2014 CIMPA School on
Real Algebraic Geometry
Villa de Leyva (Colombia)
14-25 July
Organizing committee:
• Erwan Brugallé, Ecole Polytechnique, Palaiseau (France).
• César Galindo, Universidad de Los Andes, Bogotá (Colombia).
• Eddy Pariguan, Pontificia Universidad Javeriana, Bogotá (Colombia).
• Florent Schaffhauser, Universidad de Los Andes, Bogotá (Colombia).
Scientific committee:
• Alicia Dickenstein, Universidad de Buenos Aires, Buenos Aires (Argentina).
• Johannes Huisman, Université de Brest, Brest (France).
• Chiu-Chu Melissa Liu, Columbia University, New York City (USA).
• Jean-Jacques Risler, Université Pierre et Marie Curie, Paris(France).
Contents
1. Lectures
1.1. Mini-courses
1.2. Research talks
2. Abstracts
2.1. Mini-courses
2.2. Research talks
3. Schedule
3.1. Week 1
3.2. Week 2
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2
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2
4
8
8
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1. Lectures
1.1. Mini-courses.
• Erwan Brugallé, École Polytechnique, Palaiseau (France).
• Rubén Hidalgo, Universidad Técnica Federico Santa María, Valparaíso (Chile).
• Ilia Itenberg, Université Pierre et Marie Curie, Paris (France).
• Florent Schaffhauser, Universidad de Los Andes, Bogotá (Colombia).
• Jean-Yves Welschinger, CNRS - Université Lyon-1, Lyon (France).
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2
1.2. Research talks.
• Benoît Bertrand, Université Paul Sabatier, Toulouse (France).
• Alicia Dickenstein, Universidad de Buenos Aires, Buenos Aires (Argentina).
• Vinicio Gómez, UNAM, Ciudad de México D.F. (México).
• Rubén Hidalgo, Universidad Técnica Federico Santa María, Valparaíso (Chile).
• Johannes Huisman, Université de Brest, Brest (France).
• Ilia Itenberg, Université Pierre et Marie Curie, Paris (France).
• Chiu-Chu Melissa Liu, Columbia University, New York City (USA).
• Lucía López de Medrano, UNAM, Cuernavaca (México).
• Alf Onshuus, Universidad de Los Andes, Bogotá (Colombia).
• Nicolas Puignau, Universidade Federal do Rio de Janeiro, Rio de
Janeiro (Brasil).
• Jean-Jacques Risler, Université Pierre et Marie Curie, Paris(France).
• Nermin Salepci, Université Lyon-1, Lyon (France).
• Mauricio Velasco, Universidad de Los Andes, Bogotá (Colombia).
• Shuguang Wang, University of Missouri, Columbia (USA).
2. Abstracts
2.1. Mini-courses.
• Erwan Brugallé, Tropical geometry and applications to real algebraic geometry.
A tropical curve may be seen as a combinatorial way to encode
a degeneration of complex algebraic curves. The main advantage of
tropical curves is that they are much more easier to study than complex curves. A central problem in tropical geometry is to understand
which properties of tropical objects can be lifted to classical geometry. Viro’s patchworking Theorem, which allows to "glue" plane real
algebraic curves, is an example of such theorems relating tropical to
classical geometry. In this course I will give an introduction to tropical geometry, and discuss its connections to real and enumerative
algebraic geometry in relation with the course of I. Itenberg.
• Rubén Hidalgo, Dessins d’enfants y curvas de Belyi reales.
Uno de los grupos más interesantes y poco entendido es el grupo
de Galois absoluto G = Gal(cl(Q)/Q). Como cl(Q) es la unión
de todas las extensiones de Galois finitas de Q, se tiene que G es
el límite inverso de los grupos de Galois finitos, es decir, G es un
grupo profinito. En este grupo tenemos toda la teoría clásica de
Galois. En su "Esquisse d’un Programme", Grothendieck observó
que el grupo G tiene una acción natural en ciertos objetos combinatorios llamados "Dessins d’enfants". Tal acción resulta ser una
acción fiel y la idea de Grothendieck era estudiar la estructura de G
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por medio de tal acción. La idea de este curso es definir los dessins
d’enfants, relacionarlos con superficies de Riemann (curvas de Belyi)
y funciones de Belyi (funciones meromorfas no constantes con a lo
más tres valores de ramificación) y bosquejar la acción de G en estos objetos. Casos interesantes de curvas de Belyi son aquellas que
además admiten estructuras reales, es decir, admiten una simetría
(automorfismo anticonforme de orden 2). Un resultado reciente de
Koeck-Lau-Singerman nos asegura que toda curva de Belyi real se
puede definir en la intersección de cl(Q) con los números reales. En
este curso daremos una demostración constructiva de tal hecho.
