Relevant bibliographies by topics / Hodge classe

Academic literature on the topic 'Hodge classe'

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Published: 9 March 2023

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Journal articles on the topic "Hodge classe"

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Fargues, Laurent. "G-torseurs en théorie de Hodge p-adique." Compositio Mathematica 156, no. 10 (October 2020): 2076–110. http://dx.doi.org/10.1112/s0010437x20007423.

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RésuméÉtant donné un groupe réductif $G$ sur une extension de degré fini de $\mathbb {Q}_p$ on classifie les $G$-fibrés sur la courbe introduite dans Fargues and Fontaine [Courbes et fibrés vectoriels en théorie de Hodge$p$-adique, Astérisque 406 (2018)]. Le résultat est interprété en termes de l'ensemble $B(G)$ de Kottwitz. On calcule également la cohomologie étale de la courbe à coefficients de torsion en lien avec la théorie du corps de classe local.
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Santiago, Gabrielli Stefaninni, Laura Ribeiro, Irene Da Silva Coelho, Miliane Moreira Soares de Souza, and Shana De Mattos de Oliveira Coelho. "TESTE DE HODGE MODIFICADO EM ÁGAR CLED PARA TRIAGEM DE Proteus mirabilis PRODUTORES DE CARBAPENEMASE." Revista Univap 24, no. 46 (December 17, 2018): 1. http://dx.doi.org/10.18066/revistaunivap.v24i46.397.

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Enterobacteriaceae produtoras de carbapenemase vêm sendo descritas em todo o mundo. Uma detecção precisa de bactérias produtoras de carpabenemase é necessária pois esta classe de antibióticos é usada no tratamento de infecções severas. A nível laboratorial, o método fenotípico para a detecção de produtores de carbapenemase é o teste de Hodge modificado. Entretanto, algumas enterobactérias tem grande motilidade dificultando a leitura e interpretação dos resultados desta técnica. O objetivo deste estudo foi validar um meio para se obter resultados confiáveis em bactérias com grande motilidade, como é o caso de Proteus mirabilis. O meio ágar Müller-Hinton, preconizado pelo CLSI, foi comparado ao ágar CLED que demostrou ser um bom meio para análise da produção de carbapenemase em Proteus mirabilis suspeitos de produzirem esta enzima embora todos os isolados tenham sido negativos no teste.
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Voisin, Claire. "Hodge loci and absolute Hodge classes." Compositio Mathematica 143, no. 04 (July 2007): 945–58. http://dx.doi.org/10.1112/s0010437x07002837.

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Mongardi, Giovanni, and John Christian Ottem. "Curve classes on irreducible holomorphic symplectic varieties." Communications in Contemporary Mathematics 22, no. 07 (November 15, 2019): 1950078. http://dx.doi.org/10.1142/s0219199719500780.

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We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of [Formula: see text]-type and of generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.
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van Geemen, B., and A. Verra. "Quaternionic pryms and Hodge classes." Topology 42, no. 1 (January 2003): 35–53. http://dx.doi.org/10.1016/s0040-9383(02)00004-6.

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Koike, Kenji. "Algebraicity of some Weil Hodge Classes." Canadian Mathematical Bulletin 47, no. 4 (December 1, 2004): 566–72. http://dx.doi.org/10.4153/cmb-2004-055-x.

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AbstractWe show that the Prym map for 4-th cyclic étale covers of curves of genus 4 is a dominant morphism to a Shimura variety for a family of Abelian 6-folds of Weil type. According to the result of Schoen, this implies algebraicity of Weil classes for this family.
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Cattani, Eduardo, Pierre Deligne, and Aroldo Kaplan. "On the locus of Hodge classes." Journal of the American Mathematical Society 8, no. 2 (May 1, 1995): 483. http://dx.doi.org/10.1090/s0894-0347-1995-1273413-2.

