The tight approximation property

Olivier Benoist; Olivier Wittenberg

doi:10.1515/crelle-2021-0003

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The tight approximation property

Olivier Benoist and Olivier Wittenberg

Published/Copyright: March 11, 2021

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2021 Issue 776

Abstract

This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological obstructions to it) by incorporating an approximation condition in the Euclidean topology. We prove that the tight approximation property is a stable birational invariant, is compatible with fibrations, and satisfies descent under torsors of linear algebraic groups. Its validity for a number of rationally connected varieties follows. Some concrete consequences are: smooth loops in the real locus of a smooth compactification of a real linear algebraic group, or in a smooth cubic hypersurface of dimension ≥2, can be approximated by rational algebraic curves; homogeneous spaces of linear algebraic groups over the function field of a real curve satisfy weak approximation.

A G-equivariant complex analytic spaces

In this appendix, we develop the basics of G-equivariant complex geometry, and collect the results that we need.

A.1 Definition

A G-equivariant complex analytic space is a complex analytic space (Z,𝒪Z) in the sense of [47, p. 16] whose underlying locally ringed space is endowed with an action of G such that the complex conjugation σ acts 𝐂-antilinearly on 𝒪Z. It is said to be a manifold (resp. Stein, projective, etc.) if so is the underlying complex analytic space. A G-equivariant complex manifold is nothing but a complex analytic manifold endowed with an action of G such that σ acts antiholomorphically.

If Z is a G-equivariant complex analytic space, a G-equivariant coherent sheaf on Z is a coherent sheaf on the underlying complex analytic space that is endowed with an action of G compatible with its 𝒪Z-module structure.

There are two equivalent approaches to G-equivariant complex geometry. One can consider the G-equivariant spaces defined above, as in [51, Section II.4] where they are called complex analytic spaces with an antiinvolution, as in [84, p. 250] where they are called analytic spaces over 𝐑, or as in [20, Section 2.1] where they are called real structures on complex spaces. One can also consider their quotients by the action of G in the category of locally ringed spaces: those are the Berkovich 𝐑-analytic spaces hinted at in [9, Examples 1.5.4], which were also considered by Huisman [62] under the name of real analytic spaces. The reason for our choice is that classical results of complex geometry apply more directly in the former context.

A.2 Analytification

Let (Z,𝒪Z) be a complex analytic space with structural morphism μ:𝐂→𝒪Z. We define its conjugate (Zσ,𝒪Zσ) to be equal to (Z,𝒪Z) as a locally ringed space, but with structural morphism μ∘σ. With a coherent sheaf ℱ on Z, one associates a coherent sheaf ℱσ on Zσ: it is equal to ℱ as a sheaf and its 𝒪Zσ-module structure is induced by the equality 𝒪Zσ=𝒪Z. One verifies that a G-equivariant complex analytic space is nothing but a complex analytic space Z endowed with an isomorphism α:Zσ→Z of complex analytic spaces such that α∘ασ=IdZ, and that a G-equivariant coherent sheaf on it is a coherent sheaf ℱ on the underlying complex space endowed with an isomorphism β:α*⁢ℱ→ℱσ such that βσ∘ασ⁣*⁢β=Idℱ.

If X is a variety over 𝐑, then the analytification X𝐂an of X𝐂 has a natural structure of G-equivariant complex analytic space, denoted by Xan. Similarly, if ℱ is a coherent sheaf on X, then the analytification ℱ𝐂an of ℱ𝐂 has a natural structure of a G-equivariant coherent sheaf, denoted by ℱan.

When X is proper, the analytification functor Y↦Yan induces an equivalence between the categories of closed subvarieties of X and of closed G-equivariant analytic subspaces of Xan, and the functor ℱ↦ℱan is an equivalence between the categories of coherent sheaves on X and of G-equivariant coherent sheaves on Xan. These facts follow from the above description of G-equivariant complex analytic spaces and G-equivariant coherent sheaves, together with Serre’s GAGA theorem for proper varieties [50, Exposé XII, Théorème 4.4] (see also [50, Exposé XII, Corollaire 4.6]) and Galois descent (see for instance [93, I Section 1]).

Arguing in the same way, we obtain a G-equivariant version of Riemann’s existence theorem from the non-equivariant statement [50, Exposé XII, Théorème 5.1]: for any variety X over 𝐑, the analytification functor induces an equivalence between the categories of finite étale coverings of X and of G-equivariant finite topological coverings of X⁢(𝐂).

