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The Wu relations in real algebraic geometry

Olivier Benoist, Olivier Wittenberg

TL;DR

The paper develops a comprehensive framework linking the Galois cohomology class ω to Chern classes c_i in Gal({ C}/{ R})-equivariant cohomology via stably complex G-manifolds, yielding new results in real algebraic geometry. Central technical advances include the proof that ω^3=0 for real surfaces with X({ R})=∅ and the construction of extensive relations between ω and the Chern classes, both modulo 2 and integrally, grounded in Wu classes and right A-actions. These relations propagate to topological constraints on X(C) through coindex bounds and to arithmetic questions via the level of real function fields, enabling improved Pfister-type bounds in uniruled and conic-bundle contexts and sharpened Hilbert 17th results for low-degree polynomials. Collectively, the work connects real algebraic geometry, equivariant topology, and quadratic form theory, providing new tools for understanding sums of squares, levels, and the topology of complex points of real varieties.$

Abstract

We construct and study relations between Chern classes and Galois cohomology classes in the Gal(C/R)-equivariant cohomology of real algebraic varieties with no real points. We give applications to the topology of their sets of complex points, and to sums of squares problems. In particular, we show that -1 is a sum of 2 squares in the function field of any smooth projective real algebraic surface with no real points and with vanishing geometric genus, as well as higher-dimensional generalizations of this result.

The Wu relations in real algebraic geometry

TL;DR

The paper develops a comprehensive framework linking the Galois cohomology class ω to Chern classes c_i in Gal({ C}/{ R})-equivariant cohomology via stably complex G-manifolds, yielding new results in real algebraic geometry. Central technical advances include the proof that ω^3=0 for real surfaces with X({ R})=∅ and the construction of extensive relations between ω and the Chern classes, both modulo 2 and integrally, grounded in Wu classes and right A-actions. These relations propagate to topological constraints on X(C) through coindex bounds and to arithmetic questions via the level of real function fields, enabling improved Pfister-type bounds in uniruled and conic-bundle contexts and sharpened Hilbert 17th results for low-degree polynomials. Collectively, the work connects real algebraic geometry, equivariant topology, and quadratic form theory, providing new tools for understanding sums of squares, levels, and the topology of complex points of real varieties.$

Abstract

We construct and study relations between Chern classes and Galois cohomology classes in the Gal(C/R)-equivariant cohomology of real algebraic varieties with no real points. We give applications to the topology of their sets of complex points, and to sums of squares problems. In particular, we show that -1 is a sum of 2 squares in the function field of any smooth projective real algebraic surface with no real points and with vanishing geometric genus, as well as higher-dimensional generalizations of this result.
Paper Structure (52 sections, 67 theorems, 93 equations)

This paper contains 52 sections, 67 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. The level of real algebraic surfaces
  3. The vanishing of omega 3 for surfaces
  4. Relations between tautological classes
  5. Applications to the topology of X(C)
  6. Applications to sums of squares
  7. Organization of the text
  8. Acknowledgements
  9. Notation and conventions
  10. Real algebraic surfaces with no real points
  11. The relation omega 3 = 0
  12. Levels of function fields of surfaces
  13. G-equivariant preliminaries
  14. G-equivariant vector bundles
  15. Stably complex G-manifolds
  16. ...and 37 more sections

Key Result

Theorem 1.1

Let $X$ be an irreducible smooth proper surface over ${\mathbf R}$ with $X({\mathbf R})=\varnothing$ and $H^2(X,{\mathscr O}_X)=0$. Then $-1$ is a sum of $2$ squares in ${\mathbf R}(X)$.

Theorems & Definitions (131)

  • Theorem 1.1: Theorem \ref{['th:levelsurface']}
  • Theorem 1.2: Theorem \ref{['th:omega3']}
  • Theorem 1.3: see Corollary \ref{['th:omega2n-1ouvert']}
  • Theorem 1.4: see Theorems \ref{['coindexn']} and \ref{['coindexn2']}
  • Theorem 1.5: Theorems \ref{['coindexsurfaces']} and \ref{['coindexblowup']}
  • Theorem 1.6: Corollaries \ref{['coreven']} and \ref{['cor3']}
  • Theorem 1.7: Theorem \ref{['thlowerdegree']}
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • ...and 121 more