Centre Emile Borel

In cooperation with the European Research Training Network “Real Algebraic and Analytic Geometry”

Topology of Real Algebraic Varieties;
Semi-algebraic, Subanalytic and o-minimal Geometries

Paris, 05-09 December 2005

Abstracts

Back to the programme of the workshop

Frederic Bihan : Polynomial systems supported on circuits and dessins d'enfants
We study real solutions to polynomial systems whose equations have as common support a set, called a circuit, of n + 2 points in Zⁿ. We find bounds on the number of real solutions to such systems which depend on the rank modulo 2 of the circuit and the dimension of the affine span of the minimal affinely dependent subset of the circuit. We prove that these bounds are sharp by drawing so-called dessins d'enfant on the Riemann sphere. We also obtain that the maximal number of solutions with positive coordinates to such systems is n + 1, which is very small comparatively to the bound given by the Khovanskii fewnomial theorem.
Marcin Bilski : Approximation of analytic sets with proper projection by Nash sets
Let X be a complex analytic subset of U \times C^k of pure dimension n, where U is a Runge domain in Cⁿ. Assume that X is with proper and surjective projection onto U. We show that X can be approximated by complex Nash sets of pure dimension n. The approximation is expressed in terms of the convergence of analytic cycles.
Erwan Brugallé : Viro patchworking of real algebraic trigonal curves without convexity assumption
This is a joint work with Benoit Bertrand. The Viro method (for curves) is a way to glue real algebraic curves whose Newton polygons realize a subdivision $\tau$ of a polygon $\Delta$ to obtain a real algebraic curve with a controlled topology and whose Newton polygon is $\Delta$ .
One of the hypothesis of Viro's theorem is that the subdivision $\tau$ should be convex. I. Itenberg and E. Shustin have proved that it is possible to glue pseudoholomorphically the pieces of the patchwork along a pencil of lines. Hence, one obtains at least pseudoholomorphic curves with any subdivision. In this talk, I will explain how it is actually possible to glue all these pieces algebraically in the case of trigonal curves (i.e. when $\Delta$ is the triangle with vertices (0,0), (0,3) and (3n,0)). We glue these curves using Dessins d'enfants.
Fabrizio Catanese Complex and real structures on principal torus bundles
We shall consider principal holomorphic torus bundles over a torus: these are particular cases of quotients of a complex vector space by a group of affine transformations. If they admit a real structure, a natural question is to ask to which extent the real orbifold fundamental group $\Pi^{orb}$ determines the deformation type of these real varieties. In fact, also the action of $\Pi^{orb}$ on the universal cover is an affine group of dianalytic automorphisms, and one can give explicit parametrizations , by the socalled Appell Humbert families, of these complex and real structures. An interesting open question is to determine whether the Appell Humbert family gives all the possible deformations.
The problem is settled, (joint work with P. Frediani) in dimension 3, under a certain reality condition on the fundamental group $\Pi$ .
Alexander Degtyarev A Decomposability Inequality for Trigonal Curves (joint work with Ilia Itenberg and Viatcheslav Kharlamov}
The subject of our ongoing research is the study of real elliptic surfaces (roughly, one dimensional families of elliptic curves). To much extent this is reduced to a real version of Grothendick's dessins d'enfants, which are merely graphs enhanced with a certain decoration. The curves over a rational base have been studied by S. Orevkov, who has shown that the dessin of an M-curve decomposes into simple pieces (plane cubics). We show that this is a general phenomenon: for a fixed d, the (M - d)-curves whose dessin is indecomposable have bounded degree. (In the case of rational base the inequality has the form .) Alternatively, this means that an indecomposable dessin has few ovals (about 2/3 of the maximal number in the rational case) and, probably, have simple structure.
Nicolas Dutertre : Semi-algebraic neighbourhoods of closed semi-algebraic sets
Let X be a compact algebraic set in Rⁿ. It is well-known that X is the zero set of a positive polynomial f. However f is not necessarily proper. In [Du], Durfee explained how to construct from f a proper positive polynomial g such that X =g^-1(0). Then he proved that for $\delta$ > 0 sufficiently small, the inclusion X g^-¹([0, $\delta$ ]) is a homotopy equivalence. By Lojasiewicz's work [Lo1,Lo2], this inclusion is actually a retraction. Moreover, since van den Dries and Miller [DM] proved that every closed semi-algebraic set is the zero set of a C² semi-algebraic function, it is easy to see that Durfee's construction also works if X is a compact semi-algebraic set in Rⁿ.