A.3 The Stein property

Recall that a complex analytic space Z is Stein if one has Hq⁢(Z,ℱ)=0 for all coherent sheaves ℱ on Z and all q>0. We collect here for later use G-equivariant analogues of well-known consequences of the Stein property.

For a G-equivariant coherent sheaf ℱ on a G-equivariant complex analytic space Z, the group G acts on Hq⁢(Z,ℱ), the action of σ being 𝐂-antilinear. Since the Hq⁢(Z,ℱ) are 𝐂-vector spaces, one has Hp⁢(G,Hq⁢(Z,ℱ))=0 for p>0, and the second spectral sequence of equivariant cohomology [49, Théorème 5.2.1] shows that HGq⁢(Z,ℱ)=Hq⁢(Z,ℱ)G. Thus, if Z is Stein, then HGq⁢(Z,ℱ)=0 for all q>0.

The real vector space Hq⁢(Z,ℱ)G satisfies Hq⁢(Z,ℱ)=Hq⁢(Z,ℱ)G⊗𝐑𝐂, giving rise to a real structure on Hq⁢(Z,ℱ). The cohomology long exact sequence induced by a short exact sequence of G-equivariant coherent sheaves is G-equivariant, hence is defined over 𝐑 for these real structures.

Lemma A.1.

Let Z be a Stein G-equivariant complex analytic space.

If Z has finite dimension, then a G-equivariant coherent sheaf ℱ on Z whose fibers have bounded dimensions is a quotient of a trivial G-equivariant coherent sheaf.
A short exact sequence 0→ℱ1→ℱ2→ℱ3→0 of G-equivariant coherent sheaves on Z splits G-equivariantly if ℱ3 is locally free.

Proof.

(i) Combine Cartan’s Theorem A and [73, Theorem 1] to see that ℱ is generated by finitely many global sections. The smallest G-stable sub-𝐂-vector space V⊂H0⁢(Z,ℱ) containing these sections is defined over 𝐑: an 𝐑-basis ζ1,…,ζN of V∩H0⁢(Z,ℱ)G is also a 𝐂-basis of V. The ζi induce a G-equivariant surjection 𝒪ZN→ℱ of coherent sheaves.

(ii) Consider the exact sequence 0→ℱ1⊗ℱ3∨→ℱ2⊗ℱ3∨→ℱ3⊗ℱ3∨→0. Since Z is Stein, there is an induced short exact sequence of global sections on Z, hence of the underlying real vector spaces:

0→H0⁢(Z,ℱ1⊗ℱ3∨)G→H0⁢(Z,ℱ2⊗ℱ3∨)G→H0⁢(Z,ℱ3⊗ℱ3∨)G→0.

A lift of Idℱ3∈H0⁢(Z,ℱ3⊗ℱ3∨)G in H0⁢(Z,ℱ2⊗ℱ3∨)G corresponds to a G-equivariant morphism ℱ3→ℱ2 of coherent sheaves inducing the required splitting. ∎

Proposition A.2.

Let Z be a G-equivariant complex analytic space. Any G-stable locally closed Stein subspace Y⊂Z has a G-stable Stein open neighborhood in Z.

Proof.

By Siu’s theorem [95], Y has a Stein open neighborhood Y′ in Z. The G-stable open neighborhood Y′∩σ⁢(Y′) is then Stein by [47, p. 127]. ∎

A.4 The Picard group

Let Z be a G-equivariant complex analytic space. The isomorphism classes of G-equivariant invertible sheaves on Z form a group for the tensor product: the Picard group PicG⁢(Z) of Z. Letting 𝒪Z*⊂𝒪Z be the G-equivariant subsheaf of invertible analytic functions, one has an isomorphism PicG⁢(Z)≃HG1⁢(Z,𝒪Z*). This follows from the Čech description of G-equivariant cohomology [49, Théorème 5.5.6] (for details in the topological setting see [64, p. 698] or [71, Proposition 1.1.1]).

Let 𝐙⁢(1)⊂𝐂 be the sub-G-module generated by -1. Viewing the exponential exact sequence [47, Lemma, p. 142]

(A.1)0→𝐙⁢(1)→𝒪Z→f↦exp⁡(2⁢π⁢f)𝒪Z*→0

as a short exact sequence of G-equivariant sheaves on Z yields a boundary map

cl:PicG⁢(Z)→HG2⁢(Z,𝐙⁢(1)),

the so-called equivariant cycle class map. Composing it with the restriction map

HG2⁢(Z,𝐙⁢(1))→HG2⁢(ZG,𝐙⁢(1))

and with the canonical isomorphism

HG2⁢(ZG,𝐙⁢(1))≃H1⁢(ZG,𝐙/2⁢𝐙)

described in [72, Theorem 1.3] induces the Borel–Haefliger cycle class map

cl𝐑:PicG⁢(Z)→H1⁢(ZG,𝐙/2⁢𝐙).