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Méo, Michel. "Chow forms and Hodge cohomology classes." Comptes Rendus Mathematique 352, no. 4 (April 2014): 339–43. http://dx.doi.org/10.1016/j.crma.2014.01.012.

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J. Moonen, B. J., and Yu G. Zarhin. "Hodge classes and Tate classes on simple abelian fourfolds." Duke Mathematical Journal 77, no. 3 (March 1995): 553–81. http://dx.doi.org/10.1215/s0012-7094-95-07717-5.

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Scavia, Federico. "Motivic classes and the integral Hodge Question." Comptes Rendus. Mathématique 359, no. 3 (April 20, 2021): 305–11. http://dx.doi.org/10.5802/crmath.178.

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Dissertations / Theses on the topic "Hodge classe"

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Blottière, David. "Réalisation de Hodge du polylogarithme d'un schéma abélien et dégénérescence des classes d'Eisenstein des familles modulaires de Hilbert-Blumenthal." Phd thesis, Université Paris-Nord - Paris XIII, 2006. http://tel.archives-ouvertes.fr/tel-00132405.

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La réalisation de Hodge du polylogarithme d'un schéma abélien complexe de dimension g est une (2g-1)-extension de modules de Hodge. Lorsque le schéma abélien est principalement polarisé, on en donne une description au niveau topologique. Pour cela, on utilise des courants de type "courants de Green" introduits par Levin. On applique alors ce résultat aux familles modulaires de Hilbert-Blumenthal pour montrer que certaines classes d'Eisenstein (construites à partir du polylogarithme et d'une section de torsion) dégénèrent, en l'infini, en une valeur spéciale de fonction L du corps de nombres totalement réel sous-jacent. On en déduit deux autres résultats : une version partielle du théorème de Klingen-Siegel et un résultat de non nullité pour certaines de ces classes d'Eisenstein. Ainsi, on montre que pour tout entier g plus grand que 2, il existe un schéma abélien complexe de dimension g tel que certaines de ses classes d'Eisenstein soient non nulles.
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NOVARIO, SIMONE. "LINEAR SYSTEMS ON IRREDUCIBLE HOLOMORPHIC SYMPLECTIC MANIFOLDS." Doctoral thesis, Università degli Studi di Milano, 2021. http://hdl.handle.net/2434/886303.