In this talk, we explain how this result can be generalized to the case of a closed (not necessarily compact) semi-algebraic set X in Rⁿ. More precisely, we construct a positive semi-algebraic C² function g such that X =g^-¹(0) and such that for $\delta$ > 0 sufficiently small, the inclusion X g^-¹([0, $\delta$ ]) is a retraction. As a corollary, we give a degree formula for $\chi$ (X) and a Gauss-Bonnet formula for X.
[DM] L. van den DRIES, C. MILLER : Geometric categories and o-minimal structures, Duke Math. J., 84, No. 2 (1996), 497-540.
[Du] A.H. DURFEE: Neighborhoods of algebraic sets, Trans. Am. Math. Soc. 276, No. 2 (1983), 517-530.
[Lo1] S. LOJASIEWICZ : Une propriété topologique des sous-ensembles analytiques réels, Colloques Internationaux du CNRS, Les équations aux dérivées partielles, 117, éd. B. Malgrange (Paris 1962), Publications du CNRS, Paris, 1963.
[Lo2] S. LOJASIEWICZ : Sur les trajectoires du gradient d'une fonction analytique réelle, Seminari di Geometria 1982-83, Bologna, 1984, 115-117.
Abdelhafed El Khadiri : Weierstrass division Theorem in definable $C^\infty$ germs in a polynomially bounded o-minimal structure (joint work with Hassan Sfouli)
We give some examples of polynomially bounded o-minimal expansion R of the ordered field of real numbers where the Weierstrass Division Theorem does not hold in the ring of germs, at the origin of Rⁿ, of definable $C^\infty$ functions.
Sergei Finashin : Complex and real aspects of low-dimensional topology and the quotients by the complex conjugation
The principal objects of Low-dimensional topology (Braids and Links, Contact and Symplectic manifolds) can be considered as complex in their nature. Therefore, one can refine these objects by introducing a real structure and ask the usual questions that arise in the topology of real algebraic varieties. I plan to discuss these questions paying special attention to the related quotients by the complex conjugation.
Vincent Florens : Real plane algebraic curves and signatures of colored links
In the context of 16^th Hilbert's problem, Orevkov constructed a method that reduces the question of the existence of plane non-singular algebraic curves with prescribed topology to a classical problem of link theory: what smooth surfaces in the four-ball can be bounded by a given link in S³. He obtained several results by using the Murasugi-Tristram inequality, a necessary condition given in terms of Seifert forms and link signatures.
In this talk, we present a generalisation of link signatures to colored links in terms of generalised Seifert surfaces. We show that many remarkable properties of the classical invariants extend to this several variables generalisation. This provides in particular a new method of prohibition for real algebraic curves.
Andreas Gathmann : Kontsevich's formula in tropical geometry
Given 3d - 1 general points in the complex plane, how many rational curves of degree d are there that pass through all of them? Using Gromov-Witten theory these numbers can be computed using Kontsevich's formula. In this talk we will explain how the same results can be obtained entirely in the language of tropical geometry.
Joost van Hamel : Decomposition of Galois equivariant chain complexes
Zbignew Jelonek : Exotic embeddings of smooth affine varieties
We find examples of exotic embeddings of smooth affine varieties into Rⁿ or Cⁿ in large codimensions. We show also examples of affine smooth, rational algebraic varieties X, for which there are algebraically exotic embeddings $\phi : X \to X \times \C^1$ which are holomorphically trivial. Using this we construct an infinite family {C_2p+3} (p is a prime number) of complex manifolds, such that every C_2p+3 has at least two different algebraic (quasi-affine) structures. We show also that there is a natural connection between Abhyankar-Sathaye Conjecture and the famous Quillen-Suslin Theorem.
Tobias Kaiser: An o-minimal version of the Riemann mapping theorem
Let $\Omega \subset \mathbb{C}$ be a bounded semianalytic domain. Let $\varphi : \Omega \to B(0,1)$ be a Riemann map (i.e. a biholomorphic isomorphism) from the domain into the unit ball. Assume that the angles $\sphericalangle \partial\Omega_x \in \pi(\mathbb{R} \setminus \mathbb{Q})$ for all singular boundary points of $\Omega$ . Then $\varphi$ is definable in a (new) o-minimal structure.
Viatcheslav Kharlamov : Surfaces with DIF $\ne$ DEF real structures. (joint work with Vik. Kulikov, available at arXiv)
We solve the DIF = DEF problem for real structures by constructing real surfaces with diffeomorphic, but not deformation equivalent, real structures.
