DOCSLIB.ORG

A. Algebraic Varieties and Complex Analytic Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A. Algebraic Varieties and Complex Analytic Spaces A. Algebraic Varieties and Complex Analytic Spaces In this Appendix we compile some results about algebraic varieties over C, complex analytic spaces and their interrelations, which are frequently used in this book. To any algebraic variety X over the complex numbers one can associate a complex analytic space Xhol in a natural way (see Serre [1]). This construction is functorial: for any morphism f : Z → X of algebraic varieties over C there is a natural morphism of complex analytic spaces fhol : Zhol → Xhol. Moreover, many of the important notions in both categories are preserved under X → Xhol. For example: an algebraic variety X is complete if and only if Xhol is compact, and X is smooth if and only if Xhol is a complex manifold. Conversely, a complex analytic space X is called algebraic, if there is an algebraic variety X over C such that X Xhol. In both, the algebraic and the complex analytic category one has the notion of a vector bundle and its associated locally free sheaf. For our purposes it is convenient not to distinguish between these objects: we always speak of vector bundles, even if we mean the associated locally free sheaf. In particular, for a line bundle L on a complex analytic space (respectively algebraic variety) X we denote by H i(X, L), or H i(L) if there is no ambiguity about the base space X, the i-th cohomology group of X with values in the locally free sheaf associated to L. With a similar construction as above one can associate to an algebraic vector bundle (or more generally an algebraic coherent sheaf) F on the algebraic variety X over C a holomorphic vector bundle (respectively a holomorphic coherent sheaf) Fhol over the complex analytic space Xhol. Again this process is functorial: given algebraic vector bundles (respectively coherent sheaves) F and G on X, one can associate to any algebraic homomorphism f : F → G a holomorphic homomorphism fhol : Fhol → Ghol. Not every holomorphic vector bundle on an arbitrary algebraic complex analytic space comes from an algebraic one. However in the case of complete algebraic varieties we have the following comparison theorems due to Serre [1]. Theorem A.1. Let X be a complete algebraic variety over C. a) For any holomorphic vector bundle (respectively holomorphic coherent sheaf) F on Xhol there exists a unique algebraic vector bundle (respectively algebraic coherent sheaf) F such that Fhol F. 568 A. Algebraic Varieties and Complex Analytic Spaces b) For every homomorphism f: F → G of holomorphic coherent sheaves F and G on Xhol there exists a unique homomorphism f : F → G of algebraic coherent sheaves on X such that fhol = f. Theorem A.2. Let X be a complete algebraic variety over C. For any coherent sheaf i i F on X the natural maps H (X, F ) → H (Xhol,Fhol), i ∈ Z, are isomorphisms of C-vector spaces. Thus in the case of a complete algebraic variety X we need not distinguish between algebraic vector bundles on X and holomorphic vector bundles on Xhol. Recall the following theorem due to (see Griffiths-Harris [1]). Theorem A.3. Suppose X is a complete algebraic variety over C and Z a closed analytic subset of Xhol, then there is an algebraic subvariety Z of X, such that Zhol Z. A consequence of Chow’s theorem is Corollary A.4. Suppose X and Z are complete algebraic varieties over C and f : Zhol → Xhol is a holomorphic map. Then there is an algebraic map f : Z → X with fhol = f. Moreover we need the following theorem due to Grauert-Remmert [1]. Theorem A.5. Suppose X is a normal algebraic variety over C and Z is a normal complex analytic space. If f: Z → Xhol is a finite morphism, then there exists a unique normal algebraic variety Z over C and a finite morphism f : Z → X such that Zhol = Z and fhol = f. Next let us recall a result about quotients of complex analytic spaces (see Cartan [1]). For this let G be a group acting as a group of isomorphisms on the complex analytic space X. The quotient X/G, endowed with the quotient topology, admits the structure of a ringed space in a natural way: denote by π : X → X/Gthe canonical projection. Then by definition OX/G(U), for U ⊆ X/Gopen, is the set of functions f : U → C, − for which fπ is an element of OX(π 1U). Moreover recall that the group G acts properly and discontinuously on the complex analytic space X, if for any pair of compact subsets K1,K2 of X the set {g ∈ G|gK1 ∩ K2 =∅}is finite. Then Theorem A.6. Suppose X is a complex analytic space and G is a group, acting properly and discontinuously on X. The quotient X/G is also a complex analytic space. Moreover, if X is normal, so is X/G. In the special case that X is a complex manifold there is a criterion for the quotient X/G to be also a complex manifold. Note that the action of G on X is said to be free, if gx = x for some x ∈ X and g ∈ G implies g = id. Corollary A.7. Let X be a complex manifold and suppose G is a group acting freely and properly discontinuously on X. Then X/G is also a complex manifold. Finally we give a proof of the so-called Seesaw Theorem, which we need in the following version. The same proof works also in the algebraic category. A. Algebraic Varieties and Complex Analytic Spaces 569 Seesaw Theorem A.8. Let X be a connected compact complex manifold, Z are- duced analytic space, and L a holomorphic line bundle on X × Z. a) The set Z0 ={z ∈ Z| L|X ×{z} is trivial} is Zariski-closed in Z. b) If q : X × Z0 → Z0 denotes the projection map, then there is a holomorphic line bundle M on Z0 such that ∗ L X × Z0 q M. Proof. a) Note first that a holomorphic line bundle N on a connected compact com- plex manifold is trivial if and only if h0(N )>0 and h0(N −1)>0. Consequently % & 0 0 −1 Z0 = z ∈ Z | h (L|X × {z})>0 and h (L |X × {z})>0 and the assertion follows from the Semicontinuity Theorem (see Grauert-Remmert [2] Theorem 10.5.4). b) According to the Base Change Theorem (see Grauert [1] p.2 (2)) the sheaf M := q∗(L|X × Z0) is invertible on Z0 and the canonical base change homomorphism ϕ(z) : M(z) → H 0 LX × {z} =∼ C is an isomorphism for every z ∈ Z0. We have to show that the canonical map ∗ ∗ q M = q q∗ L X × Z0 → L X × Z0 is an isomorphism. For this it suffices to show that the induced map ∗ H 0 q MX × {z} → H 0 LX × {z} is surjective for every z ∈ Z0, since L|X ×{z} is trivial. But this is a consequence of the surjectivity of ϕ(z). The following statement is an immediate consequence of the Seesaw Theorem. It is often called the Seesaw Principle. Corollary A.9. Let X and Z be compact complex manifolds and L a holomorphic line bundle on X × Z.IfL|X ×{z} is trivial for all z out of an open dense subset of Z and L|{x0}×Z is trivial for some x0 ∈ X, then L is trivial. B. Line Bundles and Factors of Automorphy In this section we outline how to describe certain line bundles by factors of automor- phy. Though the results are valid in greater generality, we restrict ourselves to the cases needed in this book. Suppose X is a complex manifold (for us X will be a complex torus or the universal abelian variety over the Siegel upper half space), and let π : X → X be the universal covering. The topological space X inherits the structure of a complex manifold in a natural way. We denote by π1(X) the fundamental group of X → X.Inthe notation we omit the base point of the fundamental group. Our aim is to describe those holomorphic line bundles L on X, whose pullback π ∗L is trivial, in terms of the cohomology of the fundamental group π1(X) acting from the left on X. Let us recall the definition of the cohomology group H 1(π (X), H 0(O∗ )) of π (X) 1 X 1 with values in the multiplicative group H 0(O∗ ) of nonvanishing holomorphic func- X tions on X: the action of the fundamental group π1(X) on X induces a π1(X)-module structure on H 0(O∗ ). A holomorphic map f : π (X) × X → C∗ satisfying the co- X 1 cycle relation f (λμ, x)˜ = f(λ,μx)f(μ,˜ x)˜ (1) for all λ, μ ∈ π (X) and x˜ ∈ X is called 1-cocycle of π (X) with values in H 0(O∗ ). 1 1 X Under multiplication these 1-cocycles form an abelian group Z1(π (X), H 0(O∗ )). 1 X For historical reasons we call the elements of the group Z1(π (X), H 0(O∗ )) fac- 1 X tors of automorphy or simply factors. The factors of the form (λ, x)˜ → h(λx)h(˜ x)˜ −1 for some h ∈ H 0(O∗ ) are called boundaries. They form the subgroup X B1(π (X), H 0(O∗ )) of Z1(π (X), H 0(O∗ )). The group H 1(π (X), H 0(O∗ )) is 1 X 1 X 1 X defined to be the quotient H 1(π (X), H 0(O∗ )) = Z1(π (X), H 0(O∗ ))/B1(π (X), H 0(O∗ )) .