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In questa tesi studiamo alcuni sistemi lineari completi associati a divisori di schemi di Hilbert di 2 punti su una superficie K3 proiettiva complessa con gruppo di Picard di rango 1, e le mappe razionali indotte. Queste varietà sono chiamate quadrati di Hilbert su superfici K3 generiche, e sono esempi di varietà irriducibili olomorfe simplettiche (varietà IHS). Nella prima parte della tesi, usando la teoria dei reticoli, gli operatori di Nakajima e il modello di Lehn–Sorger, diamo una base per il sottospazio vettoriale dell’anello di coomologia singolare a coefficienti razionali generato dalle classi di Hodge razionali di tipo (2, 2) sul quadrato di Hilbert di una qualsiasi superficie K3 proiettiva. In seguito sfruttiamo un teorema di Qin e Wang insieme a un risultato di Ellingsrud, Göttsche e Lehn per ottenere una base del reticolo delle classi di Hodge integrali di tipo (2, 2) sul quadrato di Hilbert di una qualsiasi superficie K3 proiettiva. Nella seconda parte della tesi studiamo il problema seguente: se X è il quadrato di Hilbert di una superficie K3 generica che ammette un divisore ampio D con q(D) = 2, dove q è la forma quadratica di Beauville-Bogomolov-Fujiki, descrivere geometricamente la mappa razionale indotta dal sistema lineare completo |D|. Il risultato principale della tesi mostra che tale X, tranne nel caso del quadrato di Hilbert di una superficie quartica generica di P^3, è una doppia EPW sestica, cioè il ricoprimento doppio di una EPW sestica, una ipersuperficie normale di P^5, ramificato nel suo luogo singolare. Inoltre la mappa razionale indotta da |D| coincide proprio con tale ricoprimento doppio. Gli strumenti principali per ottenere questo risultato sono la descrizione del reticolo delle classi integrali di Hodge di tipo (2, 2) della prima parte della tesi e l’esistenza di un’involuzione anti-simplettica su tali varietà per un teorema di Boissière, Cattaneo, Nieper-Wißkirchen e Sarti.
In this thesis we study some complete linear systems associated to divisors of Hilbert schemes of 2 points on complex projective K3 surfaces with Picard group of rank 1, together with the rational maps induced. We call these varieties Hilbert squares of generic K3 surfaces, and they are examples of irreducible holomorphic symplectic (IHS) manifold. In the first part of the thesis, using lattice theory, Nakajima operators and the model of Lehn–Sorger, we give a basis for the subvector space of the singular cohomology ring with rational coefficients generated by rational Hodge classes of type (2, 2) on the Hilbert square of any projective K3 surface. We then exploit a theorem by Qin and Wang together with a result by Ellingsrud, Göttsche and Lehn to obtain a basis of the lattice of integral Hodge classes of type (2, 2) on the Hilbert square of any projective K3 surface. In the second part of the thesis we study the following problem: if X is the Hilbert square of a generic K3 surface admitting an ample divisor D with q(D)=2, where q is the Beauville–Bogomolov–Fujiki form, describe geometrically the rational map induced by the complete linear system |D|. The main result of the thesis shows that such an X, except on the case of the Hilbert square of a generic quartic surface of P^3, is a double EPW sextic, i.e., the double cover of an EPW sextic, a normal hypersurface of P^5, ramified over its singular locus. Moreover, the rational map induced by |D| is a morphism and coincides exactly with this double covering. The main tools to obtain this result are the description of integral Hodge classes of type (2, 2) of the first part of the thesis and the existence of an anti-symplectic involution on such varieties due to a theorem by Boissière, Cattaneo, Nieper-Wißkirchen and Sarti.
Dans cette thèse, nous étudions certains systèmes linéaires complets associés aux diviseurs des schémas de Hilbert de 2 points sur des surfaces K3 projectives complexes avec groupe de Picard de rang 1, et les fonctions rationnelles induites. Ces variétés sont appelées carrés de Hilbert sur des surfaces K3 génériques, et sont un exemple de variété symplectique holomorphe irréductible (variété IHS). Dans la première partie de la thèse, en utilisant la théorie des réseaux, les opérateurs de Nakajima et le modèle de Lehn–Sorger, nous donnons une base pour le sous-espace vectoriel de l’anneau de cohomologie singulière à coefficients rationnels engendré par les classes de Hodge rationnels de type (2, 2) sur le carré de Hilbert de toute surface K3 projective. Nous exploitons ensuite un théorème de Qin et Wang ainsi qu’un résultat de Ellingsrud, Göttsche et Lehn pour obtenir une base du réseau des classes de Hodge intégraux de type (2, 2) sur le carré de Hilbert d’une surface K3 projective quelconque. Dans la deuxième partie de la thèse, nous étudions le problème suivant : si X est le carré de Hilbert d’une surface K3 générique tel que X admet un diviseur ample D avec q(D) = 2, où q est la forme quadratique de Beauville–Bogomolov–Fujiki, on veut décrire géométriquement la fonction rationnelle induite par le système linéaire complet |D|. Le résultat principal de la thèse montre qu’une telle X, sauf dans le cas du carré de Hilbert d’une surface quartique générique de P^3, est une double sextique EPW, c’est-à-dire le revêtement double d’une sextique EPW, une hypersurface normale de P^5, ramifié sur son lieu singulier. En plus la fonction rationnelle induite par |D| est exactement ce revêtement double. Les outils principaux pour obtenir ce résultat sont la description des classes de Hodge intégraux de type (2, 2) de la première partie de la thèse et l’existence d’une involution anti-symplectique sur de telles variétés par un théorème de Boissière, Cattaneo, Nieper-Wißkirchen et Sarti.
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Gheorghita, Iulia. "Effective classes in the projectivized k-th Hodge bundle:." Thesis, Boston College, 2021. http://hdl.handle.net/2345/bc-ir:109066.

