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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The kinematic formula in complex integral geometry
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by Theodore Shifrin PDF
Trans. Amer. Math. Soc. 264 (1981), 255-293 Request permission

Erratum: Trans. Amer. Math. Soc. 266 (1981), 667.

Abstract:

Given two nonsingular projective algebraic varieties $X,Y \subset {{\mathbf {P}}^n}$, $Y \subset {{\mathbf {P}}^n}$ meeting transversely, it is classical that one may express the Chern classes of their intersection $X \cap Y$ in terms of the Chern classes of $X$ and $Y$ and the Kähler form (hyperplane class) of ${{\mathbf {P}}^n}$. This depends on global considerations. However, by putting a hermitian connection on the tangent bundle of $X$, we may interpret the Chern classes of $X$ as invariant polynomials in the curvature form of the connection. Armed with this local formulation of Chern classes, we now consider two complex submanifolds (not necessarily compact) $X$, $Y \subset {{\mathbf {P}}^n}$, and investigate the geometry of their intersection. The pointwise relation between the Chern forms of $X \cap Y$ and those of $X$ and $Y$ is rather complicated. However, when we average integrals of Chern forms of $X \cap gY$ over all elements $g$ of the group of motions of ${{\mathbf {P}}^n}$, these can be expressed in a universal fashion in terms of integrals of Chern forms of $X$ and $Y$. This is, then, the kinematic formula for the unitary group.
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Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 264 (1981), 255-293
  • MSC: Primary 53C65
  • DOI: https://doi.org/10.1090/S0002-9947-1981-0603763-8
  • MathSciNet review: 603763