Load more

Recommended publications
  • Models of Shimura Varieties in Mixed Characteristics by Ben Moonen
    Models of Shimura varieties in mixed characteristics by Ben Moonen Contents Introduction .............................. 1 1 Shimuravarieties ........................... 5 2 Canonical models of Shimura varieties. 16 3 Integralcanonicalmodels . 27 4 Deformation theory of p-divisible groups with Tate classes . 44 5 Vasiu’s strategy for proving the existence of integral canonical models 52 6 Characterizing subvarieties of Hodge type; conjectures of Coleman andOort................................ 67 References ............................... 78 Introduction At the 1996 Durham symposium, a series of four lectures was given on Shimura varieties in mixed characteristics. The main goal of these lectures was to discuss some recent developments, and to familiarize the audience with some of the techniques involved. The present notes were written with the same goal in mind. It should be mentioned right away that we intend to discuss only a small number of topics. The bulk of the paper is devoted to models of Shimura varieties over discrete valuation rings of mixed characteristics. Part of the discussion only deals with primes of residue characteristic p such that the group G in question is unramified at p, so that good reduction is expected. Even at such primes, however, many technical problems present themselves, to begin with the “right” definitions. There is a rather large class of Shimura data—those called of pre-abelian type—for which the corresponding Shimura variety can be related, if maybe somewhat indirectly, to a moduli space of abelian varieties. At present, this seems the only available tool for constructing “good” integral models. Thus, 1 if we restrict our attention to Shimura varieties of pre-abelian type, the con- struction of integral canonical models (defined in 3) divides itself into two § parts: Formal aspects.
    [Show full text]
  • Siegel Modular Varieties and Borel-Serre Compactification of Modular Curves
    Siegel modular varieties and Borel-Serre compactification of modular curves Michele Fornea December 11, 2015 1 Scholze’s Theorem on Torsion classes To motivate the study of Siegel modular varieties and Borel-Serre compactifica- tions let me recall Scholze’s theorem on torsion classes. i We start from an Hecke eigenclass h 2 H (Xg=Γ; Fp) where Xg is the locally symmetric domain for GLg, and we want to attach to it a continuous semisimple Galois representation ρh : GQ ! GLg(Fp) such that the characteristic polyno- mials of the Frobenius classes at unramified primes are determined by the Hecke eigenvalues of h. Problem: Xg=Γ is not algebraic in general, in fact it could be a real manifold of odd dimension; while the most powerful way we know to construct Galois representations is by considering étale cohomology of algebraic varieties. Idea(Clozel): Find the cohomology of Xg=Γ in the cohomology of the boundary BS of the Borel-Serre compactification Ag of Siegel modular varieties. 1.0.1 Properties of the Borel-Serre compactification BS • Ag is a real manifold with corners. BS • The inclusion Ag ,! Ag is an homotopy equivalence (same cohomology). BS • The boundary of Ag is parametrized by parabolic subgroups of Sp2g and consists of torus bundles over arithmetic domains for the Levi subgroups of each parabolic. BS It follows from the excision exact sequence for the couple (Ag ;@) that i one can associate to the Hecke eigenclass h 2 H (Xg=Γ; Fp) another eigenclass 0 i h 2 H (Ag; Fp) whose eigenvalues are precisely related to those of h.
