Sums of squares on real algebraic surfaces

Consider real polynomials g₁, . . . , g_r in n variables, and assume that the subset K = {g₁≥0, . . . , g_r≥0} of ℝⁿ is compact. We show that a polynomial f has a representation in which the s_e are sums of squares, if and only if the same is true in every localization of the polynomial ring by a maximal ideal. We apply this result to provide large and concrete families of cases in which dim (K) = 2 and every polynomial f with f|_K≥0 has a representation (*). Before, it was not known whether a single such example exists. Further geometric and arithmetic applications are given. Support by DFG travel grant KON 1823/2002 and by the European RAAG network HPRN-CT-2001-00271 is gratefully acknowledged. Part of this work was done while the author enjoyed a stay at MSRI Berkeley. He would like to thank the institute for the invitation and the very pleasant working conditions.     

Linear pencils on real algebraic curves

Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k(R) be the smallest degree of a pencil on X (the real gonality of X). Then we can find on X a base point free pencil of degree g+1 (resp. g if X is not hyperelliptic, i.e. if k(R)>2) with an assigned geometric behaviour w.r.t. the real components of X, and if we prove that which is the same bound as for the gonality of a complex curve of even genus g. Furthermore, if the complexification of X is a k-gonal curve (k≥2) one knows that k≤k(R)≤2k−2, and we show that for any two integers k≥2 and 0≤n≤k−2 there is a real curve with real points and k-gonal complexification such that its real gonality is k+n.     

On line bundles over real algebraic curves

Let X be a geometrically irreducible smooth projective curve defined over the real numbers. Let nX be the number of connected components of the locus of real points of X. Let x₁,…,x? be real points from ? distinct components, with ?<nX. We prove that the divisor x₁+?+x? is rigid. We also give a very simple proof of the Harnack's inequality.     

Fixed points of automorphisms of real algebraic curves

We bound the number of fixed points of an automorphism of a real curve in terms of the genus and the number of connected components of the real part of the curve. Using this bound, we derive some consequences concerning the maximum order of an automorphism and the maximum order of an abelian group of automorphisms of a real curve. We also bound the full group of automorphisms of a real hyperelliptic curve. Work supported by the European Community’s Human Potential Programme under contract HPRN-CT-2001-00271, RAAG.     

Witt equivalence of algebraic function fields over real closed fields

We examine the conditions for two algebraic function fields over real closed fields to be Witt equivalent. We show that there are only two Witt classes of algebraic function fields with a fixed real closed field of constants: real and non-real ones. The first of them splits further into subclasses corresponding to the tame equivalence. This condition has a natural interpretation in terms of both: orderings (the associated Harrison isomorphism maps 1-pt fans onto 1-pt fans), and geometry and topology of associated real curves (the bijection of points is a homeomorphism and these two curves have the same number of semi-algebraically connected components). Finally, we derive some immediate consequences of those theorems. In particular we describe all the Witt classes of algebraic function fields of genus 0 and 1 over the fixed real closed field. Received: 16 February 2000; in final form: 7 December 2000 / Published online: 18 January 2002     

Absolutely split real algebraic vector bundles over a real form of projective space

Let M be a projective variety, defined over the field of real numbers, with the property that the base change MR_×C is isomorphic to CPN for some N. A real algebraic vector bundle E over M will be called absolutely split if the vector bundle ER_⊗C over MR_×C splits into a direct sum of line bundles. We classify the isomorphism classes of absolutely split vector bundles over M.     

Rational surfaces in algebraic models of smooth manifolds

Every compact smooth manifold M is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of M. We study modulo 2 homology classes represented by rational algebraic surfaces in X, as X runs through the class of all algebraic models of M. Received: 16 June 2007     

Every orientable Seifert 3-manifold is a real component of a uniruled algebraic variety

We show that any orientable Seifert 3-manifold is diffeomorphic to a connected component of the set of real points of a uniruled real algebraic variety, and prove a conjecture of János Kollár.     

Torsion cohomology classes and algebraic cycles on complex projective manifolds

Atiyah and Hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold, which are not Poincaré dual to an algebraic cycle. We notice that the order of these classes must be small compared to the dimension of the manifold. However, building upon a construction of Kollár, one can provide such examples with arbitrary high prime order, the dimension being fixed. This method also provides examples of torsion algebraic cycles, which are non trivial in the Griffiths group, and lie in a arbitrary high level of the H. Saito filtration on Chow groups.     

On Harder-Narasimhan strata in flag manifolds

This paper deals with a question of Fontaine and Rapoport which was posed in [1]. They asked for the determination of the index set of the Harder-Narasimhan vectors of the filtered isocrystals with fixed Newton- and Hodge vector. The aim of this paper is to give an answer to their question.     

On the automorphism groups of algebraic bounded domains

This paper is a part of author's doctoral thesis. It was partially supported by Sonderforschungsbereich 237 Unordnung und große Fluktuatuionen of the Deutsche Forschungsgemeinschaft     

Monodromy of real isolated singularities

Complex conjugation on complex space permutes the level sets of a real polynomial function and induces involutions on level sets corresponding to real values. For isolated complex hypersurface singularities with real defining equation we show the existence of a monodromy vector field such that complex conjugation intertwines the local monodromy diffeomorphism with its inverse. In particular, it follows that the geometric monodromy is the composition of the involution induced by complex conjugation and another involution. This topological property holds for all isolated complex plane curve singularities. Using real morsifications, we compute the action of complex conjugation and of the other involution on the Milnor fiber of real plane curve singularities.     

Computing complex and real tropical curves using monodromy

Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials with constant coefficients. These algorithms rely on homotopy continuation, monodromy loops, and Cauchy integrals. Several examples are presented which are computed using an implementation that builds on the numerical algebraic geometry software Bertini.     

Regular versus continuous rational maps

We prove that a continuous map from a compact nonsingular real algebraic variety X into the unit 2-sphere can be approximated by regular maps if and only if it is homotopic to a continuous map which is regular in the complement of a Zariski closed subvariety A of X of codimension at least 3. The assumption on the codimension of A is essential.     

Algebraically constructible functions and signs of polynomials

LetW be a real algebraic set. We show that the following families of integer-valued functions onW coincide: (i) the functions of the formω →λ(X _ω ), where X _ω are the fibres of a regular morphismf :X →W of real algebraic sets, (ii) the functions of the formω →χ(X _ω ), where X _ω are the fibres of a proper regular morphismf :X →W of real algebraic sets, (iii) the finite sums of signs of polynomials onW. Such functions are called algebraically constructible onW. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization. Research partially supported by an Australian Research Council Small Grant. Second author also partially supported by KBN 610/P3/94.