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Journal articles on the topic 'Hodge bundle'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Forni, Giovanni, Carlos Matheus, and Anton Zorich. "Zero Lyapunov exponents of the Hodge bundle." Commentarii Mathematici Helvetici 89, no. 2 (2014): 489–535. http://dx.doi.org/10.4171/cmh/325.

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12

MUÑOZ, VICENTE. "HODGE STRUCTURES OF THE MODULI SPACES OF PAIRS." International Journal of Mathematics 21, no. 11 (November 2010): 1505–29. http://dx.doi.org/10.1142/s0129167x10006604.

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Let X be a smooth projective curve of genus g ≥ 2 over ℂ. Fix n ≥ 2, d ∈ ℤ. A pair (E, ϕ) over X consists of an algebraic vector bundle E of rank n and degree d over X and a section ϕ ∈ H0(E). There is a concept of stability for pairs which depends on a real parameter τ. Let [Formula: see text] be the moduli space of τ-semistable pairs of rank n and degree d over X. Here we prove that the cohomology groups of [Formula: see text] are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure H1(X). This implies a similar result for the moduli spaces of stable vector bundles over X.

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13

Brunebarbe, Yohan. "Symmetric differentials and variations of Hodge structures." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 743 (October 1, 2018): 133–61. http://dx.doi.org/10.1515/crelle-2015-0109.

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Abstract Let D be a simple normal crossing divisor in a smooth complex projective variety X. We show that the existence on X-D of a non-trivial polarized complex variation of Hodge structures with integral monodromy implies that the pair (X,D) has a non-zero logarithmic symmetric differential (a section of a symmetric power of the logarithmic cotangent bundle). When the corresponding period map is generically immersive, we show more precisely that the logarithmic cotangent bundle is big.

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14

TEH, JYH-HAUR. "Motivic integration and projective bundle theorem in morphic cohomology." Mathematical Proceedings of the Cambridge Philosophical Society 147, no. 2 (September 2009): 295–321. http://dx.doi.org/10.1017/s0305004109002588.

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AbstractWe reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two K-equivalent varieties are the same, which implies that several conjectures of algebraic cycles are K-statements. We define stringy functions which enable us to ask stringy Grothendieck standard conjecture and stringy Hodge conjecture. We prove a projective bundle theorem in morphic cohomology for trivial bundles over any normal quasi-projective varieties.

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15

Urbanik, David. "Absolute Hodge and ℓ-adic monodromy." Compositio Mathematica 158, no. 3 (March 2022): 568–84. http://dx.doi.org/10.1112/s0010437x2200745x.

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Let $\mathbb {V}$ be a motivic variation of Hodge structure on a $K$ -variety $S$ , let $\mathcal {H}$ be the associated $K$ -algebraic Hodge bundle, and let $\sigma \in \mathrm {Aut}(\mathbb {C}/K)$ be an automorphism. The absolute Hodge conjecture predicts that given a Hodge vector $v \in \mathcal {H}_{\mathbb {C}, s}$ above $s \in S(\mathbb {C})$ which lies inside $\mathbb {V}_{s}$ , the conjugate vector $v_{\sigma } \in \mathcal {H}_{\mathbb {C}, s_{\sigma }}$ is Hodge and lies inside $\mathbb {V}_{s_{\sigma }}$ . We study this problem in the situation where we have an algebraic subvariety $Z \subset S_{\mathbb {C}}$ containing $s$ whose algebraic monodromy group $\textbf {H}_{Z}$ fixes $v$ . Using relationships between $\textbf {H}_{Z}$ and $\textbf {H}_{Z_{\sigma }}$ coming from the theories of complex and $\ell$ -adic local systems, we establish a criterion that implies the absolute Hodge conjecture for $v$ subject to a group-theoretic condition on $\textbf {H}_{Z}$ . We then use our criterion to establish new cases of the absolute Hodge conjecture.

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16

Greb, Daniel, Stefan Kebekus, Thomas Peternell, and Behrouz Taji. "Nonabelian Hodge theory for klt spaces and descent theorems for vector bundles." Compositio Mathematica 155, no. 2 (February 2019): 289–323. http://dx.doi.org/10.1112/s0010437x18007923.