    [Show full text]
  • Abelian Varieties
    Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties .
    [Show full text]
  • Arxiv:1507.05922V1
    Torsion in the Coherent Cohomology of Shimura Varieties and Galois Representations A dissertation presented by George Andrew Boxer to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts arXiv:1507.05922v1 [math.NT] 21 Jul 2015 April 2015 c 2015 – George A. Boxer All rights reserved. Dissertation Advisor: Richard Taylor George Andrew Boxer Torsion in the Coherent Cohomology of Shimura Varieties and Galois Representations Abstract We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties. The congruences are produced using some “generalized Hasse invariants” adapted to the Ekedahl-Oort stratification of the special fiber. iii Contents 1 Introduction 1 1.1 Motivation: Weight 1 Modular Forms mod p .................. 1 1.2 Constructing Congruences Using Hasse Invariants . ...... 3 1.3 GeometricSiegelModularForms ........................ 5 1.4 The Ekedahl-Oort Stratification and Generalized Hasse Invariants...... 9 1.5 HasseInvariantsattheBoundary . .. 16 1.6 ConstructingCongruences . 17 1.7 OverviewandAdvicefortheReader . 22 1.8 NotationandConventions ............................ 22 2 Good Reduction PEL Modular Varieties 25 2.1 PELDatum.................................... 25 2.2 PELModularVarieties.............................. 31 2.3 AutomorphicVectorBundles. 37 2.4 HeckeAction ................................... 38 2.5 MorphismtoSiegelSpace ............................ 39 3 Arithmetic Compactifications of PEL Modular Varieties 42 3.1 ToroidalBoundaryCharts . .. .. 43 3.2 Arithmetic Compactifications . 51 3.2.1 ToroidalCompactifications. 51 iv 3.2.2 Minimal Compactifications . 52 3.3 Extensions of Automorphic Vector Bundles . ... 54 3.3.1 Canonical and Subcanonical Extensions . 54 3.3.2 HigherDirectImages........................... 57 3.3.3 Pushforwards to the Minimal Compactification .
    [Show full text]
  • The Kuga-Satake Construction: a Modular Interpretation
    The Kuga-Satake Construction: A Modular Interpretation Alice Pozzi A Thesis for The Department of Mathematics and Statistics Presented in Partial Fulfillment of the Requirements for the Degree of Master of Science (Mathematics) at Concordia University Montreal, Quebec, Canada August 2013 c Alice Pozzi, 2013 CONCORDIA UNIVERSITY School of Graduate Studies This is to certify that the thesis prepared By: Alice Pozzi Entitled: The Kuga-Satake Construction: A Modular Interpretation and submitted in partial fulfillment of the requirements for the degree of Master of Science (Mathematics) complies with the regulations of the University and meets the accepted standards with respect to originality and quality. Signed by the final examining committee: Examiner Dr. C. Cummins Examiner Dr. C. David Thesis Supervisor Dr. A. Iovita Approved by Chair of Department or Graduate Program Director Dean of Faculty Date Abstract The Kuga-Satake Construction: A Modular Interpretation Alice Pozzi Given a polarized complex K3 surface, one can attach to it a complex abelian variety, called Kuga-Satake variety. The Kuga-Satake variety is determined by the singular cohomology of the K3 surface; on the other hand, this singular cohomology can be recovered by means of the weight 1 Hodge structure associated to the Kuga-Satake variety. Despite the transcendental origin of this construction, Kuga-Satake varieties have interesting arithmetic properties. Kuga-Satake varieties of K3 surfaces defined over number fields descend to finite extension of the field of definition. This property suggests that the Kuga-Satake construction can be interpreted as a map between moduli spaces. More precisely, one can define a morphism, called Kuga-Satake map, between the moduli space of K3 surfaces and the moduli space of abelian varieties with polarization and level structure.