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Thesis advisor: Dawei Chen
We study the classes of several loci in the projectivization of the k-th Hodge bundle over the moduli space of genus g curves and over the moduli space of genus g curves with n marked points. In particular we consider the class of the closure in the projectivization of the k-th Hodge bundle over the moduli space of genus g curves with n marked points of the codimension n locus where the n marked points are zeros of the k-differential. We compute this class when n=2 and provide a recursive formula for it when n>2. Moreover, when n=1 and k=1,2 we show its rigidity and extremality in the pseudoeffective cone. We also compute the classes of the closures in the projectivization of the k-th Hodge bundle over the moduli space of genus g curves of the loci where the k-differential has a zero at a Brill-Noether special point
Thesis (PhD) — Boston College, 2021
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Mathematics
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Schnell, Christian. "The boundary behavior of cohomology classes and singularities of normal functions." Columbus, Ohio : Ohio State University, 2008. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1218036000.

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Moya, Giusti Matias. "Sur l'existence des classes fantômes dans la cohomologie de certaines variétés de Shimura." Paris 7, 2014. http://www.theses.fr/2014PA077063.

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Dans ce travail, nous étudions l'existence des classes fantômes dans la cohomologie des variétés de Shimura associées aux groupes algébriques GSp_4 et GU(2, n-2), pour n > 3. Nous utilisons des considerations sur le poids des structures de Hodge mixtes associées aux espaces de cohomologie impliquées dans la définition de l'espace des classes fantômes
In this work we study the existence of ghost classes in the cohomology of the Shimura varieties attached to the algebraic groups GSp_4 and GU(2, n-2) for n > 3. We use considerations on the weights of the mixed Hodge structures attached to the cohomology spaces involved in the definition of the space of ghost classes
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Venturelli, Federico. "The Alexander polynomial of certain classes of non-symmetric line arrangements." Doctoral thesis, Università degli studi di Padova, 2019. http://hdl.handle.net/11577/3422691.

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The Alexander polynomial of a projective hypersurface V ϲ Pᶰ is the characteristic polynomial of the monodromy operator acting on Hᶰ¯¹(F, C), where F is the Milnor fibre of V; unless V is smooth, the problem of its computation is open. The singular hypersurfaces that have drawn the most attention are projectivisations Ᾱ of central hyperplane arrangements A C Cᶰ⁺ ¹, as one can hope to take advantage of the combinatorial nature of such objects; one can assume without loss of generality that n=2. In this Thesis we prove that the Alexander polynomials of line arrangements Ᾱ C P² belonging to some particular non-symmetric classes are trivial: this constitutes evidence in favour of the validity of a conjecture due to Papadima and Suciu. The Thesis is organised as follows. In Chapter 1 we gather some known results on which we will build upon: the discussion of mixed Hodge structures on cohomology groups of algebraic varieties and the comparison between the polar and Hodge filtration are of particular importance; the construction of cubical hyperresolutions and their use in the definition of algebraic de Rham cohomology for singular algebraic varieties will be very useful too. Chapter 2 is divided in two parts. The first one is mainly devoted to defining the Alexander polynomial and presenting a formula by Libgober for its computation in case V is a curve. The second part is a survey of known results around the problem of determining the Alexander polynomial of a line arrangement, and closes with a discussion of some interesting examples; we try to highlight how the symmetry of the arrangement affects its Alexander polynomial. In Chapter 3 we introduce some classes of non-symmetric line arrangements Ᾱ and prove that their Alexander polynomials are trivial. The methods we use are essentially two: one is the combination of Libgober's formula with an easy deformation theory argument, thanks to which we can restrict ourselves to considering a finite number of “representative arrangements”; the other relies on associating to Ᾱ a threefold T fibred in surfaces over P¹ and on studying the monodromy around a special fibre of the latter. A key step of the second method is the proof of the existence of a Gysin morphism that connects the cohomology of T to that of a hyperplane section S: this result is of independent interest, as T and S do not satisfy the hypotheses usually required in order to obtain Lefschetz-type results.
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Sacchetto, Lucas Kaufmann. "Fundamentos da geometria complexa: aspectos geométricos, topológicos e analiticos." Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-18062012-194224/.

