The topology of two-dimensional real algebraic varieties

Sunto È noto che ogni spazio analitico reale è localmente omeomorfo al cono su un poliedro con caratteristica di Eulero-Poincaré pari. Si dimostra che questa condizione è anche sufficiente affinchè un poliedro (compatto) di dimensione due P sia omeomorfo ad una varietà algebrica reale affine P. Segue inoltre dalla costruzione che la P ottenuta ha, in un certo senso, un insieme di singolarità algebriche minimale, compatibilmente con la topologia di P.

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The authors are members of the G.N.S.A.G.A.     

Comparing motives of smooth algebraic varieties

Given a perfect field of exponential characteristic e, the Cor-, $K_0^{\oplus }$-, $K_0$- and ${\mathbb{K}}_0$-motives of smooth algebraic varieties with $\mathbb{Z}[1/e]$-coefficients are shown to be locally quasi-isomorphic to each other. Moreover, it is proved that their triangulated categories of motives with $\mathbb{Z}[1/e]$-coefficients are equivalent. An application is given for the bivariant motivic spectral sequence.     

The topology of Birkhoff varieties

Our main theorem is that the inclusion of any Birkhoff subvariety in an affine flag variety is a homotopy equivalence. We also construct analogues of tubular neighborhoods for Birkhoff and Schubert varieties in affine or classical flag varieties.     

Constructive algebraic topology

The classical “computation” methods in Algebraic Topology most often work by means of highly infinite objects and in fact are not constructive. Typical examples are shown to describe the nature of the problem. The Rubio-Sergeraert solution for Constructive Algebraic Topology is recalled. This is not only a theoretical solution: the concrete computer program Kenzo has been written down which precisely follows this method. This program has been used in various cases, opening new research subjects and producing in several cases significant results unreachable by hand. In particular the Kenzo program can compute the first homotopy groups of a simply connected arbitrary simplicial set.     

On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties

We prove that for any affine variety S defined overQ there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety X(G, PO(3)) = Hom(G, PO(3))//PO(3). The subset U contains all real points of S. As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties. This research was partially supported by NSF grant DMS-96-26633 at University of Utah. This research was partially supported by NSF grant DMS-95-04193 at University of Maryland.     

Asymptotic syzygies of algebraic varieties

We study the asymptotic behavior of the syzygies of a smooth projective variety as the positivity of the embedding line bundle grows. The main result asserts that the syzygy modules are non-zero in almost all degrees allowed by Castelnuovo?CMumford regularity. We also give an effective statement for Veronese varieties that we conjecture to be optimal.     

Endomorphisms of symbolic algebraic varieties

The theorem of Ax says that any regular selfmapping of a complex algebraic variety is either surjective or non-injective; this property is called surjunctivity and investigated in the present paper in the category of proregular mappings of proalgebraic spaces. We show that such maps are surjunctive if they commute with sufficiently large automorphism groups. Of particular interest is the case of proalgebraic varieties over infinite graphs. The paper intends to bring out relations between model theory, algebraic geometry, and symbolic dynamics. Received August 3, 1998 / final version received January 22, 1999     

Algebra-geometry of piecewise algebraic varieties

Algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a kind generalization of the classical algebraic variety. This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines.     

Pure homology of algebraic varieties

We show that for a complete complex algebraic variety the pure term of the weight filtration in homology coincides with the image of intersection homology. Therefore pure homology is topologically invariant. To obtain slightly more general results we introduce image homology for noncomplete varieties.