# Conference - Real vector bundles

# Schedule

### Monday

09h00 **Welcome and coffee**

09h30 **Andrei Teleman:** Intrinsic signs and lower bounds in real algebraic geometry

10h30 **Coffee break**

11h00 **Jörg Schürmann:** Introduction to characteristic classes of singular spaces

12h30 **Lunch**

14h30 **Melissa Liu:** Yang-Mills equations over Klein surfaces

15h30 **Coffee break**

16h00 **Jacques Hurtubise:** Pseudo-real principal bundles on a compact Kähler manifold

### Tuesday

09h30 **Melissa Liu:** Yang-Mills equations over Klein surfaces

10h30 **Coffee break**

11h00 **Pedro F. dos Santos:** Bi-graded invariants for real varieties

12h30 **Lunch**

14h30 **Jörg Schürmann:** Introduction to characteristic classes of singular spaces

15h30 **Coffee break**

16h00 **Oscar Garcia-Prada:** Involutions of Higgs bundle moduli spaces

### Wednesday

09h30 **Andrei Teleman:** A wall crossing formula for degrees of real central projections

10h30 **Coffee break**

11h00 **Melissa Liu:** Yang-Mills equations over Klein surfaces

12h30 **Lunch**

14h30 **Excursion**

### Thursday

09h30 **Jörg Schürmann:** Introduction to characteristic classes of singular spaces

10h30 **Coffee break**

11h00 **Paulo Lima-Filho:** Bi-graded invariants for real varieties

12h30 **Lunch**

14h30 **Florent Schaffhauser:** Modular compactifications of moduli spaces of geometrically stable real and quaternionic vector bundles

15h30 **Coffee break**

16h00 **Wojciech Kucharz:** Stratified-algebraic vector bundles

20h00 **Dinner**

### Friday

09h30 **Michael Atiyah:** Characteristic classes for KR bundles

10h30 **Coffee break**

11h00 **Paulo Lima-Filho:** Bi-graded invariants for real varieties

12h30 **Lunch**

14h30 **Andrei Teleman:** Abelian Yang-Mills theory on Real tori
and Theta divisors of Klein surfaces

15h30 **Coffee**

All lectures will be given in Lecture room E of the Faculty of Sciences of the University of Brest, 6 avenue Le Gorgeu, Brest.

# Abstracts

### Michael Atiyah (Edingburgh): Characteristic classes for KR bundles

Abstract: Since i introduced KR bundles many years ago there has been much work on developing Chern class theory for these bundles. I will use old ideas of Grothendieck to give yet another treatment. The advantage may be that information mod powers of 2 may be useful in real algebraic geometry.

### Jean Fasel (Duisburg-Essen): A1-homotopic classification of vector bundles

### Oscar Garcia-Prada (ICMAT, Madrid): Involutions of Higgs bundle moduli spaces

Abstract: We consider the moduli space of G-Higgs bundles over a compact Riemann surface X, where G is a complex semisimple Lie group, and study various involutions of the moduli space involving conjugations in G and X. We describe the fixed points and study their relations with representations of the fundamental group of X in G. We will also comment on the relation of the fixed points with Langlands duality and mirror symmetry for Higgs bundles.

### Jacques Hurtubise (McGill, Montreal): Pseudo-real principal bundles on a compact Kähler manifold

Abstract: Let X be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let G be a connected complex reductive affine algebraic group equipped with a real form. We define pseudo-real principal G–bundles on X; these are generalizations of real algebraic principal G–bundles over a real algebraic variety. We prove that a Donaldson–Uhlenbeck–Yau type correspondence holds: a pseudo-real principal G–bundle admits a compatible Einstein-Hermitian connection if and only if it is polystable. The link with representations of the fundamental group is also considered. Joint work with Indranil Biswas and Oscar Garcia-Prada.

### Wojciech Kucharz (Jagiellonian, Krakow): Stratified-algebraic vector bundles (joint work with Krzysztof Kurdyka)

Abstract: We investigate stratified-algebraic vector bundles on a real algebraic variety X. A stratification of X is a finite partition of X into Zariski locally closed subvarieties. A topological vector bundle on X is called a stratified-algebraic vector bundle if, roughly speaking, its restriction to each stratum of some stratification of X is an algebraic vector bundle on that stratum. It turns out that stratified-algebraic vector bundles have many desirable features of algebraic vector bundles but are more flexible. Recently first significant steps have been made toward real algebraic geometry based on continuous rational functions – called regulous geometry. Stratified-algebraic vector bundles can be also regarded as the appropriate class of vector bundles in regulous geometry.