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Este trabalho tem como objetivo apresentar um estudo detalhado dos fundamentos da Geometria Complexa, ressaltando seus aspectos geométricos, topológicos e analíticos. Começando com materiais preliminares, como resultados básicos sobre funções holomorfas de uma ou mais variáveis e a definição e primeiros exemplos de variedades complexas, passamos a uma introdução à teoria de feixes e sua cohomologia, ferramenta indispensável para o restante do trabalho. Após um estudo sobre fibrados de linha e divisores damos atenção à Geometria de Kähler e alguns de seus resultados centrais, como por exemplo o Teorema da Decomposição de Hodge, o Teorema ``Difícil\'\' e o Teorema das $(1,1)$-classes de Lefschetz. Em seguida, nos dedicamos ao estudo dos fibrados vetoriais complexos e sua geometria, abordando os conceitos de conexões, curvatura e Classes de Chern. Terminamos o trabalho descrevendo alguns aspectos da topologia de variedades complexas, como o Teorema dos Hiperplanos de Lefschetz e algumas de suas consequências.
The main goal of this work is to present a detailed study of the foundations of Complex Geometry, highlighting its geometric, topological and analytical aspects. Beginning with a preliminary material, such as the basic results on holomorphic functions in one or more variables and the definition and first examples of a complex manifold, we move on to an introduction to sheaf theory and its cohomology, an essential tool to the rest of the work. After a discussion on divisors and line bundles we turn attention to Kähler Geometry and its central results, such as the Hodge Decomposition Theorem, the Hard Lefschetz Theorem and the Lefschetz Theorem on $(1,1)$-classes. After that, we study complex vector bundles and its geometry, focusing on the concepts of connections, curvature and Chern classes. Finally, we finish by describing some aspects of the topology of complex manifolds, such as the Lefschetz Hyperplane Theorem and some of its consequences.
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Schlickewei, Ulrich [Verfasser]. "Hodge classes on self-products of K3 surfaces / vorgelegt von Ulrich Schlickewei." 2009. http://d-nb.info/1000464202/34.

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Moya, Giusti Matias Victor. "Sobre la existencia de clases fantasma en la cohomología de ciertas variedades de Shimura." Doctoral thesis, 2014. http://hdl.handle.net/11086/2878.

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Tesis (Doctor en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía y Física, 2014.
En este trabajo, estudiamos la existencia de clases fantasma en la cohomología de ciertas variedades de Shimura asociadas a grupos algebráicos de rango racional 2. Utilizamos ciertos argumentos sobre los pesos de las estructuras de Hodge mixtas asociadas a los espacios de cohomología involucrados en la definición del espacio de clases fantasma.
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Books on the topic "Hodge classe"

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J, Frazer W., ed. New Testament criticism: Lectures by Dr. C.W. Hodge before the junior class, Princeton Theological Seminary. Princeton: [s.n.], 1985.

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Charles, Franc¸ois, and Christian Schnell. Notes on Absolute Hodge Classes. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0011.

