Two Or Three Things I Know About Abelian Varieties
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Canonical Heights on Varieties with Morphisms Compositio Mathematica, Tome 89, No 2 (1993), P
COMPOSITIO MATHEMATICA GREGORY S. CALL JOSEPH H. SILVERMAN Canonical heights on varieties with morphisms Compositio Mathematica, tome 89, no 2 (1993), p. 163-205 <http://www.numdam.org/item?id=CM_1993__89_2_163_0> © Foundation Compositio Mathematica, 1993, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 89: 163-205,163 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands. Canonical heights on varieties with morphisms GREGORY S. CALL* Mathematics Department, Amherst College, Amherst, MA 01002, USA and JOSEPH H. SILVERMAN** Mathematics Department, Brown University, Providence, RI 02912, USA Received 13 May 1992; accepted in final form 16 October 1992 Let A be an abelian variety defined over a number field K and let D be a symmetric divisor on A. Néron and Tate have proven the existence of a canonical height hA,D on A(k) characterized by the properties that hA,D is a Weil height for the divisor D and satisfies A,D([m]P) = m2hA,D(P) for all P ~ A(K). Similarly, Silverman [19] proved that on certain K3 surfaces S with a non-trivial automorphism ~: S ~ S there are two canonical height functions hs characterized by the properties that they are Weil heights for certain divisors E ± and satisfy ±S(~P) = (7 + 43)±1±S(P) for all P E S(K) . -
Divisors and the Picard Group
18.725 Algebraic Geometry I Lecture 15 Lecture 15: Divisors and the Picard Group Suppose X is irreducible. The (Weil) divisor DivW (X) is defined as the formal Z combinations of subvarieties of codimension 1. On the other hand, the Cartier divisor group, DivC (X), consists of subvariety locally given by a nonzero rational function defined up to multiplication by a nonvanishing function. Definition 1. An element of DivC (X) is given by 1. a covering Ui; and 2. Rational functions fi on Ui, fi 6= 0, ∗ such that on Ui \ Uj, fj = 'ijfi, where 'ij 2 O (Ui \ Uj). ∗ ∗ ∗ Another way to express this is that DivC (X) = Γ(K =O ), where K is the sheaf of nonzero rational functions, where O∗ is the sheaf of regular functions. Remark 1. Cartier divisors and invertible sheaves are equivalent (categorically). Given D 2 DivC (X), then we get an invertible subsheaf in K, locally it's fiO, the O-submodule generated by fi by construction it is locally isomorphic to O. Conversely if L ⊆ K is locally isomorphic to O, A system of local generators defines the data as above. Note that the abelian group structure on Γ(K∗=O∗) corresponds to multiplying by the ideals. ∗ ∗ ∗ ∗ Proposition 1. Pic(X) = DivC (X)=Im(K ) = Γ(K =O )= im Γ(K ). Proof. We already have a function DivC (X) = IFI ! Pic (IFI: invertible frational ideals) given by (L ⊆ K) 7! L. This map is an homomorphism. It is also onto: choosing a trivialization LjU = OjU gives an isomorphism ∼ ∗ ∗ L ⊗O⊇L K = K.No w let's look at its kernel: it consits of sections of K =O coming from O ⊆ K, which is just the same as the set of nonzero rational functions, which is im Γ(K∗) = Γ(K∗)=Γ(O∗). -
Rigid Analytic Curves and Their Jacobians
Rigid analytic curves and their Jacobians Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult¨at f¨urMathematik und Wirtschaftswissenschaften der Universit¨atUlm vorgelegt von Sophie Schmieg aus Ebersberg Ulm 2013 Erstgutachter: Prof. Dr. Werner Lutkebohmert¨ Zweitgutachter: Prof. Dr. Stefan Wewers Amtierender Dekan: Prof. Dr. Dieter Rautenbach Tag der Promotion: 19. Juni 2013 Contents Glossary of Notations vii Introduction ix 1. The Jacobian of a curve in the complex case . ix 2. Mumford curves and general rigid analytic curves . ix 3. Outline of the chapters and the results of this work . x 4. Acknowledgements . xi 1. Some background on rigid geometry 1 1.1. Non-Archimedean analysis . 