• Ilia Itenberg, Topological properties of real algebraic varieties.
The purpose of the course is to give an introduction to topology
of real algebraic varieties and to present several central results in
this domain. We will start with a study of algebraic curves in RP 2
and algebraic surfaces in RP 3 (questions about the topology of these
algebraic varieties were put by D. Hilbert in the first part of his 16th
problem). We will speak about various restrictions on the topology
of real algebraic curves and real algebraic surfaces, as well as about
constructions of these varieties. We will put a particular attention to
the Viro patchworking, a powerful construction of algebraic varieties
which is directly related to tropical geometry.
• Florent Schaffhauser, Fundamental groups in real algebraic geometry.
The algebraic fundamental group of a real algebraic variety classifies the finite étale covering spaces of said variety and is a birational
invariant of it. The goal of this course is to give an introduction to
the discrete analogue of the algebraic fundamental group of a real
algebraic variety, emphasizing the case of curves. After defining the
fundamental group of a smooth real algebraic variety, we shall study
linear representations of this group in the case of one-dimensional
varieties. We shall in particular explain the real version of a theorem
Narasimhan and Seshadri which sets up a correspondence between
unitary representations of the fundamental group of a curve and polystable vector bundles over it. In the last part of the course, we shall
study the topology of the representation varieties thus defined and
show in particular that, in good cases, they provide natural examples of real algebraic varieties for which the Milnor-Smith-Thom is
an equality.
• Jean-Yves Welschinger, Topology of random real hypersurfaces.
The topology of the vanishing locus of a polynomial with real
coefficients much depends on the choice of the coefficients of the
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polynomial, contrary to the complex case. Which topology to expect
for a polynomial chosen at random? I will explain how to estimate
the expected Betti numbers of the vanishing locus of such a random
polynomial. The L2 estimates of Hörmander in complex analysis
plays a crucial role in these estimates. This is a joint work with
Damien Gayet.
2.2. Research talks.
• Benoît Bertrand, Counting real curves with fixed cogenus.
I will describe one example of the use tropical geometry to enumerate real curves. We consider real curves of degree d and genus g
passing through a real configuration of 3d+g −1 points in general position in the plane. Using a version of floor diagrams enhanced with
signs, one can estimate the number of such curves for some generic
configurations when the cogenus (d−1)(d−2)
−g is fixed. This number,
2
which does depend on the configuration, admits as an upper bound
the corresponding number of complex curves through the configuration which is invariant. Using these tropical tools one can prove
that if the cogenus is fixed, there exist configurations such that the
asymptotics is the same as in the complex case when d tends towards
infinity.
• Alicia Dickenstein, From chemical reaction networks to Descartes’
rule of signs.
In the context of chemical reaction networks with mass-action and
other rational kinetics, a major question is to preclude or to guarantee multiple positive steady states. I will explain this motivation and
I will present necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomials maps with arbitrary
real exponents defined on the positive orthant. These conditions extend existing injectivity conditions expressed in terms of Jacobian
matrices and determinants.
I will show that in the context of real algebraic geometry, our results reveal the first partial multivariate generalization of the classical
Descartes’ rule, which bounds the number of positive real roots of a
univariate real polynomial in terms of the number of sign variations
of its coefficients.
This is joint work with Stefan Müller, Elisenda Feliu, Georg Regensburger, Anne Shiu and Carsten Conradi.
I will also present some further advances in this multivariate generalization obtained in collaboration with Frédéric Bihan.
• Vinicio Gómez, Intersections of Quadrics and Oriented Matroids.
Some examples.
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In this talk we consider some smooth manifolds that can be expressed as the intersection of quadric hypersurfaces in Rn . By a
quadric we mean the set of solutions of an equation like
a1 x1 + a2 x2 + . . . + an xn = 0.
This kind of manifolds was studied by Santiago López de Medrano,
Alberto Verjovsky, Laurent Meersseman, Samuel Gitler and others.
There is a relation with other mathematical objects, the oriented
matroids. I try to give a introduction to the subject and examine
some examples of these manifolds from this point of view.
• Rubén Hidalgo, Symmetric algebraic varieties, a computational approach.
