Two subanalytic subsets of Rn are s-equivalent at a common point, say O, if the Hausdorff distance between their intersections with the sphere centered at O of radius r goes to zero faster than rs. In the present paper we investigate the existence of an algebraic representative in every sequivalence class of subanalytic sets. First we prove that such a result holds for the zero-set V (f) of an ...

Atiyah and Hirzebruch gave the first counterexamples to the Hodge conjecture with integer coefficients. In particular, there is a smooth complex projective variety X of dimension 7 and a torsion element of H(X,Z) which is not the class of a codimension-2 algebraic cycle [4]. In this paper, we provide a more systematic explanation for their examples: for every smooth complex algebraic variety X ...

Recall that a subset of R is called semi-algebraic if it can be represented as a (finite) boolean combination of sets of the form {~ α ∈ R : p(~ α) = 0}, {~ α ∈ R : q(~ α) > 0} where p(~x), q(~x) are n-variable polynomials with real coefficients. A map from R to R is called semi-algebraic if its graph, considered as a subset of R, is so. The geometry of such sets and maps (“semi-algebraic geome...

We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes. To the memory of Masayoshi Nagata

We prove that a germ of a holomorphic map f between Cn and Cn ′ sending one real-algebraic submanifold M ⊂ Cn into another M ′ ⊂ Cn ′ is algebraic provided M ′ contains no complex-analytic discs and M is generic and minimal. We also propose an algorithm for finding complex-analytic discs in a real submanifold.

We prove that the classical integral cycle class map from algebraic cycles to étale cohomology factors through a quotient of `-adic étale cobordism over an algebraically closed field of positive characteristic. This shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah, Hirzebruch and Totaro also work in positive

Let C be an algebraic curve of genus g ≥ 3. Let us assume that C is not hyperelliptic, so that it is isomorphic to its image by the canonical map φ : C → P. Enriques proved in [1] that φ(C) is the intersection of the quadrics that contain it, except when C is trigonal (that is, it has a g 3) or C is isomorphic to a plane quintic (g = 6). The proof was completed by Babbage [2], and later Petri p...

In this paper we solve for the quantum propagator of a general time dependent system quadratic in both position and momentum, linearly coupled to an infinite bath of harmonic oscillators. We work in the regime where the quantum optical master equation is valid. We map this master equation to a Schroedinger equation on Super-Hilbert space and utilize Lie Algebraic techniques to solve for the dyn...