Giorgi Khimshiashvili On the mean topological invariants of random real polynomials
Several natural settings for calculating the mean topological invariants (i.e., the expected values of those invariants) of the fibres of random real polynomial mappings will be described. The discussion will be focused on calculating the mean mapping (topological) degree of a random polynomial endomorphism and mean Euler characteristic of a random projective hypersurface.
The main attention will be given to mappings defined by random real polynomials of fixed degree with Gaussian
coefficients. Two types of random polynomials will be discussed. Firstly, we consider the so-called standard homogeneous Gaussian random polynomial (SHGRP) in n indeterminates of degree d with the coefficients given by the independent identically distributed (i.i.d.) Gaussian random variables. Given such a SHGRP we wish to estimate the quantities EEZ(n,d) defined as the mean Euler characteristic of its projectivized level surfaces and EAD(n,d) defined as the mean absolute mapping degree of a random polynomial endomorphism
obtained by taking n i.i.d. SHGRP as above. In this setting, we'll find the rate of growth of EEZ(n,d) and EAD(n,d) as d tends to infinity. We'll also give an estimate for the mean absolute mapping degree EAGD(n,d) of the gradient of a SHGRP.
Secondly, more precise results will be presented for a GRP with the distribution of coefficients which is invariant with respect to the standard action of orthogonal group O(n) on the space of d-forms. For such GRP and n even, we'll give an exact formula for EGD(n,d) which at the same time yields an estimate for the rate of growth of EAGD(n,d).
Some applications of these results will be outlined. For example, one can give lower estimates for the average crossing number and mean self-linking number of random polygonal knots of various types. Another type of application is concerned with estimating the mean number of cusps of random planar endomorphisms. A number of related conceptual problems will be also discussed.
Krzysztof Kurdyka Algebraicity of global real analytic hypersurfaces
Let X be an algebraic manifold without compact component and let V be a compact coherent analytic hypersurface in X, with finite singular set. We prove that V is diffeotopic (in X) to an algebraic hypersurface in X if and only if the homology class represented by V is algebraic and singularities are locally analytically equivalent to Nash singularities. This allows us to construct algebraic hypersurfaces in X with prescribed Nash singularities. Joint work with Wojciech Kucharz.
Oliver Labs Real Line Arrangements and Hypersurfaces with Many Real Nodes
A long standing question is if maximum number $\mu(d)$ of nodes on a surface of degree d in P³(C) can be achieved by a surface defined over the reals which has only real singularities. The currently best known asymptotic lower bound, $\mu(d) \gtrapprox \frac{5}{12}d^3$ , is provided by Chmutov's construction from 1992 which gives surfaces whose nodes have non-real coordinates.
Using explicit constructions of certain real line arrangements we show that Chmutov's construction can be adapted to give only real singularities. All currently best known constructions which
exceed Chmutov's lower bound (i.e., for d = 3, 4, ..., 8, 10, 12) can also be realized with only real singularities. Thus, our result shows that, up to now, all known lower bounds can be attained with only real singularities.
We conclude with an application of the theory of real line arrangements which shows that our arrangements are aymptotically the best possible ones.
This proves a special case of a conjecture of Chmutov.
François Loeser : Motivic constructible functions
We shall present our joint work with Raf Cluckers on motivic constructible functions, with special emphasis on the analogies with the o-minimal setting. Our work provides in particular a new, more general, construction of the motivic measure.
Lucia Lopez de Medrano : Critical points of a T-hypersurface with respect to the hyperplanes orthogonal at one axis.
We are going to describe how to find and how to know the index, in the sense of the Morse theory, of the critical points of a T-hypersurface with respect to the intersection with the set of hyperplanes orthogonal at one chosen coordinate axis. We are going to see that these are generalizations of results by Itenberg and Shustin about the critical points of a T-polynomial.
Frédéric Mangolte : Recent advances in the topological classification of real uniruled projective 3-folds
I will talk about recent advances in the topological classification of real uniruled projective 3-folds. In particular I will prove that any connected sum of lens spaces is diffeomorphic to a real component of a uniruled real algebraic variety.
This was conjectured by János Kollár.
This new result extends a former result about the realizability of orientable Seifert 3-manifolds by projective uniruled manifolds which was also conjectured by J. Kollár.
(Joint work with J. Huisman.)
