................ 10 3 Certain rigid inner forms and Arthur packets 10 3.1 Certain rigid inner forms............................. 10 3.1.1 Rigid inner forms at real places..................... 11 3.1.2 Global rigid inner forms......................... 14 3.2 Local Arthur packets for real reductive groups having discrete series.... 17 3.2.1 Tempered parameters.......................... 17 3.2.2 Non-tempered parameters: Adams-Johnson packets.......... 20 3.2.3 Relation with Arthur packets...................... 22 3.3 Trivial inner twists................................ 23 3.4 Adèlic Arthur packets for rigid inner forms................... 25 4 Arthur's multiplicity formula for certain inner forms 27 ∗Imperial College London, United Kingdom.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Harmonic Analysis
    Mathematisches Forschungsinstitut Oberwolfach Report No. 25/2017 DOI: 10.4171/OWR/2017/25 Harmonic Analysis and the Trace Formula Organised by Werner M¨uller, Bonn Sug Woo Shin, Berkeley Birgit Speh, Ithaca Nicolas Templier, Ithaca 21 May – 27 May 2017 Abstract. The purpose of this workshop was to discuss recent results in harmonic analysis that arise in the study of the trace formula. This theme is common to different directions of research on automorphic forms such as representation theory, periods, and families. Mathematics Subject Classification (2010): Primary: 11xx; secondary: 14Lxx, 22xx. Introduction by the Organisers The Oberwolfach Workshop took place from May 21st through 27th of 2016, and brought together 50 participants. There were 25 talks of 45 minutes each. A motivation of the subject can be seen from the far-reaching reciprocity and func- toriality conjectures of Langlands, according to which all reasonable L-functions arising from Galois representations, arithmetic geometry and harmonic analysis should be automorphic. One of the main tools to study automorphic L-functions is the Arthur-Selberg trace formula. The trace formula is used to globalize local representations into global automorphic forms, and it is essential in the concept of families where one studies an object by deforming it. The trace formula continues to motivate a wide-range of techniques in algebra, representation theory on p-adic, real and adelic groups, in analysis, and in differential and algebraic geometry. Packets. Gan, Moeglin, Mezo, Cunningham, B. Xu, Kaletha, and W.-W. Li (in the order of their talks) spoke on the Langlands packets (L-packets) and Arthur packets (A-packets) for p-adic, real reductive, and adelic groups.
    [Show full text]
  • Arxiv:1306.1515V2 [Math.AG] 3 Jun 2014 At3 Uoopi Forms Automorphic 3
    THE HODGE CONJECTURE AND ARITHMETIC QUOTIENTS OF COMPLEX BALLS NICOLAS BERGERON, JOHN MILLSON, AND COLETTE MOEGLIN In memory of Raquel Maritza Gilbert beloved wife of the second author. Abstract. Let S be a closed Shimura variety uniformized by the complex n-ball. The Hodge conjecture predicts that every Hodge class in H2k(S, Q), k = 0,...,n, is algebraic. We show that this holds for all degrees k away from the neighborhood ]n/3, 2n/3[ of the middle degree. We also prove the Tate conjecture for the same degrees as the Hodge conjecture and the generalized form of the Hodge conjecture in degrees away from an interval (depending on the codimension c of the subvariety) centered at the middle dimension of S. Finally we extend most of these results to Shimura varieties associated to unitary groups of any signature. The proofs make use of the recent endoscopic classification of automorphic representations of classical groups by [1, 58]. As such our results are conditional on the stabilization of the trace formula for the (disconnected) groups GL(N) ⋊ hθi associated to base change. At present the stabilization of the trace formula has been proved only for the case of connected groups. But the extension needed is now announced, see §1.13 for more details. Contents 1. Introduction 2 Part 1. Local computations 9 2. Hermitian vector spaces over C 9 3. Cohomological unitary representations 11 4. The action of U(p) U(q) U(a) U(b) in the twisted Fock model 21 × × × arXiv:1306.1515v2 [math.AG] 3 Jun 2014 5.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Harmonic Analysis