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We generalise Simpson’s nonabelian Hodge correspondence to the context of projective varieties with Kawamata log terminal (klt) singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety$X$and any resolution of singularities, any vector bundle on the resolution that appears to come from$X$numerically, does indeed come from $X$. Furthermore, and of independent interest, a new restriction theorem for semistable Higgs sheaves defined on the smooth locus of a normal, projective variety is established.

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17

MUÑOZ, VICENTE, DANIEL ORTEGA, and MARIA-JESÚS VÁZQUEZ-GALLO. "HODGE POLYNOMIALS OF THE MODULI SPACES OF PAIRS." International Journal of Mathematics 18, no. 06 (July 2007): 695–721. http://dx.doi.org/10.1142/s0129167x07004266.

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Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic pair on X is a couple (E, ϕ), where E is a holomorphic bundle over X of rank n and degree d, and ϕ ∈ H0(E) is a holomorphic section. In this paper, we determine the Hodge polynomials of the moduli spaces of rank 2 pairs, using the theory of mixed Hodge structures. We also deal with the case in which E has fixed determinant.

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18

BISWAS, INDRANIL. "HOLOMORPHIC PRINCIPAL BUNDLES WITH AN ELLIPTIC CURVE AS THE STRUCTURE GROUP." International Journal of Geometric Methods in Modern Physics 05, no. 06 (September 2008): 851–62. http://dx.doi.org/10.1142/s0219887808003004.

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Let Λ ⊂ ℂ be the ℤ-module generated by 1 and [Formula: see text], where τ is a positive real number. Let Z := ℂ/Λ be the corresponding complex torus of dimension one. Our aim here is to give a general construction of holomorphic principal Z-bundles over a complex manifold X. Let θ1 and θ2 be two C∞ real closed two-forms on X such that the Hodge type (0, 2) component of the form [Formula: see text] vanishes, and the elements in H2(X, ℂ) represented by θ1 and θ2 are contained in the image of H2(X, ℤ). For such a pair we construct a holomorphic principal Z-bundle over X. Conversely, given any holomorphic principal Z-bundle EZ over X, we construct a pair of closed differential forms on X of the above type.

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19

Bayer, Arend, and Charles Cadman. "Quantum cohomology of [ℂN/μr]." Compositio Mathematica 146, no. 5 (June 22, 2010): 1291–322. http://dx.doi.org/10.1112/s0010437x10004793.

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AbstractWe give a construction of the moduli space of stable maps to the classifying stack Bμr of a cyclic group by a sequence of rth root constructions on $\overline {M}_{0, n}$. We prove a closed formula for the total Chern class of μr-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus-zero Gromov–Witten theory of stacks of the form [ℂN/μr]. We deduce linear recursions for genus-zero Gromov–Witten invariants.

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20

Looijenga, Eduard. "Goresky–Pardon lifts of Chern classes and associated Tate extensions." Compositio Mathematica 153, no. 7 (May 3, 2017): 1349–71. http://dx.doi.org/10.1112/s0010437x17007175.

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Let $X$ be an irreducible complex-analytic variety, ${\mathcal{S}}$ a stratification of $X$ and ${\mathcal{F}}$ a holomorphic vector bundle on the open stratum ${X\unicode[STIX]{x0030A}}$. We give geometric conditions on ${\mathcal{S}}$ and ${\mathcal{F}}$ that produce a natural lift of the Chern class $\operatorname{c}_{k}({\mathcal{F}})\in H^{2k}({X\unicode[STIX]{x0030A}};\mathbb{C})$ to $H^{2k}(X;\mathbb{C})$, which, in the algebraic setting, is of Hodge level ${\geqslant}k$. When applied to the Baily–Borel compactification $X$ of a locally symmetric variety ${X\unicode[STIX]{x0030A}}$ and an automorphic vector bundle ${\mathcal{F}}$ on ${X\unicode[STIX]{x0030A}}$, this refines a theorem of Goresky–Pardon. In passing we define a class of simplicial resolutions of the Baily–Borel compactification that can be used to define its mixed Hodge structure. We use this to show that the stable cohomology of the Satake ($=$ Baily–Borel) compactification of ${\mathcal{A}}_{g}$ contains nontrivial Tate extensions.

