Relevant bibliographies by topics / Hodge bundle

Academic literature on the topic 'Hodge bundle'

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Published: 9 March 2023
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Journal articles on the topic "Hodge bundle"

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FUJITA, HAJIME. "ON THE FUNCTORIALITY OF THE CHERN–SIMONS LINE BUNDLE AND THE DETERMINANT LINE BUNDLE." Communications in Contemporary Mathematics 08, no. 06 (December 2006): 715–35. http://dx.doi.org/10.1142/s0219199706002271.

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We investigate functorial properties of two hermitian line bundles over the moduli space of flat SU(n)-connections on a closed oriented surface; that is, of the Chern–Simons line bundle and the determinant line bundle. We investigate actions of cyclic subgroups of the mapping class group on them. As a consequence, we show that if we modify the determinant line bundle by the Hodge bundle over the moduli space of Riemann surfaces, then these line bundles are functorially isomorphic. This implies two quantum Hilbert spaces defined by the Chern–Simons line bundle and the modified determinant line bundle are functorially isomorphic.
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Fujino, Osamu. "A canonical bundle formula for certain algebraic fiber spaces and its applications." Nagoya Mathematical Journal 172 (2003): 129–71. http://dx.doi.org/10.1017/s0027763000008679.

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AbstractWe investigate period maps of polarized variations of Hodge structures of weight one or two. We treat the case when the period domains are bounded symmetric domains. We deal with a relationship between canonical extensions of some Hodge bundles and automorphic forms. As applications, we obtain a canonical bundle formula for certain algebraic fiber spaces, such as Abelian fibrations, K3 fibrations, and solve Iitaka’s famous conjecture Cn,m for some algebraic fiber spaces.
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Buchdahl, Nicholas, and Georg Schumacher. "Polystable bundles and representations of their automorphisms." Complex Manifolds 9, no. 1 (January 1, 2022): 78–113. http://dx.doi.org/10.1515/coma-2021-0131.

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Abstract Using a quasi-linear version of Hodge theory, holomorphic vector bundles in a neighbourhood of a given polystable bundle on a compact Kähler manifold are shown to be (poly)stable if and only if their corresponding classes are (poly)stable in the sense of geometric invariant theory with respect to the linear action of the automorphism group of the bundle on its space of in˝nitesimal deformations.
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Avila, Artur, Alex Eskin, and Martin Möller. "Symplectic and isometric SL(2,#x211D;)-invariant subbundles of the Hodge bundle." Journal für die reine und angewandte Mathematik (Crelles Journal) 2017, no. 732 (November 1, 2017): 1–20. http://dx.doi.org/10.1515/crelle-2014-0142.

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Abstract Suppose N is an affine {\mathrm{SL}(2,{\mathbb{R}})} -invariant submanifold of the moduli space of pairs (M,\omega) , where M is a curve, and ω is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximal {\mathrm{SL}(2,{\mathbb{R}})} -invariant isometric subbundle of the Hodge bundle of N) is always flat and is always orthogonal to the tangent space of N. As a corollary, it follows that the Hodge bundle of N is semisimple.
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Kouvidakis, Alexis. "Theta line bundles and the determinant of the Hodge bundle." Transactions of the American Mathematical Society 352, no. 6 (February 14, 2000): 2553–68. http://dx.doi.org/10.1090/s0002-9947-00-02619-2.

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VAFA, CUMRUN. "EXTENDING MIRROR CONJECTURE TO CALABI–YAU WITH BUNDLES." Communications in Contemporary Mathematics 01, no. 01 (February 1999): 65–70. http://dx.doi.org/10.1142/s0219199799000043.

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We define the notion of mirror of a Calabi–Yau manifold with a stable bundle in the context of type II strings in terms of supersymmetric cycles on the mirror. This allows us to relate the variation of Hodge structure for cohomologies arising from the bundle to the counting of holomorphic maps of Riemann surfaces with boundary on the mirror side. Moreover it opens up the possibility of studying bundles on Calabi–Yau manifolds in terms of supersymmetric cycles on the mirror.
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Smith, Ivan. "Lefschetz fibrations and the Hodge bundle." Geometry & Topology 3, no. 1 (July 14, 1999): 211–33. http://dx.doi.org/10.2140/gt.1999.3.211.