Proposition A.3.

The map cl:PicG⁢(Z)→HG2⁢(Z,Z⁢(1)) is an isomorphism if Z is a Stein G-equivariant complex analytic space. So is clR:PicG⁢(Z)→H1⁢(ZG,Z/2⁢Z) if Z is a Stein G-equivariant complex manifold of pure dimension 1.

Proof.

If Z is Stein, then HGq⁢(Z,𝒪Z)=0 for q>0 (see Section A.3). The long exact sequence of G-equivariant cohomology induced by the exponential exact sequence then shows that cl is an isomorphism.

It remains to prove that the restriction map HG2⁢(Z,𝐙⁢(1))→HG2⁢(ZG,𝐙⁢(1)) is an isomorphism if Z is a Stein G-equivariant complex manifold of pure dimension 1. To do so, we let i:Z∖ZG↪Z be the inclusion and we let i! denote the extension by zero. Since G acts antiholomorphically on Z, the fixed point set ZG⊂Z is a one-dimensional 𝒞∞ closed submanifold, and the quotient Z/G is a 𝒞∞ manifold with boundary ZG and interior (Z∖ZG)/G. We denote by j:(Z∖ZG)/G↪Z/G the inclusion, and by 𝐙~ the sheaf on (Z∖ZG)/G induced by the G-equivariant sheaf 𝐙⁢(1) on Z∖ZG. Since G acts antiholomorphically on Z, it reverses its orientation, so that j!⁢𝐙~ is the orientation sheaf of Z/G in the sense of [17, V, Definition 9.1].

In view of the long exact sequence of relative equivariant cohomology

HG2⁢(Z,i!⁢𝐙⁢(1))→HG2⁢(Z,𝐙⁢(1))→HG2⁢(ZG,𝐙⁢(1))→HG3⁢(Z,i!⁢𝐙⁢(1))⁢,

it suffices to show that HGq⁢(Z,i!⁢𝐙⁢(1))=0 for q≥2. By the first spectral sequence of equivariant cohomology [49, Théorème 5.2.1], HGq⁢(Z,i!⁢𝐙⁢(1))≃Hq⁢(Z/G,j!⁢𝐙~). By Poincaré duality, we have Hq⁢(Z/G,j!⁢𝐙~)≃H2-qBM⁢(Z/G,𝐙), where H*BM denotes Borel–Moore homology (apply [17, V, Theorem 9.3] to X=Z/G, ℳ=𝐙, and with Φ the family of closed subsets of X). This group obviously vanishes if q≥3. It also vanishes if q=2 as Z/G has no compact component since Z is Stein. ∎

A.5 Meromorphic functions and ramified coverings

If Z is a complex manifold, we let ℳ⁢(Z) be the ring of meromorphic functions on Z. It is the product of the fields of meromorphic functions of the connected components of Z.

Proposition A.4.

Let Z be a complex manifold of pure dimension 1 with finitely many connected components. Associating with π:Z′→Z the M⁢(Z)-algebra M⁢(Z′) induces an equivalence between the categories of finite morphisms π:Z′→Z of complex manifolds of pure dimension 1 and of finite étale M⁢(Z)-algebras.

Proof.

When Z is connected, and if one restricts to the subcategories of finite morphisms π:Z′→Z with Z′ non-empty and connected, and of finite field extensions of ℳ⁢(Z), this is [92, Chapter 1, Section 4.14, Corollary 4]. The general case follows at once. ∎

If Z is a G-equivariant complex manifold, let ℳ⁢(Z)G be the ring of G-equivariant meromorphic functions on Z. If Z is connected, one has ℳ⁢(Z)G⁢(-1)=ℳ⁢(Z). If Z has two connected components Z′ and σ⁢(Z′), one has ℳ⁢(Z)G≃ℳ⁢(Z′). In general, ℳ⁢(Z)G is the product of the fields of G-equivariant meromorphic functions on the G-orbits of connected components of Z.

Proposition A.5.