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21

Koskivirta, Jean-Stefan. "Canonical sections of the Hodge bundle over Ekedahl–Oort strata of Shimura varieties of Hodge type." Journal of Algebra 449 (March 2016): 446–59. http://dx.doi.org/10.1016/j.jalgebra.2015.11.030.

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22

Chiodo, Alessandro. "Towards an enumerative geometry of the moduli space of twisted curves and rth roots." Compositio Mathematica 144, no. 6 (November 2008): 1461–96. http://dx.doi.org/10.1112/s0010437x08003709.

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AbstractThe enumerative geometry of rth roots of line bundles is crucial in the theory of r-spin curves and occurs in the calculation of Gromov–Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In a previous paper, we constructed a compact moduli stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford’s formula for the Chern character of the Hodge bundle to the direct image of the universal rth root in K-theory.

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23

Loi, Andrea, Roberto Mossa, and Fabio Zuddas. "Some remarks on the Gromov width of homogeneous Hodge manifolds." International Journal of Geometric Methods in Modern Physics 11, no. 09 (October 2014): 1460029. http://dx.doi.org/10.1142/s0219887814600299.

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We provide an upper bound for the Gromov width of compact homogeneous Hodge manifolds (M, ω) with b2(M) = 1. As an application we obtain an upper bound on the Seshadri constant ϵ(L) where L is the ample line bundle on M such that [Formula: see text].

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24

Getzler, E., and R. Pandharipande. "Virasoro constraints and the Chern classes of the Hodge bundle." Nuclear Physics B 530, no. 3 (October 1998): 701–14. http://dx.doi.org/10.1016/s0550-3213(98)00517-3.

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25

Tayou, Salim, and Nicolas Tholozan. "Equidistribution of Hodge loci II." Compositio Mathematica 159, no. 1 (January 2023): 1–52. http://dx.doi.org/10.1112/s0010437x22007795.

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Let $\mathbb {V}$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$ . In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, the pull–push form. In particular, it is always analytically dense when the pull–push form does not vanish. When the weight is two, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$ , we prove that the typical locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$ , where $c_q$ is the $q$ th Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in $\mathcal {A}_g$ , equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in $\mathcal {A}_g$ . These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull–push form appears in this greater generality, we provide several tools to determine it, and we compute it in many examples.

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26

Bruzzo, Ugo, Igor Mencattini, Vladimir N. Rubtsov, and Pietro Tortella. "Nonabelian holomorphic Lie algebroid extensions." International Journal of Mathematics 26, no. 05 (May 2015): 1550040. http://dx.doi.org/10.1142/s0129167x15500408.

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We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the Hochschild–Serre spectral sequence for Lie algebras to the holomorphic Lie algebroid setting. As an application, we show that the hypercohomology of the Atiyah algebroid of a line bundle has a natural Hodge structure.

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27

Matheus, Carlos. "The Teichmüller geodesic flow and the geometry of the Hodge bundle." Séminaire de théorie spectrale et géométrie 29 (2011): 73–95. http://dx.doi.org/10.5802/tsg.286.

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28

Lu, Xin. "Family of curves with large unitary summand in the Hodge bundle." Mathematische Zeitschrift 291, no. 3-4 (December 6, 2018): 1381–87. http://dx.doi.org/10.1007/s00209-018-2181-3.

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29

Zborowski, Grzegorz. "A-manifolds on a principal torus bundle over an almost Hodge A-manifold base." Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica 69, no. 1 (November 30, 2015): 109. http://dx.doi.org/10.17951/a.2015.69.1.109.

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An A-manifold is a manifold whose Ricci tensor is cyclic-parallel, equivalently it satisfies ∇<sub>X</sub> Ric(X, X) = 0. This condition generalizes the Einstein condition. We construct new examples of A-manifolds on r-torus bundles over a base which is a product of almost Hodge A-manifolds.

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30

GONZÁLEZ–MARTÍNEZ, CRISTIAN. "HODGE POLYNOMIALS OF SOME MODULI SPACES OF COHERENT SYSTEMS." International Journal of Mathematics 24, no. 03 (March 2013): 1350014. http://dx.doi.org/10.1142/s0129167x13500146.