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Geer, Gerard van Der, and Alexis Kouvidakis. "The Hodge Bundle on Hurwitz Spaces." Pure and Applied Mathematics Quarterly 7, no. 4 (2011): 1297–308. http://dx.doi.org/10.4310/pamq.2011.v7.n4.a10.

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ZÚÑIGA ROJAS, RONALD A. "A BRIEF SURVEY OF HIGGS BUNDLES." Revista de Matemática: Teoría y Aplicaciones 26, no. 2 (July 12, 2019): 197–214. http://dx.doi.org/10.15517/rmta.v26i2.38315.

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Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.
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FORNI, GIOVANNI, CARLOS MATHEUS, and ANTON ZORICH. "Lyapunov spectrum of invariant subbundles of the Hodge bundle." Ergodic Theory and Dynamical Systems 34, no. 2 (April 2012): 353–408. http://dx.doi.org/10.1017/etds.2012.148.

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AbstractWe study the Lyapunov spectrum of the Kontsevich–Zorich cocycle on SL(2,ℝ)-invariant subbundles of the Hodge bundle over the support of SL(2,ℝ)-invariant probability measures on the moduli space of Abelian differentials. In particular, we prove formulas for partial sums of Lyapunov exponents in terms of the second fundamental form (the Kodaira–Spencer map) of the Hodge bundle with respect to the Gauss–Manin connection and investigate the relations between the central Oseledets subbundle and the kernel of the second fundamental form. We illustrate our conclusions in two special cases.
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Dissertations / Theses on the topic "Hodge bundle"

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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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RIVA, ENEA. "Slope inequalities for fibred surfaces and fibreed threefolds." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2022. http://hdl.handle.net/10281/374266.

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Su una varieta algebrica fibrata si definesce un invariante relativo detto slope che ne specifica la natura. Per queste fibrazioni un ruolo importante è svolto del fibrato di Hodge e dagli invarianti geometrici delle fibre generiche. In particulare in questa tesi ci concentreremo su superfici e threefold fibrati su curve, dando un stima dal basso della slope che dipenda del rango unitario del fibrato di hodge e da: - indice di clifford cella curva generale, nel caso di superfici; -dal genere geometrico ($p_{g}$) della superficie generale nel caso di threefold. infine sfrutteremo i risultati ottenuti sui threefold per definere un upper bound del rango unitario $u_{f}$ in funzione di $p_{g}$ sotto l'ipotesi che il genere della curva base sia zero o uno.
On a fibred algebraic variety, is defined a relative invariant called slope which classifies the variety itself. For these fibration a main character is played by the Hodge bundle and by the geometric invariants of the general fibers. In particular in this thesis we focus on surfaces and threefolds fibred over curves, and we give a lower bound for the slope which depends on the unitary rank of the hodge bundle and on: -the clifford index of the general curve, in case of fibred surfaces; - the geometric genus ($p_{g}$) of the general surface, in case of threefolds. Finally we use these results on fibred threefolds to make a new upper bound for the unitary rank $u_{f}$ depending on $p_{g}$ under the hypothesis that the genus of the base curve is zero or one.
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Huang, Pengfei. "Théorie de Hodge non-abélienne et des spécialisations." Thesis, Université Côte d'Azur, 2020. http://www.theses.fr/2020COAZ4029.