    [Show full text]
  • Arxiv:Math/0605346V2 [Math.AG] 21 May 2007 Ru SL(2 Group Fhceoeaos Rmtefuircoefficients
    SIEGEL MODULAR FORMS GERARD VAN DER GEER Abstract. These are the lecture notes of the lectures on Siegel modular forms at the Nordfjordeid Summer School on Modular Forms and their Applications. We give a survey of Siegel modular forms and explain the joint work with Carel Faber on vector-valued Siegel modular forms of genus 2 and present evidence for a conjecture of Harder on congruences between Siegel modular forms of genus 1 and 2. 1. Introduction Siegel modular forms generalize the usual modular forms on SL(2, Z) in that the group SL(2, Z) is replaced by the automorphism group Sp(2g, Z) of a unimodular symplectic form on Z2g and the upper half plane is replaced by the Siegel upper half plane g. The integer g 1 is called the degree or genus. Siegel pioneeredH the generalization≥ of the theory of elliptic modular forms to the modular forms in more variables now named after him. He was motivated by his work on the Minkowski-Hasse principle for quadratic forms over the rationals, cf., [95]. He investigated the geometry of the Siegel upper half plane, determined a fundamental domain and its volume and proved a central result equating an Eisenstein series with a weighted sum of theta functions. No doubt, Siegel modular forms are of fundamental importance in number theory and algebraic geometry, but unfortunately, their reputation does not match their importance. And although vector-valued rather than scalar-valued Siegel modular forms are the natural generalization of elliptic modular forms, their reputation amounts to even less. A tradition of ill-chosen notations may have contributed to this, but the lack of attractive examples that can be handled decently seems to be the main responsible.
    [Show full text]
  • Kudla's Modularity Conjecture and Formal
    Forum of Mathematics, Pi (2015), Vol. 3, e7, 30 pages 1 doi:10.1017/fmp.2015.6 KUDLA’S MODULARITY CONJECTURE AND FORMAL FOURIER–JACOBI SERIES JAN HENDRIK BRUINIER1 and MARTIN WESTERHOLT-RAUM2 1 Fachbereich Mathematik, Technische Universitat¨ Darmstadt, Schlossgartenstraße 7, D-64289 Darmstadt, Germany; email: [email protected] 2 Chalmers tekniska hogskola¨ och Goteborgs¨ Universitet, Institutionen for¨ Matematiska vetenskaper, SE-412 96 Goteborg,¨ Sweden; email: [email protected] Received 2 January 2015; accepted 11 August 2015 Abstract We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogs of Fourier–Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties. 2010 Mathematics Subject Classification: 11F46 (primary); 14C25 (secondary) 1. Introduction Fourier Jacobi expansions are one of the major tools to study Siegel modular forms. For example, they appeared prominently in the proof of the Saito– Kurokawa conjecture [1, 24–26, 38]. Also the work of Kohnen and Skoruppa on spinor L-functions [18, 19] features Fourier–Jacobi expansions. We formalize the notion of Fourier–Jacobi expansions by combining two features of Siegel modular forms: Fourier–Jacobi coefficients are Siegel–Jacobi forms, and Fourier expansions of genus-g Siegel modular forms have symmetries with respect to GLg.Z/. For simplicity, we restrict this exposition to classical Siegel modular forms of even weight. Suppose that f is a Siegel modular form of even weight k for the c The Author(s) 2015.