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When k < n, we study the coherent systems that come from a BGN extension in which the quotient bundle is strictly semistable. In this case we describe a stratification of the moduli space of coherent systems. We also describe the strata as complements of determinantal varieties and we prove that these are irreducible and smooth. These descriptions allow us to compute the Hodge polynomials of this moduli space in some cases. In particular, we give explicit computations for the cases in which (n, d, k) = (3, d, 1) and d is even, obtaining from them the usual Poincaré polynomials.

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31

Popescu, Paul, Vladimir Rovenski, and Sergey Stepanov. "The Weitzenböck Type Curvature Operator for Singular Distributions." Mathematics 8, no. 3 (March 6, 2020): 365. http://dx.doi.org/10.3390/math8030365.

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We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior derivatives and their L 2 adjoint operators on tensors. Then, we introduce the Weitzenböck type curvature operator on tensors, prove the Weitzenböck type decomposition formula, and derive the Bochner–Weitzenböck type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian. The assumptions used in the results are reasonable, as illustrated by examples with f-manifolds, including almost Hermitian and almost contact ones.

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32

González-Alonso, Víctor, Lidia Stoppino, and Sara Torelli. "On the rank of the flat unitary summand of the Hodge bundle." Transactions of the American Mathematical Society 372, no. 12 (July 8, 2019): 8663–77. http://dx.doi.org/10.1090/tran/7868.

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33

Popovici, Dan. "Non-Kähler Mirror Symmetry of the Iwasawa Manifold." International Mathematics Research Notices 2020, no. 23 (November 7, 2018): 9471–538. http://dx.doi.org/10.1093/imrn/rny256.

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Abstract We propose a new approach to the mirror symmetry conjecture in a form suitable to possibly non-Kähler compact complex manifolds whose canonical bundle is trivial. We apply our methods by proving that the Iwasawa manifold $X$, a well-known non-Kähler compact complex manifold of dimension $3$, is its own mirror dual to the extent that its Gauduchon cone, replacing the classical Kähler cone that is empty in this case, corresponds to what we call the local universal family of essential deformations of $X$. These are obtained by removing from the Kuranishi family the two “superfluous” dimensions of complex parallelisable deformations that have a similar geometry to that of the Iwasawa manifold. The remaining four dimensions are shown to have a clear geometric meaning including in terms of the degeneration at $E_2$ of the Frölicher spectral sequence. On the local moduli space of “essential” complex structures, we obtain a canonical Hodge decomposition of weight $3$ and a variation of Hodge structures, construct coordinates and Yukawa couplings while implicitly proving a local Torelli theorem. On the metric side of the mirror, we construct a variation of Hodge structures parametrised by a subset of the complexified Gauduchon cone of the Iwasawa manifold using the sGG property (which means that all the Gauduchon metrics are strongly Gauduchon) of all the small deformations of this manifold proved in earlier joint work of the author with L. Ugarte. Finally, we define a mirror map linking the two variations of Hodge structures and we highlight its properties.

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34

Hunsicker, Eugénie. "Hodge and signature theorems for a family of manifolds with fibre bundle boundary." Geometry & Topology 11, no. 3 (July 23, 2007): 1581–622. http://dx.doi.org/10.2140/gt.2007.11.1581.

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35

Matheus, Carlos, and Gabriela Weitze-Schmithüsen. "Some Examples of Isotropic SL(2, ℝ)-Invariant Subbundles of the Hodge Bundle." International Mathematics Research Notices 2015, no. 18 (November 19, 2014): 8657–79. http://dx.doi.org/10.1093/imrn/rnu207.

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36

Li, Hongjun, Chunhui Qiu, and Weixia Zhu. "Laplacians for the holomorphic tangent bundles with g-nature metrics on complex Finsler manifolds." International Journal of Mathematics 28, no. 09 (August 2017): 1740011. http://dx.doi.org/10.1142/s0129167x17400110.