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La premiére partie de cette thèse est la géométrie de la théorie de Hodge non-Abélienne, en particulier l’étude des propriétés géométriques des espaces de modules.Le premier résultat principal de cette partie est la construction d’un système dynamique sur l’espace de modules des fibrés de Higgs, nous montrons que les points fixes de ce système dynamique sont exactement ceux fixés par l’action de C* sur l’espace de modules des fibrés de Higgs, c’est-àdire tous les C-VHS dans l’espace de modules. Dans le même temps, nous étudions sa première variation et son comportement asymptotique.Le deuxième résultat principal de cette partie est la preuve d’une conjecture (forme faible) par Simpson sur la stratification de l’espace de modules des fibrés plats, nous prouvons que la strata d’opérateurs est la strata fermée unique de dimension minimale en étudiant l’espace de modules des chaînes holomorphes de type donné.Le troisième résultat principal de cette partie est une généralisation de la construction par Deligne en l’espace de twistor de Hitchin dans le cas de surface de Riemann, nous construisons des sections holomorphes pour ce nouvel espace de twistor, c’est-à-dire les sections de de Rham. Nous calculons les fibrés normals de ces sections, et nous avons constaté que les sections de de Rham dans l’espace de twistor de Deligne–Hitchin ont également la propriété wight 1, donc ce sont des courbes rationnelles amples. Dans le même temps, nous montrons le théorème de type Torelli pour l’espace de twistor.La deuxième partie de cette thèse est l’étude de certaines spécialisations de la correspondance de Hodge non-Abélienne. Celui-ci comprend principalement deux chapitres, le premier est une preuve fondamentale d’une conjecture liée aux représentations de carquois proposée par Reineke en 2003, nous montrons pour les représentations de carquois de type An , il existe un système de poids tel que les représentations stables par rapport à ce système de poids sont précisément celles indécomposables. Pour la deuxième, nous construisons la correspondance de Kobayashi–Hitchin pour les fibrés de carquois sur les variétés Kähleriennes généralisées
The first part of this thesis is the geometry of non-Abelian Hodge theory, especially the study of geometric properties of moduli spaces.The first main result of this part is the construction of a dynamical system on the moduli space of Higgs bundles, we show that fixed points of this dynamical system are exactly those fixed by the C*-action on the moduli space of Higgs bundles, that is, all C-VHS in the moduli space. At the same time, we study its first variation and asymptotic behaviour.The second main result of this part is the proof of a conjecture (weak form) by Simpson on the stratification of the moduli space of flat bundles, we prove that the oper stratum is the unique closed stratum of minimal dimension by studying the moduli space of holomorphic chains of given type.The third main result of this part is a generalization of Deligne’s construction of Hitchin twistor space in Riemann surface case, we construct holomorphic sections for this new twistor space, namely the de Rham sections. We calculate the normal bundles of these sections, and we found that de Rham sections in the Deligne–Hitchin twistor space also have wight 1 property, so they are ample rational curves. We also show the Torelli-type theorem for this new twistor space
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Stelzig, Jonas [Verfasser], and Christopher [Akademischer Betreuer] Deninger. "Double complexes and Hodge structures as vector bundles / Jonas Stelzig ; Betreuer: Christopher Deninger." Münster : Universitäts- und Landesbibliothek Münster, 2018. http://d-nb.info/1165650959/34.

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Damjanovic, Nikola. "Arakelov inequalities and semistable families of curves uniformized by the unit ball." Thesis, Bordeaux, 2018. http://www.theses.fr/2018BORD0079/document.

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L'objet principal de cette thèse est de démontrer une inégalité d'Arakelov qui consiste à borner le degré d'un sous-faisceau inversible de l'image directe d'un faisceau relatif pluricanonique d'une famille semi-stable de courbes. Un problème naturel qui apparaît est la caractérisation des familles pour lesquelles sont satisfaites le cas d'égalité dans l'inégalité d'Arakelov, i.e. le cas d'égalité d'Arakelov. Peu d'exemples de telles familles sont connus. Dans cette thèse nous en proposons plusieurs en prouvant que le faisceau relatif bicanonique d'une famille semi-stable de courbes uniformisée par la boule unité et dont toutes les fibres singulières sont totalement géodésiques contient un sous-faisceau inversible qui satisfait l'égalité d'Arakelov
The main object of study in this thesis is an Arakelov inequality which bounds the degree of an invertible subsheaf of the direct image of the pluricanonical relative sheaf of a semistable family of curves. A natural problem that arises is the characterization of those families for which the equality is satisfied in that Arakelov inequality, i.e. the case of Arakelov equality. Few examples of such families are known. In this thesis we provide some examples by proving that the direct image of the bicanonical relative sheaf of a semistable family of curves uniformized by the unit ball, all whose singular fibers are totally geodesic, contains an invertible subsheaf which satisfies Arakelov equality
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Spinaci, Marco. "Déformations des applications harmoniques tordues." Phd thesis, Grenoble, 2013. http://tel.archives-ouvertes.fr/tel-00877310.