    [Show full text]
  • The Cohomology of the Moduli Space of Abelian Varieties Van Der Geer, G
    UvA-DARE (Digital Academic Repository) The cohomology of the moduli space of Abelian varieties van der Geer, G. Publication date 2013 Document Version Submitted manuscript Published in The handbook of moduli. - Volume 1 Link to publication Citation for published version (APA): van der Geer, G. (2013). The cohomology of the moduli space of Abelian varieties. In G. Farkas, & I. Morrison (Eds.), The handbook of moduli. - Volume 1 (pp. 415-458). (Advanced Lectures in Mathematics; No. 24). International Press. http://www.maa.org/publications/maa- reviews/handbook-of-moduli-volume-i General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:30 Sep 2021 The Cohomology of the Moduli Space of Abelian Varieties Gerard van der Geer Introduction That the moduli spaces of abelian varieties are a rich source of arithmetic and geometric information slowly emerged from the work of Kronecker, Klein, Fricke and many others at the end of the 19th century.
    [Show full text]
  • Integral Cohomology of the Siegel Modular Variety of Degree Two and Level Three Mustafa Arslan Louisiana State University and Agricultural and Mechanical College
    Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2006 Integral cohomology of the Siegel modular variety of degree two and level three Mustafa Arslan Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Applied Mathematics Commons Recommended Citation Arslan, Mustafa, "Integral cohomology of the Siegel modular variety of degree two and level three" (2006). LSU Doctoral Dissertations. 2504. https://digitalcommons.lsu.edu/gradschool_dissertations/2504 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. INTEGRAL COHOMOLOGY OF THE SIEGEL MODULAR VARIETY OF DEGREE TWO AND LEVEL THREE A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial ful¯llment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Mustafa Arslan B.S. in Math., Middle East Technical University, 1996 May 2006 Acknowledgements I would like to thank Dr. Jerome W. Ho®man for his guidance and invaluable help, Dr. Leonard Richardson for constant reminders and enormous patience, Dr. Gestur Olafsson for giving his time when needed and committee members Dr. Robert F. Lax, Dr. Richard A. Litherland, Dr. Frank Neubrander, Dr. Carlos Slawson, and Dr. Paul van Wamelen for their service. I would also like to thank my wife Reena Kothari for her encouragement and support.
    [Show full text]
  • MULTI-TENSORS of DIFFERENTIAL FORMS on the SIEGEL MODULAR VARIETY and on ITS SUBVARIETIES by Shigeaki Tsuyumine
    TSUKUBA J. MATH. Vol. 11 No. 1 (1987). 107-119 MULTI-TENSORS OF DIFFERENTIAL FORMS ON THE SIEGEL MODULAR VARIETY AND ON ITS SUBVARIETIES By Shigeaki Tsuyumine Let An=Hn/rn> where Hn is the Siegel space and rn―Sp2n{Z). An is called a Siegel modular variety, which is the coarse moduli variety of n-dimen- sional principallypolarized abelian varieties over C. Let An be a projective non-singular model of An. An is shown to be of general type for n2i9 by Tai [10] (n=8 by Freitag [6], n―1 by Mumford [9]). Subvarieties of An are ex- pected to have the same property if they are not too special. Freitag [7] showed that An carries many global sections of Symmd(i2f~1), JV=n(n+l)/2, if n is bounded from below by some n0, and that any subvariety in An (n^n0) of codimension one is of type 'G' which is some weakened notion of "general type". He conjectured that n0 would be taken to be ten,in connec- tion with the argument of extensibility of holomorphic differentials to a non- singular model which is similar to Tai's one [9]. In this paper, we show that for n^lO, An carries many global sections of (Q*~1)Rr for some r (Theorem 1), and show the following; Theorem 2. Let n^lO. Then any subvariety in An of codimension one it nf aanam I tvfia. We have the following corollary to this theorem (cf. Freitag [7]). We denote by /＼≫(/),the principal congruence subgroup {Me/7n|M=l2n mod/}, l2n being the identity matrix of size 2n, and by AnA, the quotient space Hn/Fn{l).