1 1.2. Affinoid varieties . 2 1.3. Admissible coverings and rigid analytic varieties . 3 1.4. The reduction of a rigid analytic variety . 4 1.5. Adic topology and complete rings . 5 1.6. Formal schemes . 9 1.7. Analytification of an algebraic variety . 11 1.8. Proper morphisms . 12 1.9. Etale´ morphisms . 13 1.10. Meromorphic functions . 14 1.11. Examples . 15 2. The structure of a formal analytic curve 17 2.1. Basic definitions . 17 2.2. The formal fiber of a point . 17 2.3. The formal fiber of regular points and double points . 22 2.4. The formal fiber of a general singular point . 23 2.5. Formal blow-ups . 27 2.6. The stable reduction theorem . 29 2.7. Examples . 31 3. Group objects and Jacobians 33 3.1. Some definitions from category theory . 33 3.2. Group objects . 35 3.3. Central extensions of group objects . -
Arxiv:2008.05229V2 [Math.AG]
A TORELLI THEOREM FOR MODULI SPACES OF PARABOLIC VECTOR BUNDLES OVER AN ELLIPTIC CURVE THIAGO FASSARELLA AND LUANA JUSTO Abstract. Let C be an elliptic curve, w ∈ C, and let S ⊂ C be a finite subset of cardinality at least 3. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle OC (w) over (C,S) which are semistable with respect to a weight vector 1 1 2 ,..., 2 . 1. Introduction Let C be a smooth complex curve of genus g ≥ 2 and fix w ∈ C. Let M be the corresponding moduli space of semistable rank two vector bundles having OC (w) as determinant line bundle. A classical Torelli type theorem of D. Mumford and P. Newstead [MN68] says that the isomorphism class of M determines the isomorphism class of C. This result has been extended first in [Tyu70, NR75, KP95] to higher rank and later to the parabolic context, which we now describe. We now assume g ≥ 0. Let S ⊂ C be a finite subset of cardinality n ≥ 1. Let Ma be the moduli space of rank two parabolic vector bundles on (C,S) with fixed determinant line bundle OC (w), and which are µa-semistable, see Section 2. The subscript a refers to a particular choice of a weight vector a = (a1, . , an) of real numbers, 0 ≤ ai ≤ 1, which gives the slope-stability condition. The moduli 1 1 space associated to the central weight aF = 2 ,..., 2 is particularly interesting, for instance when g = 0 and n ≥ 5 it is a Fano variety that is smooth if n is odd and has isolated singularities if n is even, see [Muk05, Cas15, AM16, AFKM19]. -
A Torelli Theorem for Moduli Spaces of Principal Bundles Over a Curve Tome 62, No 1 (2012), P
R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Indranil BISWAS & Norbert HOFFMANN A Torelli theorem for moduli spaces of principal bundles over a curve Tome 62, no 1 (2012), p. 87-106. <http://aif.cedram.org/item?id=AIF_2012__62_1_87_0> © Association des Annales de l’institut Fourier, 2012, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 62, 1 (2012) 87-106 A TORELLI THEOREM FOR MODULI SPACES OF PRINCIPAL BUNDLES OVER A CURVE by Indranil BISWAS & Norbert HOFFMANN (*) 0 Abstract. — Let X and X be compact Riemann surfaces of genus > 3, and 0 d let G and G be nonabelian reductive complex groups. If one component MG(X) of the coarse moduli space for semistable principal G–bundles over X is isomorphic d0 0 0 to another component MG0 (X ), then X is isomorphic to X . Résumé. — Soient X et X0 des surfaces de Riemann compactes de genre au moins 3, et G et G0 des groupes complexes réductifs non abéliens. -
Algebraic Cycles and Fibrations