Grigory Mikhalkin : Planar log-fronts and Harnack curves
Dorota Mozyrska : Real analytic Nullstellensatz in
In 1952 S. Lang [6] extended Hilbert’s Nullstellensatz to polynomials of infinitely many variables. On the other hand, W. Rückert [11], in 1932, instead of polynomials took germs of complex analytic functions of finitely many variables; his Nullstellensatz involved germs of complex analytic sets. The real case for polynomials of finitely many variables was independently solved by J.-L. Krivine [5], D.W. Dubois [4] and J.-J. Risler [9] in the sixties. In their theorem of zeros, the ordinary radical of an ideal, used in the complex version, had to be changed for the real radical. Finally, in 1976 Risler [10] proved finite-dimensional real analytic counterpart of Hilbert’s Nullstellensatz. It is important to notice that the ring of the germs of analytic functions of finitely many complex or real variables (at some point) is Noetherian. Hence, if I is an ideal of this ring, then the germ of zero-set of I is well defined as I is finitely generated.
In this talk we study the infinite-dimensional analytic case, where an analytic function depends (like Lang’s polynomials) only on a finite number of variables (is finitely presented). But, as the number of all variables is infinite, the ring of the germs of such functions is no longer Noetherian and the germs of the zero sets of ideals cannot be defined in the standard way. Moreover there is no topology in the infinite-dimensional space of all real sequences that would give required properties of the germs of sets. We have been interested there in “local” solutions of infinitely many analytic equations in infinitely many variables. Such equations describe, for instance, indistinguishable states of a (control) system with output and are related to observability and local observability of the system (see e.g. [1, 2] for the finite-dimensional case and [7, 8] for the infinite-dimensional one). In particular, it is important in local observability whether such equations have locally only one solution (which is the point at which we localize the system and the equations).
Instead of using topology to define the germ of a set, we consider special families of finite dimensional set-germs (systems of germs) which approximate in some sense what we want to be an infinite-dimensional set-germ. Systems of germs give rise to a concept of multigerm — the equivalence class of such systems under a natural equivalence relation. We show that it is the right language in this infinite-dimensional world. We can manipulate with multigerms in the same way as we do with finite-dimensional germs. In particular, we can define multigerm of zeros corresponding to an ideal of the ring of germs of finitely presented analytic functions and, conversely, zero ideal of a multigerm. We consider the real and the complex cases. The main result consists of real theorem of zeros, where we show that the real radical of an ideal consists exactly of the germs of finitely presented analytic functions that vanish on the multigerm of zeros of the ideal.
[1] Z. Bartosiewicz, Local observability of nonlinear systems, Systems & Control Letters 25 (1995), 295–298.
[2] Z. Bartosiewicz, Real analytic geometry and local observability, Proc. Sympos. Pure Math. 64 (1998), Amer. Math. Soc., Providence, RI.
[3] J. Bochnak, M. Coste, and M.-F. Roy, Géométrie algébrique réelle, Berlin Heidelberg New York: Springer-Verlag 1987.
[4] D.W. Dubois, A Nullstellensatz for ordered fields, Ark. Mat. 8 (1969), 111-114.
[5] J.-L. Krivine, Anneaux prÂ´eordonne, J. Analyse Math. 12 (1964), 307- 326.
[6] S. Lang, Hilbert’s Nullstellensatz in infinite-dimensional space, Proc. Amer. Math. Soc. 3 (1952), 407–410.
[7] D. Mozyrska, Local observability of infinitely-dimensional finitely presented dynamical systems with output (in Polish), Ph.D. thesis, Technical University of Warsaw, Poland, 2000.
Wieslaw Pawlucki : A continuous linear extension operator for Lipschitz functions on o-minimal sets.
Let E be a subset of an Euclidean space definable in an o-minimal structure. A continuous linear extension operator for Lipschitz functions on E preserving definability will be presented.
Grigory M. Polotovskiy : On arrangements of a plane real quintic curve with respect to a pair of straight lines
A classification of arrangements from the title is a fragment of a more general problem about topology of real decomposable curves. This last problem belongs in turn to the topics of the Hilbert 16th problem.
Let RC_n be the set of real points of a plane real curve C_n of degree n in the real projective plane RP². Our problem consists in the topological classification of quadruples

(1)
(RP², RC₅ $\cup$ RC₁ $\cup$ RC₁', RC₅ $\cup$ RC₁, RC₅)

under the following conditions:
1. C₅ is an M-curve, i.e. RC₅ consists of the odd branch J and six ovals lying outside each other.
2. The curves RC₅, RC₁, RC₁' are in general position, i.e. they have transversal intersection and have no concurrent points.