    [Show full text]
  • Genus Two Quasi-Siegel Modular Forms and Gromov-Witten Theory of Toric Calabi-Yau Threefolds
    Genus Two Quasi-Siegel Modular Forms and Gromov-Witten Theory of Toric Calabi-Yau Threefolds Yongbin Ruan, Yingchun Zhang and Jie Zhou Abstract We first develop theories of differential rings of quasi-Siegel modular and quasi-Siegel Jacobi forms for genus two. Then we apply them to the Eynard-Orantin topological re- cursion of certain local Calabi-Yau threefolds equipped with branes, whose mirror curves are genus-two hyperelliptic curves. By the proof of the Remodeling Conjecture, we prove that the corresponding open- and closed- Gromov-Witten potentials are essentially quasi-Siegel Jacobi and quasi-Siegel modular forms for genus two, respectively. Contents 1 Introduction 2 1.1 Differential ring of quasi-Siegel modular forms for genus two . .2 1.2 Modularity in topological recursion and in open-closed GW theory . .5 2 Review on genus two curves and their Jacobians 7 2.1 Hyperellipticity of genus two algebraic curves and their Jacobians . .7 2.2 Weierstrass s-function and functional field of hyperelliptic Jacobian . .9 2.3 Families of genus two curves . 11 3 Differential rings of quasi-modular and quasi-Jacobi forms for genus two 15 3.1 Quasi-modular forms . 15 3.2 Quasi-Jacobi forms . 30 4 Topological recursion for genus two mirror curve families and applications 33 4.1 Geometry of genus two curves constructed from mirror symmetry . 33 arXiv:1911.07204v3 [math.AG] 21 Feb 2020 4.2 Modularity in topological recursion . 36 4.3 Applications to open-closed Gromov-Witten theory . 40 A Dependence of Weierstrass }-function on frame and marking 42 A.1 Jacobian and Abel-Jacobi map .
    [Show full text]
  • Notes on Low Dimensional Modular Varieties
    Notes on Low Dimensional Modular Varieties Donu Arapura May 10, 2019 Contents 1 Elliptic curves in a nutshell3 1.1 Elliptic curves: elementary approach................3 1.2 Elliptic curves: analytic theory...................4 1.3 Analytic theory continued: theta functions.............6 1.4 Elliptic curves over arbitrary fields.................9 2 Modular curves 13 2.1 The action of SL2(R)........................ 13 2.2 The modular group SL2(Z)..................... 14 2.3 Modular forms............................ 16 2.4 Modular curves............................ 19 2.5 Dimension of spaces of modular forms............... 22 2.6 Moduli interpretation........................ 23 2.7 Models over number fields...................... 24 3 Hilbert and Picard modular surfaces 26 3.1 The Hilbert modular group..................... 26 3.2 Hilbert modular surfaces: topology................. 27 3.3 Hilbert modular forms........................ 29 3.4 Riemann-Roch for surfaces..................... 32 3.5 Picard modular surfaces....................... 35 4 Abelian varieties 37 4.1 Abelian varieties........................... 37 4.2 Jacobians............................... 40 4.3 Siegel modular varieties....................... 43 4.4 Siegel modular varieties are moduli spaces............. 46 5 The endomorphism algebra 48 5.1 Endomorphisms of elliptic curves.................. 48 5.2 Poincar´ereducibility......................... 49 5.3 The Rosati involution........................ 50 5.4 Division rings with involution.................... 51 1 6 Introduction to Shimura varieties 54 6.1 Hilbert modular varieties....................... 54 6.2 Shimura varieties of PEL type.................... 56 6.3 Representation theoretic viewpoint................. 57 6.4 Deligne's axioms........................... 60 7 Siegel modular varieties of small genus 62 7.1 Genus 2 curves and abelian surfaces................ 62 7.2 Explicit models for A2 ........................ 65 7.3 Compactification of A2 ........................ 68 7.4 Genus 3 curves...........................
    [Show full text]