    Mathematisches Forschungsinstitut Oberwolfach Report No. 25/2017 DOI: 10.4171/OWR/2017/25 Harmonic Analysis and the Trace Formula Organised by Werner M¨uller, Bonn Sug Woo Shin, Berkeley Birgit Speh, Ithaca Nicolas Templier, Ithaca 21 May – 27 May 2017 Abstract. The purpose of this workshop was to discuss recent results in harmonic analysis that arise in the study of the trace formula. This theme is common to different directions of research on automorphic forms such as representation theory, periods, and families. Mathematics Subject Classification (2010): Primary: 11xx; secondary: 14Lxx, 22xx. Introduction by the Organisers The Oberwolfach Workshop took place from May 21st through 27th of 2016, and brought together 50 participants. There were 25 talks of 45 minutes each. A motivation of the subject can be seen from the far-reaching reciprocity and func- toriality conjectures of Langlands, according to which all reasonable L-functions arising from Galois representations, arithmetic geometry and harmonic analysis should be automorphic. One of the main tools to study automorphic L-functions is the Arthur-Selberg trace formula. The trace formula is used to globalize local representations into global automorphic forms, and it is essential in the concept of families where one studies an object by deforming it. The trace formula continues to motivate a wide-range of techniques in algebra, representation theory on p-adic, real and adelic groups, in analysis, and in differential and algebraic geometry. Packets. Gan, Moeglin, Mezo, Cunningham, B. Xu, Kaletha, and W.-W. Li (in the order of their talks) spoke on the Langlands packets (L-packets) and Arthur packets (A-packets) for p-adic, real reductive, and adelic groups.
    [Show full text]
  • THE NOETHER-LEFSCHETZ CONJECTURE and GENERALIZATIONS Contents 1. Introduction 1 2. a General Theorem on Arithmetic Manifolds
    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 [3], here we extend the results of [3] 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. A general theorem on arithmetic manifolds associated to orthogonal groups 6 3. Consequences of the general theorem 11 4. Automorphic description of the cohomology groups of YK 15 5. Arthur's classification theory 17 6. Residual representations 23 7. Theta correspondence for orthogonal groups 29 8. Proof of Theorem 2.14 33 References 36 1. Introduction 1.1. The Noether-Lefschetz conjecture. The study of Picard groups of moduli problems was started by Mumford [52] in the 1960's. For the moduli space Mg of genus g curves, Mumford and Harer (cf. [53, 25]) showed that the Picard group 2 Pic(Mg) of Mg is isomorphic to its second cohomology group H (Mg; Z), which is a finitely generated abelian group of rank one for g ≥ 3. Moreover, the generator 2 of H (Mg; Q) is the first Chern class of the Hodge bundle on Mg. In higher dimensional moduli theory, a quasi-polarized K3 surface of genus g is a two dimensional analogue of the genus g smooth projective curve.
    [Show full text]
  • ENDOSCOPY and COHOMOLOGY in a TOWER of CONGRUENCE MANIFOLDS for U(N, 1)
    Forum of Mathematics, Sigma (2019), vol. , e1, 46 pages 1 doi: ENDOSCOPY AND COHOMOLOGY IN A TOWER OF CONGRUENCE MANIFOLDS FOR U(n; 1) SIMON MARSHALL1 and SUG WOO SHIN2 1Department of Mathematics University of Wisconsin – Madison 480 Lincoln Drive Madison WI 53706, USA; [email protected] 2Department of Mathematics University of California, Berkeley 901 Evans Hall, Berkeley, CA 94720, USA / Korea Institute for Advanced Study, Dongdaemun-gu, Seoul 130-722, Republic of Korea; [email protected] Abstract By assuming the endoscopic classification of automorphic representations on inner forms of unitary groups, which is currently work in progress by Kaletha, Minguez, Shin, and White, we bound the growth of cohomology in congruence towers of locally symmetric spaces associated to U(n; 1). In the case of lattices arising from Hermitian forms, we expect that the growth exponents we obtain are sharp in all degrees. 2010 Mathematics Subject Classification: 32N10 (primary); 32M15 (secondary) 1. Introduction This paper studies the limit multiplicity problem for cohomological automorphic forms on arithmetic quotients of U(N − 1; 1). Let F be a totally real number field with ring of integers OF. Write A for the ring of adeles over F. Let U(N) = UE=F(N) denote the quasi-split unitary group with respect to a totally imaginary quadratic extension E of F. Let G be a unitary group over F that is an inner form of UE=F(N). We assume that G has signature (N − 1; 1) at one real place and compact factors at all other real places. Let S be a finite set of c The Author(s) 2019.
    [Show full text]