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This chapter surveys the theory of absolute Hodge classes. First, the chapter recalls the construction of cycle maps in de Rham cohomology, which is then used in the definition of absolute Hodge classes. The chapter then deals with variational properties of absolute Hodge classes. After stating the variational Hodge conjecture, the chapter proves Deligne's principle B and discusses consequences of the algebraicity of Hodge bundles and of the Galois action on relative de Rham cohomology. Finally, the chapter provides some important examples of absolute Hodge classes: a discussion of the Kuga–Satake correspondence as well as a full proof of Deligne's theorem which states that Hodge classes on abelian varieties are absolute.
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Voisin, Claire. Review of Hodge theory and algebraic cycles. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691160504.003.0002.

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This chapter provides the background for the studies to be undertaken in succeeding chapters. It reviews Chow groups, correspondences and motives on the purely algebraic side, cycle classes, and (mixed) Hodge structures on the algebraic–topological side. Emphasis is placed on the notion of coniveau and the generalized Hodge conjecture which states the equality of geometric and Hodge coniveau. The chapter first follows the construction of Chow groups, the application of the localization exact sequence, the functoriality and motives of Chow groups, and cycle classes. It then turns to Hodge structures; pursuing related topics such as polarization, Hodge classes, standard conjectures, mixed Hodge structures, and Hodge coniveau.
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Cattani, Eduardo, Fouad El Zein, Phillip A. Griffiths, and Lê Dung Tráng. Hodge Theory (MN-49). Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.001.0001.

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This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch–Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and does not require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch–Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's theorem on absolute Hodge cycles), and variation of mixed Hodge structures.
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Huybrechts, D. K3 Surfaces. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.003.0010.

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After abelian varieties, K3 surfaces are the second most interesting special class of varieties. These have a rich internal geometry and a highly interesting moduli theory. Paralleling the famous Torelli theorem, results from Mukai and Orlov show that two K3 surfaces have equivalent derived categories precisely when their cohomologies are isomorphic weighing two Hodge structures. Their techniques also give an almost complete description of the cohomological action of the group of autoequivalences of the derived category of a K3 surface. The basic definitions and fundamental facts from K3 surface theory are recalled. As moduli spaces of stable sheaves on K3 surfaces are crucial for the argument, a brief outline of their theory is presented.
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Katz, Sanford N. Family Law in America. 3rd ed. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780197554319.001.0001.

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This book examines the present state of family law in America. The third edition captures recent developments, including the transformation of the institution of marriage from being a relationship between a man and a woman to encompassing same-sex marriage. In this regard, the book includes a full discussion and analysis of Obergefell v. Hodges. Obergefell v. Hodges is the U.S. Supreme Court case that held in a 5-4 decision that the bans on same-sex marriage in Michigan, Kentucky, Ohio, and Tennessee were unconstitutional. The Court held that the right to marry a person of the same sex is protected by the Due Process and Equal Protection Clauses of the Fourteenth Amendment, and therefore may not be denied in any state.
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Series, Michigan Historical Reprint. A commentary on the Confession of faith. With questions for theological students and Bible classes. By the Rev. Archibald Alexander Hodge ... Scholarly Publishing Office, University of Michigan Library, 2005.

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Book chapters on the topic "Hodge classe"

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Charles, François, and Christian Schnell. "Chapter Eleven. Notes on Absolute Hodge Classes." In Hodge Theory, 469–530. Princeton: Princeton University Press, 2014. http://dx.doi.org/10.1515/9781400851478.469.

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Murty, V. Kumar. "Hodge and Well Classes on Abelian Varieties." In The Arithmetic and Geometry of Algebraic Cycles, 83–115. Dordrecht: Springer Netherlands, 2000. http://dx.doi.org/10.1007/978-94-011-4098-0_4.

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Voisin, Claire. "Integral Hodge Classes, Decompositions of the Diagonal, and Rationality Questions." In Trends in Contemporary Mathematics, 137–49. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-05254-0_11.

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Maillot, Vincent, and Damian Roessler. "On the Order of Certain Characteristic Classes of the Hodge Bundle of Semi-Abelian Schemes." In Number Fields and Function Fields—Two Parallel Worlds, 287–310. Boston, MA: Birkhäuser Boston, 2005. http://dx.doi.org/10.1007/0-8176-4447-4_14.