Documenta Math. 1521 Algebraic Cycles and Fibrations Charles Vial Received: March 23, 2012 Revised: July 19, 2013 Communicated by Alexander Merkurjev Abstract. Let f : X → B be a projective surjective morphism between quasi-projective varieties. The goal of this paper is the study of the Chow groups of X in terms of the Chow groups of B and of the fibres of f. One of the applications concerns quadric bundles. When X and B are smooth projective and when f is a flat quadric fibration, we show that the Chow motive of X is “built” from the motives of varieties of dimension less than the dimension of B. 2010 Mathematics Subject Classification: 14C15, 14C25, 14C05, 14D99 Keywords and Phrases: Algebraic cycles, Chow groups, quadric bun- dles, motives, Chow–K¨unneth decomposition. Contents 1. Surjective morphisms and motives 1526 2. OntheChowgroupsofthefibres 1530 3. A generalisation of the projective bundle formula 1532 4. On the motive of quadric bundles 1534 5. Onthemotiveofasmoothblow-up 1536 6. Chow groups of varieties fibred by varieties with small Chow groups 1540 7. Applications 1547 References 1551 Documenta Mathematica 18 (2013) 1521–1553 1522 Charles Vial For a scheme X over a field k, CHi(X) denotes the rational Chow group of i- dimensional cycles on X modulo rational equivalence. Throughout, f : X → B will be a projective surjective morphism defined over k from a quasi-projective variety X of dimension dX to an irreducible quasi-projective variety B of di- mension dB, with various extra assumptions which will be explicitly stated. Let h be the class of a hyperplane section in the Picard group of X. -
Abelian Varieties and Theta Functions Associated to Compact Riemannian Manifolds; Constructions Inspired by Superstring Theory
ABELIAN VARIETIES AND THETA FUNCTIONS ASSOCIATED TO COMPACT RIEMANNIAN MANIFOLDS; CONSTRUCTIONS INSPIRED BY SUPERSTRING THEORY. S. MULLER-STACH,¨ C. PETERS AND V. SRINIVAS MATH. INST., JOHANNES GUTENBERG UNIVERSITAT¨ MAINZ, INSTITUT FOURIER, UNIVERSITE´ GRENOBLE I ST.-MARTIN D'HERES,` FRANCE AND TIFR, MUMBAI, INDIA Resum´ e.´ On d´etailleune construction d^ue Witten et Moore-Witten (qui date d'environ 2000) d'une vari´et´eab´elienneprincipalement pola- ris´eeassoci´ee`aune vari´et´ede spin. Le th´eor`emed'indice pour l'op´erateur de Dirac (associ´e`ala structure de spin) implique qu'un accouplement naturel sur le K-groupe topologique prend des valeurs enti`eres.Cet ac- couplement sert commme polarization principale sur le t^oreassoci´e. On place la construction dans un c^adreg´en´eralce qui la relie `ala ja- cobienne de Weil mais qui sugg`ereaussi la construction d'une jacobienne associ´ee`an'importe quelle structure de Hodge polaris´eeet de poids pair. Cette derni`ereconstruction est ensuite expliqu´eeen termes de groupes alg´ebriques,utile pour le point de vue des cat´egoriesTannakiennes. Notre construction depend de param`etres,beaucoup comme dans la th´eoriede Teichm¨uller,mais en g´en´erall'application de p´eriodes n'est que de nature analytique r´eelle. Abstract. We first investigate a construction of principally polarized abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000. The index theorem for the Dirac operator associated to the spin structure implies integrality of a natural skew pairing on the topological K-group. The latter serves as a principal polarization. -
The Chiral De Rham Complex of Tori and Orbifolds
The Chiral de Rham Complex of Tori and Orbifolds Dissertation zur Erlangung des Doktorgrades der Fakult¨atf¨urMathematik und Physik der Albert-Ludwigs-Universit¨at Freiburg im Breisgau vorgelegt von Felix Fritz Grimm Juni 2016 Betreuerin: Prof. Dr. Katrin Wendland ii Dekan: Prof. Dr. Gregor Herten Erstgutachterin: Prof. Dr. Katrin Wendland Zweitgutachter: Prof. Dr. Werner Nahm Datum der mundlichen¨ Prufung¨ : 19. Oktober 2016 Contents Introduction 1 1 Conformal Field Theory 4 1.1 Definition . .4 1.2 Toroidal CFT . .8 1.2.1 The free boson compatified on the circle . .8 1.2.2 Toroidal CFT in arbitrary dimension . 12 1.3 Vertex operator algebra . 