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1. Chern classes of automorphic vector bundles, II

   Esnault, Hélène; Harris, Michael (September 2019, Épijournal de géométrie algébrique)

   We prove that the l-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over \bar{Q}_p, descend to classes in the l-adic cohomology of the minimal compactifications. These are invariant under the Galois group of the p-adic field above which the variety and the bundle are defined. 

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2. Chern classes of automorphic vector bundles, II

   Esnault, H.; Harris, M. (October 2019, Épijournal de géométrie algébrique)

   We prove that the ℓ-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over ℚ¯𝑝, descend to classes in the ℓ-adic cohomology of the minimal compactifications. These are invariant under the Galois group of the 𝑝-adic field above which the variety and the bundle are defined. 

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3. Rational homotopy type of projectivization of the tangent bundle of certain spaces

   https://doi.org/10.1108/AJMS-02-2024-0029  

   Gastinzi, Jean Baptiste; Ndlovu, Meshach (August 2024, Arab Journal of Mathematical Sciences)

   PurposeThe paper aims to determine the rational homotopy type of the total space of projectivized bundles over complex projective spaces using Sullivan minimal models, providing insights into the algebraic structure of these spaces. Design/methodology/approachThe paper utilises techniques from Sullivan’s theory of minimal models to analyse the differential graded algebraic structure of projectivized bundles. It employs algebraic methods to compute the Sullivan minimal model of $P(E)$and establish relationships with the base space. FindingsThe paper determines the rational homotopy type of projectivized bundles over complex projective spaces. Of great interest is how the Chern classes of the fibre space and base space, play a critical role in determining the Sullivan model ofP(E). We also provide the homogeneous space ofP(E)whenn = 2. Finally, we prove the formality ofP(E)over a homogeneous space of equal rank. Research limitations/implicationsLimitations may include the complexity of computing minimal models for higher-dimensional bundles. Practical implicationsUnderstanding the rational homotopy type of projectivized bundles facilitates computations in algebraic topology and differential geometry, potentially aiding in applications such as topological data analysis and geometric modelling. Social implicationsWhile the direct social impact may be indirect, advancements in algebraic topology contribute to broader mathematical knowledge, which can underpin developments in science, engineering, and technology with societal benefits. Originality/valueThe paper’s originality lies in its application of Sullivan minimal models to determine the rational homotopy type of projectivized bundles over complex projective spaces, offering valuable insights into the algebraic structure of these spaces and their associated complex vector bundles. 

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4. Approximate and discrete Euclidean vector bundles

   https://doi.org/10.1017/fms.2023.16  

   Scoccola, Luis; Perea, Jose A. (January 2023, Forum of Mathematics, Sigma)

   Abstract We introduce $$\varepsilon $$ -approximate versions of the notion of a Euclidean vector bundle for $$\varepsilon \geq 0$$ , which recover the classical notion of a Euclidean vector bundle when $$\varepsilon = 0$$ . In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $$\varepsilon $$ -approximate vector bundles can be used to represent classical vector bundles when $$\varepsilon> 0$$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples. 

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5. Chern-Simons theory, decomposition, and the A model

   https://doi.org/10.1007/JHEP10(2024)112  

   Pantev, Tony; Sharpe, Eric; Yu, Xingyang (October 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> In this paper, we discuss how gauging one-form symmetries in Chern-Simons theories is implemented in an A-twisted topological open string theory. For example, the contribution from a fixed H/Z bundle on a three-manifold M, arising in a BZ gauging of H Chern-Simons, for Z a finite subgroup of the center of H, is described by an open string worldsheet theory whose bulk is a sigma model with target a Z-gerbe (a bundle of one-form symmetries) over T^∗M, of characteristic class determined by the H/Z bundle. We give a worldsheet picture of the decomposition of one-form-symmetry-gauged Chern-Simons in three dimensions, and we describe how a target-space constraint on bundles arising in the gauged Chern-Simons theory has a natural worldsheet realization. Our proposal provides examples of the expected correspondence between worldsheet global higher-form symmetries, and target-space gauged higher-form symmetries. 

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