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On étudie les déformations des applications harmoniques $f$ tordues par rapport à une représentation. Après avoir construit une application harmonique tordue "universelle", on donne une construction de toute déformations du premier ordre de $f$ en termes de la théorie de Hodge ; on applique ce résultat à l'espace de modules des représentations réductives d'un groupe de Kähler, pour démontrer que les points critiques de la fonctionnelle de l'énergie $E$ coïncident avec les représentations de monodromie des variations complexes de structures de Hodge. Ensuite, on procède aux déformations du second ordre, où des obstructions surviennent ; on enquête sur l'existence de ces déformations et on donne une méthode pour les construire. En appliquant ce résultat à la fonctionnelle de l'énergie comme ci-dessus, on démontre (pour n'importe quel groupe de présentation finie) que la fonctionnelle de l'énergie est strictement pluri sous-harmonique sur l'espace des modules des représentations. En assumant de plus que le groupe soit de Kähler, on étudie les valeurs propres de la matrice hessienne de $E$ aux points critiques.
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Bandara, Lashi. "Geometry and the Kato square root problem." Phd thesis, 2013. http://hdl.handle.net/1885/10690.

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The primary focus of this thesis is to consider Kato square root problems for various divergence-form operators on manifolds. This is the study of perturbations of second-order differential operators by bounded, complex, measurable coefficients. In general, such operators are not self-adjoint but uniformly elliptic. The Kato square root problem is then to understand when the square root of such an operator, which exists due to uniform ellipticity, is comparable to its unperturbed counterpart. A remarkably adaptable operator-theoretic framework due to Axelsson, Keith and McIntosh sits in the background of this work. This framework allows us to take a powerful first-order perspective of the problems which we consider in a geometric setting. Through a well established procedure, we reduce these problems to the study of quadratic estimates. Under a set of natural conditions, we prove quadratic estimates for a class of operators on vector bundles over complete measure metric spaces. The first kind of estimates we prove are global, and we establish them on trivial vector bundles when the underlying measure grows at most polynomially. The second kind are local, and there, we allow the vector bundle to be non-trivial but bounded in an appropriate sense. Here, the measure is allowed to grow exponentially. An important consequence of obtaining quadratic estimates on measure metric spaces is that it allows us to consider subelliptic operators on Lie groups. The first-order perspective allows us to reduce the subelliptic problem to a fully elliptic one on a sub-bundle. As a consequence, we are able to solve a homogeneous Kato square root problem for perturbations of subelliptic operators on nilpotent Lie groups. For general Lie groups we solve a similar inhomogeneous problem. In the situation of complete Riemannian manifolds, we consider uniformly elliptic divergence-form operators arising from connections on vector bundles. Under a set of assumptions, we show that the Kato square root problem can be solved for such operators. As a consequence, we solve this problem on functions under the condition that the Ricci curvature and injectivity radius are bounded. Assuming an additional lower bound for the curvature endomorphism on forms, we solve a similar problem for perturbations of inhomogeneous Hodge-Dirac operators. A theorem for tensors is obtained by additionally assuming boundedness of a second-order Riesz transform. Motivated by the study of these Kato problems, where for technical reasons it is useful to know the density of compactly supported functions in the domains of operators, we study connections and their divergence on a vector bundle. Through a first-order formulation, we show that this density property holds for the domains of these operators if the metric and connection are compatible and the underlying manifold is complete. We also show that compactly supported functions are dense in the second-order Sobolev space on complete manifolds under the sole assumption that the Ricci curvature is bounded below, improving a result that previously required an additional lower bound on the injectivity radius.
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CAUCCI, FEDERICO. "The basepoint-freeness threshold, derived invariants of irregular varieties, and stability of syzygy bundles." Doctoral thesis, 2020. http://hdl.handle.net/11573/1355436.

