Hard Lefschetz Theorem for Valuations, Complex Integral Geometry, and Unitarily Invariant Valuations

Abstract

We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others. 0 Introduction.In this paper we obtain new results on the structure of the space of even translation invariant continuous valuations on convex sets. In particular we prove a version of hard Lefschetz theorem for them and introduce certain natural duality operator which establishes an isomorphism between the space of such valuations on a linear space V and on its dual V * (with an appropriate twisting). Then we obtain an explicit geometric classification of unitarily invariant translation invariant continuous valuations on a Hermitian space C n . This classification is used to deduce new integral geometric formulas for real submanifolds in Hermitian spaces generalizing the classical kinematic formulas in Euclidean spaces due to Poincaré, Chern, Santaló, and others.

A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic line

“…This finishes the proof in the case where 1 and 2 are of type (2). By linearity of both sides, (3) holds true for linear combinations of such valuations.…”

Hermitian integral geometry

“…where π E is the orthogonal projection to E. This follows from the AleskerBernstein theorem [9] (compare also §1 in [1]). …”

“…In this form, the Alesker-Fourier transform was denoted by D in [1], [13] and in several other papers.…”

“…More recently, S. Alesker [1] gave another proof of the theorem of [17] as part of a far-reaching reconceptualization of the theory of convex valuations. He showed that if G is a compact Lie group acting transitively on the sphere of a euclidean space V , then the space Val G (V ) of continuous convex valuations invariant under the group G := G V , generated by translations and the action of G, is finite dimensional.…”

“…In these terms, the result may be stated as follows. Moreover, in [1] Alesker gave an explicit basis (in fact two of them) for the space Val U (n) (C n ) of unitary-invariant valuations on C n . Although this in itself gives a lot of information about the kinematic formulas, a complete determination of the formulas using the template method appears intractable (although H. Park [28] used it successfully in the cases n = 2, 3).…”