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Let [Formula: see text] be a strongly pseudoconvex compact complex Finsler manifold. We first introduce a class of [Formula: see text]-nature metric [Formula: see text] for the slit holomorphic tangent bundle [Formula: see text] on [Formula: see text]. Then, we define the complex horizontal Laplacian [Formula: see text], and complex vertical Laplacian [Formula: see text], and obtain a precise relationship among [Formula: see text], [Formula: see text] and the Hodge–Laplace operator [Formula: see text] on [Formula: see text]. As an application, we discuss the holomorphic Killing vector fields associated to [Formula: see text].

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37

Owens, Bryson, Seamus Somerstep, and Renzo Cavalieri. "Boundary expression for Chern classes of the Hodge bundle on spaces of cyclic covers." Involve, a Journal of Mathematics 14, no. 4 (October 23, 2021): 571–94. http://dx.doi.org/10.2140/involve.2021.14.571.

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38

Eskin, Alex, Maxim Kontsevich, and Anton Zorich. "Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow." Publications mathématiques de l'IHÉS 120, no. 1 (November 16, 2013): 207–333. http://dx.doi.org/10.1007/s10240-013-0060-3.

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39

Fougeron, Charles. "Lyapunov exponents of the Hodge bundle over strata of quadratic differentials with large number of poles." Mathematical Research Letters 25, no. 4 (2018): 1213–25. http://dx.doi.org/10.4310/mrl.2018.v25.n4.a8.

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Loi, Andrea, Roberto Mossa, and Fabio Zuddas. "The log-term of the Bergman kernel of the disc bundle over a homogeneous Hodge manifold." Annals of Global Analysis and Geometry 51, no. 1 (July 7, 2016): 35–51. http://dx.doi.org/10.1007/s10455-016-9522-4.

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41

Ekedahl, Torsten, and Gerard van der Geer. "Cycles representing the top Chern class of the Hodge bundle on the moduli space of abelian varieties." Duke Mathematical Journal 129, no. 1 (July 2005): 187–99. http://dx.doi.org/10.1215/s0012-7094-04-12917-3.

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42

Molcho, S., R. Pandharipande, and J. Schmitt. "The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves." Compositio Mathematica 159, no. 2 (February 2023): 306–54. http://dx.doi.org/10.1112/s0010437x22007874.

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We bound from below the complexity of the top Chern class $\lambda _g$ of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for $\lambda _g$ in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove $\lambda _g$ lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for $\lambda _g$ on the moduli of curves after log blow-ups.

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Ekedahl, Torsten, and Gerard Geer. "The order of the top Chern class of the Hodge bundle on the moduli space of abelian varieties." Acta Mathematica 192, no. 1 (2004): 95–109. http://dx.doi.org/10.1007/bf02441086.

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Munoz-Price, L. Silvia, Mary K. Hayden, Karen Lolans, Sarah Won, Karen Calvert, Michael Lin, Alexander Sterner, and Robert A. Weinstein. "Successful Control of an Outbreak ofKlebsiella pneumoniaeCarbapenemase—ProducingK. pneumoniaeat a Long-Term Acute Care Hospital." Infection Control & Hospital Epidemiology 31, no. 4 (April 2010): 341–47. http://dx.doi.org/10.1086/651097.

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Objective.To determine the effect of a bundle of infection control interventions on the horizontal transmission ofKlebsiella pneumoniaecarbapenemase (KPC)-producingK. pneumoniaeduring an outbreak.Design.Quasi-experimental study.Setting.Long-term acute care hospital.Intervention.On July 23,2008, a bundled intervention was implemented: daily 2% Chlorhexidine gluconate baths for patients, enhanced environmental cleaning, surveillance cultures at admission, serial point prevalence surveillance (PPS), isolation precautions, and training of personnel. Baseline PPS was performed before the intervention was implemented. Any gram-negative rod isolate suspected of KPC production underwent a modified Hodge test and, if results were positive, confirmatory polymerase chain reaction testing. Clinical cases were defined to occur for patients whose samples yielded KPC-positive gram-negative rods in clinical cultures.Results.Baseline PPS performed on June 17, 2008, showed a prevalence of colonization with KPC-producing isolates of 21% (8 of 39 patients screened). After implementation of the intervention, monthly PPS was performed 5 times, which showed prevalences of colonization with KPC-producing isolates of 12%, 5%, 3%, 0%, and 0% (P< .001). From January 1, 2008, until the intervention, 8 KPC-positive clinical cases—suspected to be due to horizontal transmission—were detected. From implementation of the intervention through December 31, 2008, only 2 KPC-positive clinical cases, both in August 2008, were detected. From January 1 through December 31, 2008, 8 patients were detected as carriers of KPC-producing isolates at admission to the institution, 4 patients before and 4 patients after the intervention.Conclusion.A bundled intervention was successful in preventing horizontal spread of KPC-producing gram-negative rods in a long-term acute care hospital, despite ongoing admission of patients colonized with KPC producers.