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My Ph.D. thesis consists of three distinct parts, concerning various aspects of abelian and irregular varieties.The first two are related each other by the common use of the generic vanishing theory as one of the principal tools, while the third one deals with the stability of special vector bundles on an abelian variety. The first chapter regards syzygies of polarized abelian varieties. The study of equations defining projective varieties comes out very naturally in algebraic geometry and it has received considerable attention over the years. In the 80s, Green realized that classical results of Castelnuovo, Mumford, Saint-Donat and Fujita for a smooth projective curve C endowed with a line bundle L, could be unified and generalized to a very satisfying statement about syzygies. After Green's result on curves appeared, several works focused on finding extensions of it to higher dimensional varieties. A natural candidate is the class of abelian varieties. Indeed, based on classical facts of Koizumi, Mumford and Kempf, and motivated by the result of Green on higher syzygies for curves, Lazarsfeld conjectured that, for an ample line bundle L on an abelian variety, L^m satisfies the property (N_p) if m is at least p+3. This was proved by Pareschi in 2000 with an argument working in characteristic zero. The property (N_p) means that the first p modules of syzygies of the section algebra of L are "linear", i.e. as simple as possible. Intuitively, these notions consist of an increasing sequence of "positivity" properties of L. We showed a general result that, in particular, provides at the same time a surprisingly quick proof of Lazarsfeld's conjecture, extending it to abelian varieties defined over a ground field of arbitrary characteristic, and a new proof of a criterion of Lazarsfeld-Pareschi-Popa, regarding syzygies of line bundles that are not necessarily multiples of other ones. Our main result is that the basepoint-freeness threshold, introduced by Jiang-Pareschi for the numerical class of an ample line bundle L, encodes information about the syzygies of L. Namely, if it is less than 1/(p+2) for some non-negative integer p, then the property (N_p) holds for L. The proof works over algebraically closed fields of arbitrary characteristic. In addition to syzygies, we prove that the basepoint-freeness threshold also gives information on the (local) positivity of the polarization L, indeed it controls the k-jet ampleness of L, and it is related to its Seshadri constant. The second chapter contains the results of a paper joint with G. Pareschi, published in the journal Algebraic Geometry . An extension of our main theorem to pushforward of pluricanonical bundles is also included. Given a smooth complex projective variety X, its derived category is a triangulated category naturally associated to X, whose objects are bounded complexes of coherent sheaves on X. Given another variety Y, we say that X is derived equivalent to Y if there exists an exact equivalence between their derived categories. It is very natural to ask which geometric information are preserved under derived equivalence. Over the past 20 years, a big deal of interest grew on this type of question, thanks especially to the works of Bondal, Orlov, Kawamata, Kontsevich, Popa, Schnell. The main conjecture in the field, often attributed to Kontsevich, predicts that derived equivalent varieties have the same Hodge numbers. However, up to now, the derived invariance of the Hodge numbers h^{0,j} is not even known. We prove a general result in this direction: the derived invariance of the cohomology ranks of the pushforward under the Albanese map of the canonical line bundle. In particular, in the case of varieties of maximal Albanese dimension, this settles in the affirmative conjectures of Popa and Lombardi-Popa, and it proves that the Hodge numbers h^{0,j} are derived invariants, for all j. The third chapter concerns a joint work (in progress) with Martí Lahoz. We settled a conjecture of Ein-Lazarsfeld-Mustopa, in the case of abelian varieties. Let (X, L) be a polarized smooth variety over an algebraically closed field. Ein-Lazarsfeld-Mustopa proved, in dimension 2, the slope stability of the kernel of the evaluation morphism of global sections, namely the syzygy bundle, of the line bundle L^d with respect to L, for d sufficiently large. They also conjectured that such result should hold for any smooth projective variety. The main result of the third chapter is basically a proof of this conjecture in the case of abelian varieties. Namely, let (A, L) be a polarized abelian variety defined over an algebraically closed field of arbitrary characteristic, and let d at least 2. Then the syzygy bundle M_{L^d} is Gieseker semistable with respect to L.
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Books on the topic "Hodge bundle"

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Mochizuki, Takuro. Asymptotic behaviour of tame harmonic bundles and an application to pure twister D-modules. Providence, RI: American Mathematical Society, 2007.

