Harnack-Thom Inequalities for Mappings of Real Algebraic Varieties

Abstract

We consider the interaction of an atomic beam with a single-mode quantized radiation field inside a cavity and an injected classical field. Under suitable conditions, this can be used as a scheme to measure fluctuations, correlations and the average photon number of the quantum field. The proposed scheme is intended for a microwave regime, though optical implementation may be possible. No assumption is required on the type of state of the quantum field.

Picard and Lefschetz numbers of real algebraic surfaces

“…(19) follow from the definition of the Lefschetz numbers, and Eq. (20) was established in [6]. The proof of the theorem is complete.…”

“…The equations (see [6]) dim H*(X(R), F2) = 2 + dimH2(X(C), ~"2) --2k, dim H*(X(R), F2) = 2 + dimH' (G, H2(X(C), F2))…”

“…(7) Let H+ and H_ be, respectively, the g-invariant and the g-anti-invariant subgroup of H. Then from (6) and (7) we obtain p(X) = rk(n_ n HtJ(X(C))).…”

Topological properties of real algebraic varieties: du coté de chez Rokhlin

“…If A.1.3(3) turns into an equality (which is equivalent to Im(tr * + in * ) ⊃ Ker(1 + c * )), c is called (Z 2 -)Galois maximal. (This terminology is introduced by V. A. Krasnov [Kr1]. R. Thom [Th1] calls a dimension p ∈ N regular for (X, c) if Im(tr p + in p ) ⊃ Ker(1 + c p ).)…”

Smith theory and irreducible holomorphic symplectic manifolds

“…degenerates (see Krasnov [22,23] for some examples of GM and non-GM varieties). In this section, we prove some degeneracy conditions that will be useful for certain holomorphic symplectic varieties.…”