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BRASSELET, JEAN-PAUL, JÖRG SCHÜRMANN, and SHOJI YOKURA. "HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES." Journal of Topology and Analysis 02, no. 01 (March 2010): 1–55. http://dx.doi.org/10.1142/s1793525310000239.

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In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformationmCy: K0( var /X) → G0(X) ⊗ ℤ[y], which generalizes the total λ-class λy(T*X) of the cotangent bundle to singular spaces. Here K0( var /X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G0(X) is the Grothendieck group of coherent sheaves of [Formula: see text]-modules. A first construction of mCy is based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mCy is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito. We define a natural transformation Ty* : K0( var /X) → H*(X) ⊗ ℚ[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Ty* is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y = -1), the Todd class transformation in the singular Riemann-Roch theorem of Baum–Fulton–MacPherson (for y = 0) and the L-class transformation of Cappell-Shaneson (for y = 1). We also explain the relation among the "stringy version" of our characteristic classes, the elliptic class of Borisov–Libgober and the stringy Chern classes of Aluffi and De Fernex–Lupercio–Nevins–Uribe. All our results can be extended to varieties over a base field k of characteristic 0.

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Belhaj, A., M. Bensed, Z. Benslimane, M. B. Sedra, and A. Segui. "Qubit and fermionic Fock spaces from type II superstring black holes." International Journal of Geometric Methods in Modern Physics 14, no. 06 (May 4, 2017): 1750087. http://dx.doi.org/10.1142/s0219887817500876.

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Using Hodge diagram combinatorial data, we study qubit and fermionic Fock spaces from the point of view of type II superstring black holes based on complex compactifications. Concretely, we establish a one-to-one correspondence between qubits, fermionic spaces and extremal black holes in maximally supersymmetric supergravity obtained from type II superstring on complex toroidal and Calabi–Yau compactifications. We interpret the differential forms of the [Formula: see text]-dimensional complex toroidal compactification as states of [Formula: see text]-qubits encoding information on extremal black hole charges. We show that there are [Formula: see text] copies of [Formula: see text] qubit systems which can be split as [Formula: see text]. More precisely, [Formula: see text] copies are associated with even [Formula: see text]-brane charges in type IIA superstring and the other [Formula: see text] ones correspond to odd [Formula: see text]-brane charges in IIB superstring. This correspondence is generalized to a class of Calabi–Yau manifolds. In connection with black hole charges in type IIA superstring, an [Formula: see text]-qubit system has been obtained from a canonical line bundle of [Formula: see text] factors of one-dimensional projective space [Formula: see text]

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RODRIGUES, WALDYR A., and QUINTINO A. G. SOUZA. "AN AMBIGUOUS STATEMENT CALLED THE "TETRAD POSTULATE" AND THE CORRECT FIELD EQUATIONS SATISFIED BY THE TETRAD FIELDS." International Journal of Modern Physics D 14, no. 12 (December 2005): 2095–150. http://dx.doi.org/10.1142/s0218271805008157.