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2

Gerverdinck, Tjebbe. Wetenschappelijk bijdragen: Bundel ter gelegenheid van het 35-jarig bestaan van het wetenschappelijk bureau van de Hoge Raad der Nederlanden. Den Haag: Boom Juridische uitgevers, 2014.

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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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Noether-lefschetz Problems For Degeneracy Loci. American Mathematical Society, 2003.

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Cattani, Eduardo. Introduction to Variations of Hodge Structure. 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/9780691161341.003.0007.

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This chapter emphasizes the theory of abstract variations of Hodge structure (VHS) and, in particular, their asymptotic behavior. It first studies the basic correspondence between local systems, representations of the fundamental group, and bundles with a flat connection. The chapter then turns to analytic families of smooth projective varieties, the Kodaira–Spencer map, Griffiths' period map, and a discussion of its main properties: holomorphicity and horizontality. These properties motivate the notion of an abstract VHS. Next, the chapter defines the classifying spaces for polarized Hodge structures and studies some of their basic properties. Finally, the chapter deals with the asymptotics of a period mapping with particular attention to Schmid's orbit theorems.
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Huybrechts, D. Fourier-Mukai Transforms in Algebraic Geometry. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.001.0001.

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This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.
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Faltings, Gerd. Facsimile : A p-adic Simpson correspondence. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691170282.003.0007.

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This chapter presents the facsimile of Gerd Faltings' article entitled “A p-adic Simpson Correspondence,” reprinted from Advances in Mathematics 198(2), 2005. In this article, an equivalence between the category of Higgs bundles and that of “generalized representations” of the étale fundamental group is constructed for curves over a p-adic field. The definition of “generalized representations” uses p-adic Hodge theory and almost étale coverings, and it includes usual representations which form a full subcategory. The equivalence depends on the choice of an exponential function for the multiplicative group. The method used in the proofs is the theory of almost étale extensions. A nonabelian Hodge–Tate theory is also developed.
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Abbes, Ahmed, Michel Gros, and Takeshi Tsuji. The p-adic Simpson Correspondence (AM-193). Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691170282.001.0001.

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The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra—namely Higgs bundles. This book undertakes a systematic development of the theory following two new approaches. It mainly focuses on generalized representations of the fundamental group that are p-adically close to the trivial representation. The first approach relies on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J. M. Fontaine. The second approach introduces a crystalline-type topos and replaces the notion of Higgs bundles with that of Higgs isocrystals. The book shows the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the book contains results of wider interest in p-adic Hodge theory. The reader will find a concise introduction to Faltings' theory of almost étale extensions and a chapter devoted to the Faltings topos. Though this topos is the general framework for Faltings' approach in p-adic Hodge theory, it remains relatively unexplored.
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9

Courbes et Fibres Vectoriels en Theorie de Hodge $p$-Adique. American Mathematical Society, 2018.

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10

Asymptotic Behaviour of Tame Harmonic Bundles and an Application to Pure Twistor $D$-Modules, Part 1 (Memoirs of the American Mathematical Society) (Memoirs of the American Mathematical Society). American Mathematical Society, 2006.

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Book chapters on the topic "Hodge bundle"

1

Lin, Bo, and Martin Ulirsch. "Towards a Tropical Hodge Bundle." In Fields Institute Communications, 353–68. New York, NY: Springer New York, 2017. http://dx.doi.org/10.1007/978-1-4939-7486-3_16.

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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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Aldrovandi, Ruben, and José Geraldo Pereira. "Hodge Dual for Soldered Bundles." In Teleparallel Gravity, 83–87. Dordrecht: Springer Netherlands, 2013. http://dx.doi.org/10.1007/978-94-007-5143-9_8.