3. Each of the lines RC₁, RC₁' intersects the curve RC₅ at five points lying on the odd branch J of the curve RC₅.
The set RP² \ (RC₁ RC₁') consists of two open disk. Denote their closures as D₁ and D₂. Each of the sets D_i J, i in {1,2}, consists of five arcs whose terminal points lie on the lines RC₁ and RC₁'. Let a_i be the numbers of arcs in the disk D_i whose terminal points do not belong to the same line. The case a₁ = a₂ = 1 was solved in [1]. Here the case a₁=1, a₂=3 will be discussed.
There are 124 admissible topological models for quadruples (1). We advanced in the realization of 29 and in prohibitions of 63 models. The question about realizability of the rest 32 models is still open.
The restrictions are obtained by means of the Orevkov's approach based on the links theory and the method of constructions is patchworking. The constructions belong to A.B. Korchagin and this work is joint work with him.
[1] A. B. Korchagin, G. M. Polotovskii, On arrangements of a plane real quintic curve with respect to a pair of lines, Communication in contemporary math., Vol. 5, No.1 (2003), 1--24.
Jean-Philippe Rolin : Non oscillating curves and o-minimality
This talk will be a survey on the relationship between non oscillating curves (especially solutions of analytic differential equations), Hardy fields and o-minimality. We will recall recall some
recent results and a few open questions.
Boris Shapiro : On conjecture of total reality in the case of higher genera curves
The 1-dimensional case of the conjecture on total reality (sometimes referred to as the Shapiro&Shapiro conjecture) claims that if a meromorphic function on a given real algebraic curve has all real critical values then the function itself is real up to a linear-fractional transformation in the image. This conjecture in the case of a rational underlying curve was settled by Eremenko and Gabrielov about 6 years ago. We present some partial results in the case of underlying curves of higher genera. In particular, we prove it for the hyperelliptic case and for small degrees of the function in the case when the genus is fixed. (Joint with T.Ekedahl and M.Shapiro).
Domingo Toledo : Hyperbolic Geometry and Invariants of Real Cubic Surfaces
This lecture reports on joint work with Allcock and Carlson. We explain how classical invariants of real cubic surfaces can be computed from the hyperbolic geometry of the moduli space. In particular we compute the monodromy of each component of the real moduli space and compare these computations with the classical computations by Segre.
Pilar Velez : Semilinear sets over ordered fields: the convergence of Convex Geometry and Semialgebraic Geometry
Semilinear sets are sets defined by boolean combinations of linear inequalities, that is, the family of sets obtained from closed halfspaces using finite unions, intersections and complement, and constitute the smallest o-minimal family over an ordered field K.
Some properties of classical convex geometry over the real numbers do not require any topological assumption and therefore remain valid over any ordered field, but some others are only true over the reals. For instance, a classical result of real convex geometry, which is false over arbitrary K, is that two closed disjoint convex subsets can be separated by a hyperplane. We rescue these kind of properties restricting ourselves to the subfamily of semilinear sets defined over K and using topological notions from convex geometry, as core point or linearly accessible point. By the way we show, using these convex geometry tools, some other results from semialgebraic geometry, as finiteness property of semilinear sets.
Victor Vinnikov : Lax Conjecture, Linear Matrix Inequality Representation of Convex Sets, and Generalizations
A homogeneous polynomial P in R[x₀, x₁,..., x_m] of degree n is called hyperbolic in direction c in P^m_R if every real line through c intersects the real hypersurface P(x) = 0 in real points only.
If P is irreducible and the hypersurface P(x) = 0 is smooth, this is equivalent to P(x) = 0 being rigidly isotopic to a nest of [n/2] spheres around c (and an additional hyperplane if n is odd).
In 1958, Lax conjectured that for m = 2, a polynomial is hyperbolic in direction c if and only if P(x₀, x₁, x₂) = det(x₀A₀ + x₁A₁ + x₂A₂) where A₀, A₁, A₂ are n n real symmetric matrices with c₀A₀ + c₁A₁ + c₂A₂> 0. This was recently established by Lewis, Parillo, and Ramana based on a result of Helton and Vinnikov (a preliminary result along these lines has been obtained earlier by Dubrovin). A closely related question that came to be of central importance in control theory is when can one represent a convex set in R² with an algebraic boundary as the positivity set of a linear matrix pencil. The proofs are based on a careful analysis of the real structure on the Jacobian variety.
I shall discuss these results and speculate about some related issues and especially about higher dimensional generalizations.