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The names tetrad, tetrads, cotetrads have been used with many different meanings in the physics literature, not all of them equivalent from the mathematical point of view. In this paper, we introduce unambiguous definitions for each of those terms, and show how the old miscellanea made many authors introduce in their formalism an ambiguous statement called the "tetrad postulate," which has been the source of much misunderstanding, as we show explicitly by examining examples found in the literature. Since formulating Einstein's field equations intrinsically in terms of cotetrad fields θa, a = 0, 1, 2, 3 is a worthy enterprise, we derive the equation of motion of each θausing modern mathematical tools (the Clifford bundle formalism and the theory of the square of the Dirac operator). Indeed, we identify (giving all details and theorems) from the square of the Dirac operator some noticeable mathematical objects, namely, the Ricci, Einstein, covariant D'Alembertian and the Hodge Laplacian operators, which permit us to show that each θasatisfies a well-defined wave equation. Also, we present for completeness a detailed derivation of the cotetrad wave equations from a variational principle. We compare the cotetrad wave equation satisfied by each θawith some others appearing in the literature, and which are unfortunately in error.

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LUSANNA, LUCA. "CLASSICAL YANG-MILLS THEORY WITH FERMIONS II: DIRAC’S OBSERVABLES." International Journal of Modern Physics A 10, no. 26 (October 20, 1995): 3675–757. http://dx.doi.org/10.1142/s0217751x95001753.

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For pure Yang-Mills theory on Minkowski space-time, formulated in functional spaces where the covariant divergence is an elliptic operator without zero modes, and for a trivial principal bundle over the fixed time Euclidean space with a compact, semisimple, connected and simply connected structure Lie group, a Green function for the covariant divergence has been found. It allows one to solve the first class constraints associated with Gauss’ laws and to identify a connection-dependent coordinatization of the trivial principal bundle. In a neighborhood of the global identity section, by using canonical coordinates of the first kind on the fibers, one has a symplectic implementation of the Lie algebra of the small gauge transformations generated by Gauss’ laws and one can make a generalized Hodge decomposition of the gauge potential one-forms based on the BRST operator. This decomposition singles out a pure gauge background connection (the BRST ghost as Maurer-Cartan one-form on the group of gauge transformations) and a transverse gauge-covariant magnetic gauge potential. After an analogous decomposition of the electric field strength into the transverse and the longitudinal part, Dirac’s observables associated with the transverse electric and magnetic components are identified as their restriction to the global identity section of the trivial principal bundle. The longitudinal part of the electric field can be re-expressed in terms of these electric and magnetic transverse parts and of the constraints without Gribov ambiguity. The physical Lagrangian, Hamiltonian, non-Abelian and topological charges have been obtained in terms of transverse Dirac’s observables, also in the presence of fermion fields, after a symplectic decoupling of the gauge degrees of freedom; one has an explicit realization of the abstract “Riemannian metric” on the orbit space. Both the Lagrangian and the Hamiltonian are nonlocal and nonpolynomial; like in the Coulomb gauge they are not Lorentz-invariant, but the invariance can be enforced on them if one introduces Wigner covariance of the observables by analyzing the various kinds of Poincare orbits of the system and by reformulating the theory on suitable spacelike hypersurfaces, following Dirac. By extending to classical relativistic field theory the problems associated with the Lorentz noncovariance of the canonical (presymplectic) center of mass for extended relativistic systems, in the sector of the field theory with P2>0 and W2≠0 one identifies a classical invariant intrinsic unit of length, determined by the Poincare Casimirs, whose quantum counterpart is the ultraviolet cutoff looked for by Dirac and Yukawa: it is the Compton wavelength of the field configuration (in an irreducible Poincare representation) multiplied by the value of its spin.

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Goldring, Wushi, and Jean-Stefan Koskivirta. "Automorphic vector bundles with global sections on -schemes." Compositio Mathematica 154, no. 12 (October 31, 2018): 2586–605. http://dx.doi.org/10.1112/s0010437x18007467.

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A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of $G$-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type $A_{1}^{n}$, $C_{2}$, and $\mathbf{F}_{p}$-split groups of type $A_{2}$ (this includes all Hilbert–Blumenthal varieties and should also apply to Siegel modular $3$-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type.

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Dhillon, Ajneet. "On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of SLn." Canadian Journal of Mathematics 58, no. 5 (October 1, 2006): 1000–1025. http://dx.doi.org/10.4153/cjm-2006-038-8.

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AbstractWe compute some Hodge and Betti numbers of the moduli space of stable rank r, degree d vector bundles on a smooth projective curve. We do not assume r and d are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank r, degree d vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of SLn is one.

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