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"Hodge Classes." In Hodge Theory and Complex Algebraic Geometry I, 263–89. Cambridge University Press, 2002. http://dx.doi.org/10.1017/cbo9780511615344.012.

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Green, Mark, Phillip Griffiths, and Matt Kerr. "Arithmetic of Period Maps of Geometric Origin." In Mumford-Tate Groups and Domains. Princeton University Press, 2012. http://dx.doi.org/10.23943/princeton/9780691154244.003.0009.

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This chapter considers some arithmetic aspects of period maps with a geometric origin. It focuses on the situation Φ‎ : S(ℂ) → Γ‎\D, where S parametrizes a family X → S of smooth, projective varieties defined over a number field k. The chapter recalls the notion of absolute Hodge classes (AH) and strongly absolute Hodge classes (SAH). The particular case when the Noether-Lefschetz locus consists of isolated points is alluded to in the discussion of complex multiplication Hodge structures (CM Hodge structures). A related observation is that one may formulate a variant of the “Grothendieck conjecture” in the setting of period maps and period domains. The chapter also describes a behavior of fields of definition under the period map, along with the existence and density of CM points in a motivic variation of Hodge structure.
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7

"Chern class theory." In A Survey of the Hodge Conjecture, 103–18. Providence, Rhode Island: American Mathematical Society, 2016. http://dx.doi.org/10.1090/crmm/010/08.

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8

Green, Mark, Phillip Griffiths, and Matt Kerr. "Classification of Mumford-Tate Subdomains." In Mumford-Tate Groups and Domains. Princeton University Press, 2012. http://dx.doi.org/10.23943/princeton/9780691154244.003.0008.

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This chapter develops an algorithm for determining all Mumford-Tate subdomains of a given period domain. The result is applied to the classification of all complex multiplication Hodge structures (CM Hodge structures) of rank 4 and when the weight n = 1 and n = 3, to an analysis of their Hodge tensors and endomorphism algebras, and the number of components of the Noether-Lefschetz locus. The result is that one has a complex but very rich arithmetic story. Of particular note is the intricate structure of the components of the Noether-Lefschetz loci in D and in its compact dual, and the two interesting cases where the Hodge tensors are generated in degrees 2 and 4. One application is that a particular class of period maps appearing in mirror symmetry never has image in a proper subdomain of D.
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Green, Mark, Phillip Griffiths, and Matt Kerr. "The Mumford-Tate Group of a Variation of Hodge Structure." In Mumford-Tate Groups and Domains. Princeton University Press, 2012. http://dx.doi.org/10.23943/princeton/9780691154244.003.0004.

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This chapter deals with the Mumford-Tate group of a variation of Hodge structure (VHS). It begins by presenting a definition of VHS, which consists of a connected complex manifold and a locally liftable, holomorphic mapping that is an integral manifold of the canonical differential ideal. The moduli space of Γ‎-equivalence classes of polarized Hodge structures is also considered, along with a generic point for the VHS and the monodromy group of the VHS. Associated to a VHS is its Mumford-Tate group. The chapter proceeds by discussing the structure theorem for VHS, where S is a quasi-projective algebraic variety, referred to as global variations of Hodge structure. It concludes by describing an application of Mumford-Tate groups, along with the Noether-Lefschetz locus.
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"Murray, J., concurring." In What Obergefell v. Hodges Should Have Said, edited by Jack M. Balkin, 202–20. Yale University Press, 2020. http://dx.doi.org/10.12987/yale/9780300221558.003.0010.

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Today, a majority of the Court strikes down laws banning the performance and recognition of same-sex marriages on the ground that such laws constitute caste or class legislation in violation of the Equal Protection Clause of the Fourteenth Amendment. In so doing, the Court reiterates that the right to marry is a fundamental right and denominates sexual orientation a quasi-suspect classification subject to heightened scrutiny....
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