Let X be a symmetric and irreducible complex algebraic variety
with a symmetry L, both defined over a subfield Q of C. When
Q is invariant under conjugation, as a consequence of Weil’s Galois
descent theorem, it can be seen that there exist a complex algebraic
variety Y defined over Q ∩ R and an isomorphism R : X → Y .
Unfortunately, Weil’s theorem does not provide a simple method to
compute R or Y in an explicit manner. In this talk I will explain
how to compute R explicitly; so Y .
• Johannes Huisman, Chern-Stiefel-Whitney classes of real vector
bundles.
Let X be a real algebraic variety and F a real vector bundle over X.
I will define Chern-Stiefel-Whitney classes of F with values in certain
hypercohomology groups on the quotient topological space X(C)/G,
where G is the Galois group of C/R. These classes unify the ordinary
characteristic classes in the sense that they induce the Chern classes
of F (C), on the one hand, and the Stiefel-Whitney classes of F (R),
on the other hand. The construction sheds a seemingly new light on
the fact that the mod-2 cohomology ring of a real Grassmannian is
the reduction modulo 2 of the integral cohomology ring of a complex
Grassmannian after dividing all degrees by 2.
• Ilia Itenberg, Hurwitz numbers for real polynomials.
We introduce a signed count of real polynomials which gives rise
to a real analog of Hurwitz numbers in the case of polynomials. The
invariants obtained allow one to show the abundance of real solutions
in the corresponding enumerative problems: in many cases, the number of real solutions is asymptotically equivalent (in the logarithmic
scale) to the number of complex solutions.
This is a joint work with Dimitri Zvonkine.
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• Chiu-Chu Melissa Liu, The Kirwan surjectivity for real algebraic
vector bundles over a real algebraic curve.
By the work of Atiyah-Bott, the moduli space of stable algebraic
vector bundles over a non-singular complex projective curve can be
viewed as a symplectic reduction of the infinite-dimensional symplectic space of all holomorphic strucures on a smooth complex vector
bundle over a Riemann surface, and the Kirwan map is surjective.
We study surjectivity of a real version of the Kirwan map and its
implication on the topology of moduli spaces of stable real algebraic
vector bundles over a non-singular real algebraic curve.
This is based on joint work with Florent Schaffhauser.
• Lucía López de Medrano, Maximally inflected real curves.
A real curve of degree d is called maximally inflected if it contains
exactly d(d − 2) disctints real inflection points. During this talk,
we will see some results about maximally inflected real curves. In
particular, we will see how to construct theses curves using tropical
geometry and Hilbert’s method. To finish, we will give some classifications about the distribution of the real inflection points among the
ovals of a curve. This results are joint work with Aubin Arroyo and
Erwan Brugallé.
• Alf Onshuus, Some aspects of o-minimality that relate to real algebraic geometry.
The model theoretic notion of o-minimality presents an analytic
setting over the real field that behaves in many ways like semialgebraic structures. This has turned into applications close to real
algebraic geometry in two at least ways: One, some results known
to work in the real semialgebraic setting can be generalized to ominimal settings. Second, certain algebraic and semialgebraic structures (such as a modular curve or, more general, Shimura Varieties)
can be analyzed in analytic settings, an analysis which can be carried
out in an o-minimal setting with some important consequences.
In this talk I will introduce o-minimal theories, give some intuition about their reach and limitations, and give some examples of
applications.
• Nicolas Puignau, On Welschinger invariants of real symplectic 4manifolds.
Recently, significant progress have been made in the study of
Welschinger invariants of real symplectic 4-manifolds, with particular attention paid to manifolds with disconnected real part. This
has become possible through the development of real versions of the
symplectic sum formula. We will explain this method and describe
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some applications. Among them, vanishing and divisibility by a large
power of 2 of some modified Welschinger invariants. This is a joint
work with E. Brugallé.
• Jean-Jacques Risler, Real Algebraic sets and Total Curvature.
Let X ⊂ Rn+1 be a smooth algebraic hypersurface of degree
d,
R
CX ⊂ Cn+1 its complexification.
Then
the
total
curvature
|k|
of
X
R
X (resp. the total curvature CX |K| of CX) is the "volume" of the
Gauss map g : X → RP n (respR Cg : CX → CPRn ). We prove that
there is a universal inequality X |k| ≤ α(n, d) CX |K|. We study
the sharpness of this inequality, and of a similar one in the tropical
setting.
• Nermin Salepci, Real Lefschetz fillings of real open books.
It is known that every 3-manifold M , can be seen as a surface bundle over circle in the exterior of a closed 1-dimensional sub-manifold.