### Paulo Lima-Filho (Texas A&M): Bi-graded invariants for real varieties

### Melissa Liu (Columbia, New York): Yang-Mills equations over Klein surfaces

Abstract: In "The Yang-Mills equations over Riemann surfaces," Atiyah and Bott studied Yang-Mills functional over a Riemann surface from the point of view of Morse theory, and derived results on topology of the moduli space of algebraic bundles over a complex algebraic curve. In these lectures, I will discuss Yang-Mills functional over a Klein surface from the point of view of Morse theory, and derive recursive and closed formulae of mod 2 Poincare polynomials of the moduli space of real or quaternionic vector bundles over a real algebraic curve. These lectures are based on "The Yang-Mills equations over Klein surfaces" by Florent Schaffhauser and myself.

### Pedro F. dos Santos (Lisbon): Bi-graded invariants for real varieties

### Florent Schaffhauser (Los Andes, Bogota): Modular compactifications of moduli spaces of geometrically stable real and quaternionic vector bundles

Abstract: In a paper published in 1967, Seshadri introduced the notion now known as S-equivalence on categories of semi-stable holomorphic vector bundles of fixed slope, which enabled him to construct a modular compactification of Mumford's moduli spaces of stable holomorphic vector bundles of fixed rank and degree. The goal of the present talk is to explain the analogous notion for semi-stable real and quaternionic vector bundles. We then study the relationship between this construction and symplectic geometry.

### Jörg Schürmann (Münster): Introduction to characteristic classes of singular spaces

Abstract: First we explain the Lagrangian approach to Stiefel-Whitney resp. Chern classes of singular real resp. complex algebraic or analytic varieties, motivated by Poincare–Hopf index theorems for such singular spaces. Based on stratified Morse theory for constructible functions, we discuss the corresponding calculus of characteristic cycles and its relation to these characteristic classes, e.g. functoriality under proper pushdown as well as specialization. We also explain the relation between these characteristic classes for algebraic or analytic varieties defined over the reals.

Then we move to "motivic constructible" functions, i.e. to relative Grothendieck groups of algebraic varieties, and introduce the motivic Chern and Hirzebruch class transformation, where the later unifies the MacPherson Chern class, the Baum-Fulton-MacPherson Todd class, as well as the Cappell-Shaneson L-class transformation for singular spaces.

Finally we compute all these characteristic classes for singular toric varieties. If time permits, we also explain the relation to weighted lattice points counting in lattice polytopes.

### Andrei Teleman (Marseille): Intrinsic signs and lower bounds in real algebraic geometry (joint with Ch. Okonek)

Abstract: A classical result due to Segre states that on a real cubic
surface in **P**^{3} there exists two kinds of
real lines: elliptic and hyperbolic lines. These two kinds of real
lines are defined in an intrinsic way, i.e., their definition does not
depend on any choices of orientation data. Segre's classification of
smooth real cubic surfaces also shows that any such surface contains
at least 3 real lines. Starting from these remarks and inspired by the
classical problem mentioned above - I will explain a general principle
which leads to lower bounds in real algebraic geometry, - I will
explain the reason for the appearance of intrinsic signs in the
classical problem treated by Segre, showing that the same phenomenon
occurs in a large class of enumerative problems in real algebraic
geometry. - I will illustrate these principles in the enumerative
problem for real lines in real hypersurfaces of degree 2m−3 in
**P**^{3}.

### Andrei Teleman (Marseille): A wall crossing formula for degrees of real central projections (joint with Ch. Okonek)

Abstract: I will discuss a recent result in real algebraic geometry: a
wall crossing formula for central projections defined on submanifolds
of a real projective space. This formula gives the jump of the degree
of such a projection when the center of the projection varies. The
fact that the degree depends on the projection is a new phenomenon,
specific to real algebraic geometry. We illustrate this phenomenon in
many interesting situations. The crucial assumption on the class of
maps we consider is relative orientability, a condition which allows
us to define a **Z**-valued degree map in a coherent way. I will
discuss several consequences of this result.

### Andrei Teleman (Marseille): Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces (joint with Ch. Okonek)

Abstract: I will explain a method to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry.