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In this book, Claire Voisin provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The volume is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by Voisin. The book focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by Voisin looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

We show an arithmetic generalization of the recent work of Lazarsfeld–Mustaţǎ which uses Okounkov bodies to study linear series of line bundles. As applications, we derive a log-concavity inequality on volumes of arithmetic line bundles and an arithmetic Fujita approximation theorem for big line bundles.

Let A be a local Artinian ring with residue field k(A). Let X be a curve over A and let be X′ = X ×spec A spec k(A) the fiber of X over k(A). Consider ℒ an invertible sheaf on X and ℒ ′ = ϕ*ℒ ∈ Pic(X′), where ϕ : X′ → X is the natural map. Let C and C′ be the algebraic geometric codes constructed using the groups of cohomology Γ(X, ℒ) and Γ(X′, ℒ ′) respectively. In this note, we first give the complete relation between Γ(X, ℒ) and Γ(X′, ℒ ′) without any condition and finally, we provide relations between C ⊗ A k(A) and C′ using exact sequences and dimensional theory. Therefore we extend, some results of Walker [18, 20] giving the characterization of WAG codes over rings.

Let L be a big line bundle on a compact complex manifold X . Given a non-pluripolar compact subset K of X and a continuous Hermitian metric e − φ on L , we define the energy at equilibrium of ( K , φ ) as the Monge-Ampère energy of the extremal psh weight associated to ( K , φ ). We prove the differentiability of the energy at equilibrium with respect to φ , and we show that this energy describes the asymptotic behaviour as k →∞ of the volume of the sup-norm unit ball induced by ( K , k φ ) on the space of global holomorphic sections H 0 ( X , kL ). As a consequence of these results, we recover and extend Rumely’s Robin-type formula for the transfinite diameter. We also obtain an asymptotic description of the analytic torsion, and extend Yuan’s equidistribution theorem for algebraic points of small height to the case of a big line bundle.

Let$(M,g)$be a closed Riemannian$4$-manifold and let E be a vector bundle over M with structure group G , where G is a compact Lie group. We consider a new higher order Yang–Mills–Higgs functional, in which the Higgs field is a section of$\\Omega ^0(\\text {ad}E)$. We show that, under suitable conditions, solutions to the gradient flow do not hit any finite time singularities. In the case that E is a line bundle, we are able to use a different blow-up procedure and obtain an improvement of the long-time result of Zhang [‘Gradient flows of higher order Yang–Mills–Higgs functionals’, J. Aust. Math. Soc. 113 (2022), 257–287]. The proof relies on properties of the Green function, which is very different from the previous techniques.

Given several sequences of Hermitian holomorphic line bundles { ( L k p , h k p ) } p = 1 ∞ , we establish the distribution of common zeros of random holomorphic sections of L kp with respect to singular measures. We also study the dimension growth for a sequence of pseudo-effective line bundles.

We associate to certain filtrations of a graded linear series of a big line bundle a concave function on its Okounkov body, whose law with respect to the Lebesgue measure describes the asymptotic distribution of the jumps of the filtration. As a consequence, we obtain a Fujita-type approximation theorem in this general filtered setting. We then specialize these results to the filtrations by minima in the usual context of Arakelov geometry (and for more general adelically normed graded linear series), thereby obtaining in a simple way a natural construction of an arithmetic Okounkov body, the existence of the arithmetic volume as a limit and an arithmetic Fujita approximation theorem for adelically normed graded linear series. We also obtain an easy proof of the existence of the sectional capacity previously obtained by Lau, Rumely and Varley.

We present a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map. Several examples are provided as well as theorems addressing choices in the construction.

By combining modular invariance of characteristic forms and the family index theory, we obtain some new anomaly cancellation formulas for any dimension under the not top degree component. For a fiber bundle of dimension (4l−2), we obtain the anomaly cancellation formulas for the determinant line bundle. For the fiber bundle with a dimension of (4l−3), we derive the anomaly cancellation formulas of the index gerbes. For the fiber bundle of dimension (4l−1), we obtain some results of the eta invariants. Moreover, we give some anomaly cancellation formulas of the Chen–Connes character and the higher elliptic genera.