Galois action on algebraic matrix groups, Chern classes, and the Euler class

In this paper we give explicit formulas for differential characteristic classes of principal $G$-bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and the differential Euler class. Furthermore, we show that the differential Chern class is the unique natural transformation from (Simons–Sullivan) differential $K$-theory to (Cheeger–Simons) differential characters that is compatible with curvature and characteristic class. We also give the explicit formula for the differential Chern class on Freed–Lott differential $K$-theory. Finally, we discuss the odd differential Chern classes.

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Multiplicities and Chern Classes in Local Algebra

10.1017/cbo9780511529986 ◽

1998 ◽

Cited By ~ 28

Author(s):

Paul C. Roberts

Keyword(s):

Local Algebra ◽

Chern Classes

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Poincaré Bundle and Chern Classes

Recent Topics in Differential and Analytic Geometry ◽

10.2969/aspm/01810271 ◽

2018 ◽

Author(s):

Mitsuhiro Itoh

Keyword(s):

Chern Classes

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Tensors with eigenvectors in a given subspace

Rendiconti del Circolo Matematico di Palermo (1952 -) ◽

10.1007/s12215-021-00600-2 ◽

2021 ◽

Author(s):

Giorgio Ottaviani ◽

Zahra Shahidi

Keyword(s):

Algebraic Geometry ◽

Linear Subspace ◽

Chern Classes ◽

Symmetric Tensors

AbstractThe first author with B. Sturmfels studied in [16] the variety of matrices with eigenvectors in a given linear subspace, called the Kalman variety. We extend that study from matrices to symmetric tensors, proving in the tensor setting the irreducibility of the Kalman variety and computing its codimension and degree. Furthermore, we consider the Kalman variety of tensors having singular t-tuples with the first component in a given linear subspace and we prove analogous results, which are new even in the case of matrices. Main techniques come from Algebraic Geometry, using Chern classes for enumerative computations.

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EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s147474802100027x ◽

2021 ◽

pp. 1-66

Author(s):

Tom Bachmann ◽

Kirsten Wickelgren

Keyword(s):

Commutative Algebra ◽

Vector Bundles ◽

Euler Class ◽

Complete Intersections ◽

Bilinear Forms ◽

Euler Numbers ◽

Coherent Sheaves ◽

Linear Subspaces ◽

Algebraic Vector

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

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Euler classes of vector bundles over manifolds

Mathematica Slovaca ◽

10.1515/ms-2017-0461 ◽

2021 ◽

Vol 71 (1) ◽

pp. 199-210

Author(s):

Aniruddha C. Naolekar

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Euler Class ◽

Homology Sphere ◽

Rational Homology ◽

Rational Homology Sphere ◽

Orientable Manifolds

Abstract Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

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