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Transactions of the American Mathematical Society

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The kinematic formula in complex integral geometry

Author: Theodore Shifrin
Journal: Trans. Amer. Math. Soc. 264 (1981), 255-293
MSC: Primary 53C65
Erratum: Trans. Amer. Math. Soc. 266 (1981), 667.
MathSciNet review: 603763
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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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Keywords: Hermitian geometry, connections, curvature, Chern classes, Grassmannians, vector bundles, kinematic formula
Article copyright: © Copyright 1981 American Mathematical Society