The particular choice of 1-dimensional sub-manifold and the projection giving the bundle structure is called an open book decomposition on M . A Lefschetz filling of an open book decomposition is
a 4-manifold having M as boundary and admitting a Lefschetz fibration (a fiber bundle like structure on 4-manifolds) such that the
projections defining open book decomposition and the Lefschetz fibration match on M . We will first discuss the condition for an open
book decomposition to admit a Lefschetz filling. Then we discuss the
filling problem in the existence of an extra structure (called a real
structure) on both manifolds. Roughly, a real structure is a certain
action of a cyclic group of order 2 and we also want this action to be
compatible with the existent structures on the manifolds. We give
an example which underlines certain differences between real and
non-real filling problems. This is a joint work with Ferit Öztürk.
• Mauricio Velasco, Nonnegative sections and sums of squares on
real projective varieties.
If a homogeneous polynomial of degree 2d in n variables can be
written as a sum of squares then it must be a nonnegative polynomial. Hilbert proved that there are nonnegative polynomials which
cannot be written as a sum of squares unless the degree 2d is at most
two or the number of variables is at most two or in the exceptional
case of quartic polynomials in three variables. We give a classification
theorem for all real projective varieties for which nonnegative quadratic forms and sums of squares coincide and prove that this occurs
precisely for varieties of minimal degree. Our result gives a geometric
explanation for Hilbert’s Theorem and its natural generalization to
many other contexts (for instance multihomogeneous forms). In the
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talk I will discuss this Theorem as well as ongoing work to quantify
the difference between nonnegative polynomials and sums of squares
in varieties which are not of minimal degree. The results in this talk
are joint work with G. Blekherman and G.G. Smith and with G.
Blekherman, S. Iliman and M.Kubitzke.
• Shuguang Wang, Seiberg-Witten theory and real structures.
We discuss results from a recent paper of Gang Tian/Shuguang
Wang, concerning a Seiberg-Witten integer invariant for a Kaehler
surface together with a real structure. Additionally, a preliminary
discussion is given to a Seiberg-Witten theory for higher dimensional
foliated transverse real Kaehler manifolds.
3. Schedule
3.1. Week 1.
Monday
Tuesday
Wednesday
Thursday
Friday
9:30-10:30
I. Itenberg - 1
9:30-10:30
I. Itenberg - 3
9:30-10:30
R. Hidalgo - 2
9:30-10:30
I. Itenberg - 4
9:30 -10:30
I. Itenberg - 5
Coffee break
Coffee break
Coffee break
Coffee break
Coffee break
11:00-12:00
F. Schaffhauser - 1
11:00-12:00
R. Hidalgo - 1
11:00-12:00
E. Brugallé - 2
11:00-12:00
R. Hidalgo - 3
11:00-12:00
F. Schaffhauser - 5
15:00-16:00
I. Itenberg - 2
15:00-16:00
E. Brugallé - 1
15:00-16:00
E. Brugallé - 3
15:00-16:00
E. Brugallé - 4
Coffee break
Coffee break
Coffee break
Coffee break
Free afternoon
16:30-17:30
16:30-17:30
F. Schaffhauser - 2 F. Schaffhauser - 3
16:30-17:30
16:30-17:30
F. Schaffhauser - 4 J.Y. Welschinger - 1
3.2. Week 2.
Monday
Tuesday
Wednesday
Thursday
Friday
9:30-10:30
E. Brugallé - 5
9:30-10:30
J.Y. Welschinger - 4
9:30-10:30
J.J. Risler
9:30-10:30
A. Dickenstein
9:30 -10:30
A. Onshuus
Coffee break
Coffee break
Coffee break
Coffee break
Coffee break
11:00-12:00
J.Y. Welschinger - 2
11:00-12:00
R. Hidalgo - 5
11:00-12:00
L. López de Medrano
11:00-12:00
R. Hidalgo
11:00-12:00
N. Salepci
13:45-14:45
M. Liu
13:45-14:45
S. Wang
13:45-14:45
B. Bertand
15:00-16:00
R. Hidalgo - 4
15:00-16:00
J.Y. Welschinger - 5
15:00-16:00
J. Huisman
15:00-16:00
V. Gómez
15:00-16:00
N. Puignau
Coffee break
Coffee break
Coffee Break
Coffee break
Coffee break
16:30-17:30
J.Y. Welschinger - 3
16:30-17:30
M. Velasco
16:30-17:30
I. Itenberg