13 1.3.1 Complex multiplication . 15 2 Superconformal field theory 17 2.1 Definition . 17 2.2 Ising model . 21 2.3 Dirac fermion and bosonization . 23 2.4 Toroidal SCFT . 25 2.5 Elliptic genus . 26 3 Orbifold construction 29 3.1 CFT orbifold construction . 29 3.1.1 Z2-orbifold of toroidal CFT . 32 3.2 SCFT orbifold . 34 3.2.1 Z2-orbifold of toroidal SCFT . 36 3.3 Intersection point of Z2-orbifold and torus models . 38 3.3.1 c = 1...................................... 38 3.3.2 c = 3...................................... 40 4 Chiral de Rham complex 41 4.1 Local chiral de Rham complex on CD ...................... 41 4.2 Chiral de Rham complex sheaf . 44 4.3 Cechˇ cohomology vertex algebra . 49 4.4 Identification with SCFT . 49 4.5 Toric geometry . 50 5 Chiral de Rham complex of tori and orbifold 53 5.1 Dolbeault type resolution . 53 5.2 Torus . -
SMOOTH PROJECTIVE SURFACES 1. Surfaces Definition 1.1. Let K Be
SMOOTH PROJECTIVE SURFACES SCOTT NOLLET Abstract. This talk will discuss some basics about smooth projective surfaces, leading to an open question. 1. Surfaces Definition 1.1. Let k be an algebraically closed field. A surface is a two dimensional n nonsingular closed subvariety S ⊂ Pk . Example 1.2. A few examples. (a) The simplest example is S = P2. (b) The nonsingular quadric surfaces Q ⊂ P3 with equation xw − yz = 0 has Picard group Z ⊕ Z generated by two opposite rulings. (c) The zero set of a general homogeneous polynomial f 2 k[x; y; z; w] of degree d gives rise to a smooth surface of degree d in P3. (d) If C and D are any two nonsingular complete curves, then each is projective (see previous lecture's notes), and composing with the Segre embedding we obtain a closed embedding S = C × D,! Pn. 1.1. Intersection theory. Given a surface S, let DivS be the group of Weil divisors. There is a unique pairing DivS × DivS ! Z denoted (C; D) 7! C · D such that (a) The pairing is bilinear. (b) The pairing is symmetric. (c) If C; D ⊂ S are smooth curves meeting transversely, then C · D = #(C \ D). (d) If C1 ∼ C2 (they are linearly equivalent), then C1 · D = C2 · D. Remark 1.3. When k = C one can describe the intersection pairing with topology. The divisor C gives rise to the line bundle L = OS(C) which has a first Chern class 2 c1(L) 2 H (S; Z) and similarly D gives rise to a class D 2 H2(S; Z) by triangulating the 0 ∼ components of D as real surfaces. -
Abel-Jacobi Maps Associated to Smooth Cubic Threefolds
ABEL-JACOBI MAPS ASSOCIATED TO SMOOTH CUBIC THREEFOLDS JOE HARRIS, MIKE ROTH, AND JASON STARR Contents 1. Introduction 1 2. Review of the Abel-Jacobi Map 3 3. Lines, Conics and Plane Cubics 4 4. Twisted cubics 8 5. Cubic scrolls and applications 9 6. Quartic Rational Curves 12 7. Quintic Elliptics 13 8. Double Residuation and Unirationality of the Fibers 14 9. Quintic Rational Curves 18 10. Quintic Curves of Genus 2 18 References 20 Abstract. In this article we consider the spaces d,g(X) parametrizing curves of degree d and genus g on a H smooth cubic threefold X 4, with regard in particular to the Abel-Jacobi map u : d,g(X) J3(X) to ⊂ P d H → the intermediate Jacobian J3(X) of X. Our principle result is that for all d 5 the map u coincides with ≤ d the maximal rationally connected fibration of d,g(X). H 1. Introduction In this paper we will study spaces parametrizing curves on a smooth cubic threefold X P4. There has been a great deal of work in recent years on the geometry of spaces parametrizing rational cur⊂ves on a variety X. Most of it has focussed on the enumerative geometry of these spaces: the description of their Chow rings and the evaluation of certain products in their Chow rings. Here we will be concerned with a very different sort of question: we will be concerned with the birational geometry of the spaces M. We should start by explaining, at least in part, our motivation. The central object in curve theory—the one that links together every aspect of the theory, and whose study yields the majority of theorems in the subject—is the Abel-Jacobi map. -
