International Congress of Mathematicians

Algebraic Structures on Valuations, Their Properties and Applications 761By the results of Section 1 (Val(V)) sm is a subalgebra of (PVal(V)) sm . It iseasy to see that the algebra structure is compatible with the grading, namelyIn particular we have(Vali(V)) sm CS) (Valj(V)) sm —• (Val ì+ j(Vj) sm .(Vali(Vj) sm ® (Val n -i(Vj) sm —• Dens(V).A version of the Poincaré duality theorem says that this is a perfect pairing. Aloreprecisely2.1.4 Theorem ([5]). The induced mapis an isomorphism.(Vali(Vj) sm —• (Val n -i(V)*) sm ® Dens(V)2.2. Even translation invariant continuous valuationsLet us denote by Val ev (V) the subspace of even translation invariant continuousvaluations. Then clearly (Val ev (V)) sm is a subalgebra of (Val(V)) sm . It turnsout that it satisfies a version of the hard Lefschetz theorem which we are going todescribe.Let us fix on V a scalar product. Let D denote the unit ball with respect tothis product. Let us define an operator A : Val(V) —y Val(V). For a valuation £ Val(V) set(AcP)(K):=^\ e=Q cP(K+ eD).(Note that by a result of P. McMullen [17] (K + sD) is a polynomial in e > 0 ofdegree at most n.) It is easy to see that A preserves the parity of valuations anddecreases the degree of homogeneity by 1. In particularA:Vair(V)^Val^Li(V).The following result is a version of the hard Lefschetz theorem.2.2.1 Theorem ([4]). Let k > n/2. ThenA2k-n . (Vall v (V)y m —• (Val e fi_ k (Vj) smis an isomorphism. In particular for 1 < i < 2k — n the mapis injective.A* : (Val e k V (V)) sm —• (VallfifiV)) 8 " 1Note that the proof of this result is based on the solution of the cosine transformproblem due to J. Bernstein and the author [6], which is the problem from(Gelfand style) integral geometry motivated by stochastic geometry and going backto G. Alatheron [16].