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4

Scholze, Peter, and Jared Weinstein. "Mixed-characteristic shtukas." In Berkeley Lectures on p-adic Geometry, 90–97. Princeton University Press, 2020. http://dx.doi.org/10.23943/princeton/9780691202082.003.0011.

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This chapter looks at mixed-characteristic shtukas. Much of the theory of mixed-characteristic shtukas is motivated by the structures appearing in (integral) p-adic Hodge theory. The chapter assesses Drinfeld's shtukas and local shtukas. In the mixed characteristic setting, X will be replaced with Spa Zp. The test objects S will be drawn from Perf, the category of perfectoid spaces in characteristic p. For an object, a shtuka over S should be a vector bundle over an adic space, together with a Frobenius structure. The product is not meant to be taken literally (if so, one would just recover S), but rather it is to be interpreted as a fiber product over a deeper base. Motivated by this, the chapter then defines an analytic adic space and shows that its associated diamond is the appropriate product of sheaves on Perf.
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5

"Vector bundles." In A Survey of the Hodge Conjecture, 17–27. Providence, Rhode Island: American Mathematical Society, 2016. http://dx.doi.org/10.1090/crmm/010/02.

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6

"Line bundles." In A Survey of the Hodge Conjecture, 41–55. Providence, Rhode Island: American Mathematical Society, 2016. http://dx.doi.org/10.1090/crmm/010/04.

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Yuan, Xinyi, Shou-Wu Zhang, and Wei Zhang. "Decomposition of the Geometric Kernel." In The Gross-Zagier Formula on Shimura Curves. Princeton University Press, 2012. http://dx.doi.org/10.23943/princeton/9780691155913.003.0007.

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This chapter describes the decomposition of the geometric kernel. It considers the assumptions on the Schwartz function and decomposes the height series into local heights using arithmetic models. The intersections with the Hodge bundles are zero, and a decomposition to a sum of local heights by standard results in Arakelov theory is achieved. The chapter proceeds by reviewing the definition of the Néeron–Tate height and shows how to compute it by the arithmetic Hodge index theorem. When there is no horizontal self-intersection, the height pairing automatically decomposes to a summation of local pairings. The chapter proves that the contribution of the Hodge bundles in the height series is zero. It also compares two kernel functions and states the computational result. It concludes by deducing the kernel identity.
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8

Bradlow, Steven, Oscar García-Prada, Peter Gothen, and Jochen Heinloth. "Irreducibility of Moduli of Semi-Stable Chains and Applications to U(p, q)-Higgs Bundles." In Geometry and Physics: Volume II, 455–70. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198802020.003.0018.

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This chapter gives necessary and sufficient conditions for moduli spaces of semi-stable chains on a curve to be irreducible and non-empty. This gives information on the irreducible components of the nilpotent cone of GLn-Higgs bundles and of moduli of systems of Hodge bundles on curves. As it does not impose coprimality restrictions, it can apply this to prove connectedness for moduli spaces of U(p, q)-Higgs bundles.
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9

Brennan, Matt. "Studious drummers, selling drum outfits, standardization, and stardom." In Kick It, 105–52. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780190683863.003.0004.

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Literacy, education, and standardization were key steps forward in consolidating the drum kit’s legitimacy in the 1930s. This chapter examines the biographies of many early drummers and how they learnt to play the drum kit. Arguments over how to play the drum kit were inseparable from the changing form of the drum kit itself, as manufacturers like Ludwig, Slingerland, Leedy, and Gretsch competed to sell standardized, pre-bundled drum kits in their catalogues rather than the hodge-podge, self-assembled drum kits of the past. This chapter discusses the creation of an international market for drum kits through a combination of instrument innovation, education, and old-fashioned hucksterism. Drum manufacturers created their own newsletters as a way of convincing drummers to buy their product. The chapter also examines the career of swing era drummer Gene Krupa, comparing him with African-American drummer Chick Webb, an influential but less well known drumming bandleader.
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