Mirror Symmetry of Abelian Variety and Multi Theta Functions
1 Mirror symmetry of Abelian variety and Multi Theta functions by Kenji FUKAYA (深谷賢治) Department of Mathematics, Faculty of Science, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto Japan Table of contents § 0 Introduction. § 1 Moduli spaces of Lagrangian submanifolds and construction of a mirror torus. § 2 Construction of a sheaf from an affine Lagrangian submanifold. § 3 Sheaf cohomology and Floer cohomology 1 (Construction of a homomorphism). § 4 Isogeny. § 5 Sheaf cohomology and Floer cohomology 2 (Proof of isomorphism). § 6 Extension and Floer cohomology 1 (0 th cohomology). § 7 Moduli space of holomorphic vector bundles on a mirror torus. § 8 Nontransversal or disconnected Lagrangian submanifolds. ∞ § 9 Multi Theta series 1 (Definition and A formulae.) § 10 Multi Theta series 2 (Calculation of the coefficients.) § 11 Extension and Floer cohomology 2 (Higher cohomology). § 12 Resolution and Lagrangian surgery. 2 § 0 Introduction In this paper, we study mirror symmetry of complex and symplectic tori as an example of homological mirror symmetry conjecture of Kontsevich [24], [25] between symplectic and complex manifolds. We discussed mirror symmetry of tori in [12] emphasizing its “noncom- mutative” generalization. In this paper, we concentrate on the case of a commutative (usual) torus. Our result is a generalization of one by Polishchuk and Zaslow [42], [41], who studied the case of elliptic curve. The main results of this paper establish a dictionary of mirror symmetry between symplectic geometry and complex geometry in the case of tori of arbitrary dimension. We wrote this dictionary in the introduction of [12]. We present the argument in a way so that it suggests a possibility of its generalization. -
Weighted Fano Varieties and Infinitesimal Torelli Problem 3
WEIGHTED FANO VARIETIES AND INFINITESIMAL TORELLI PROBLEM ENRICO FATIGHENTI, LUCA RIZZI AND FRANCESCO ZUCCONI Abstract. We solve the infinitesimal Torelli problem for 3-dimensional quasi-smooth Q-Fano hypersurfaces with at worst terminal singularities. We also find infinite chains of double coverings of increasing dimension which alternatively distribute themselves in examples and counterexamples for the infinitesimal Torelli claim and which share the analogue, and in some cases the same, Hodge-diagram properties as the length 3 Gushel- Mukai chain of prime smooth Fanos of coindex 3 and degree 10. 1. Introduction Minimal model program suggests us to formulate inside the category of varieties with terminal singularities many questions which were initially asked for smooth varieties. The construction of the period map, and the related Torelli type problems are definitely among these problems. 1.1. Infinitesimal Torelli. A Q-Fano variety is a projective variety X such that X has at worst Q-factorial terminal singularities, −KX is ample and Pic(X) has rank 1; cf. [CPR00]. In this paper we give a full answer to the infinitesimal Torelli problem in the case of quasi- smooth Q-Fano hypersurfaces of dimension 3 with terminal singularities and with Picard number 1. In subsections 2.1 of this paper the reader can find a basic dictionary and up to date references needed to understand the statement of the infinitesimal Torelli problem and where to find the meaning of infinitesimal variation of Hodge structures (IVHS; to short). Here we stress only that the method à la Griffiths based on the extension of the Macaulay’s theorem in algebra to the weighted case, cf.