Let Z be a G-equivariant complex manifold of pure dimension 1 with finitely many connected components. Associating with π:Z′→Z the M⁢(Z)G-algebra M⁢(Z′)G induces an equivalence between the categories of finite morphisms π:Z′→Z of G-equivariant complex manifolds of pure dimension 1 and of finite étale M⁢(Z)G-algebras.

Proof.

This follows from Proposition A.4 and from the description of G-equivariant complex analytic spaces Z given in Section A.2, as complex analytic spaces Z endowed with the datum of an isomorphism α:Zσ→Z such that α∘ασ=IdZ. ∎

A.6 Meromorphic functions and cohomological dimension

The next proposition is attributed to Artin by Guralnick [52].

Proposition A.6.

Let Z be a connected complex manifold of dimension 1. Then the field M⁢(Z) has cohomological dimension 1.

Proof.

If Z is compact, this is Tsen’s theorem. If Z is not compact, let L be a finite extension of ℳ⁢(Z). By Proposition A.4, it is the field of meromorphic functions of some connected complex manifold of dimension 1. As a consequence, the Brauer group of L vanishes by [52, Proposition 3.7]. The proposition then follows from [90, II 3.1, Proposition 5]. ∎

Recalling that a field k is said to have virtual cohomological dimension 1 if k⁢(-1) has cohomological dimension 1, we deduce from Proposition A.6:

Corollary A.7.

Let Z be a G-equivariant complex manifold of dimension 1 such that Z/G is connected. Then the field M⁢(Z)G has virtual cohomological dimension 1.

Corollary A.7 can be refined if ZG=∅.

Proposition A.8.

Let Z be a G-equivariant complex manifold of dimension 1 such that Z/G is connected. If ZG=∅, the field M⁢(Z)G has cohomological dimension 1.

Proof.

We claim that -1 is a sum of two squares in ℳ⁢(Z)G. It follows that ℳ⁢(Z)G cannot be ordered, so that its absolute Galois group contains no element of finite order by the Artin–Schreier theorem [5]. The main theorem of [88] and Corollary A.7 then imply that ℳ⁢(Z)G has cohomological dimension 1.

It remains to prove the claim. If Z is compact, then ℳ⁢(Z)G is the function field of a smooth projective connected curve over 𝐑 with no 𝐑-point by GAGA (see Section A.2), and the claim is due to Witt [104, Satz 22]. We assume from now on that Z is not compact, hence Stein [47, p. 134].

Since H1⁢(Z,𝒪Z)=0 because Z is Stein, and since H2⁢(Z,𝐙)≃H0BM⁢(Z,𝐙)=0 by Poincaré duality [17, Chapter V, Theorem 9.3] and because Z has no compact component, the exponential exact sequence (A.1) yields H1⁢(Z,𝒪Z*)=0. Since HG2⁢(Z,𝒪Z)=0 because Z is Stein (see Section A.3), and since HG3⁢(Z,𝐙⁢(1))=H3⁢(Z/G,𝐙~)=0 (the first equality stems from the first spectral sequence of equivariant cohomology [49, Théorème 5.2.1] and the second from the fact that Z/G is a surface), the exponential exact sequence (A.1) shows that HG2⁢(Z,𝒪Z*)=0.

The second spectral sequence of equivariant cohomology [49, Théorème 5.2.1], that is, E2p,q=Hp⁢(G,Hq⁢(Z,𝒪Z*))⇒HGp+q⁢(Z,𝒪Z*), now shows that H2⁢(G,𝒪⁢(Z)*)=0. By [89, VIII Section 4], we have H2(G,𝒪(Z)*)=(𝒪(Z)*)G/{f⋅σ(f),f∈𝒪(Z)*}. Thus we deduce from this vanishing that there exists f∈𝒪⁢(Z)* such that -1=f⋅σ⁢(f). It follows that -1=(f+σ⁢(f)2)2+(f-σ⁢(f)2⁢-1)2 is a sum of two squares in ℳ⁢(Z)G, as desired. ∎

Let Z be a G-equivariant complex manifold of dimension 1 such that Z/G is connected. If x∈ZG and if t∈ℳ⁢(Z)G is a uniformizer at x, expanding in power series at x yields an inclusion ℳ⁢(Z)G⊂𝐑⁢((t)). Restricting to ℳ⁢(Z)G the unique ordering < of the field 𝐑⁢((t)) for which t>0 gives rise to an ordering ≺x,t of the field ℳ⁢(Z)G. It is easily verified, using the fact that Z is either projective or Stein, that ≺x,t and ≺x′,t′ coincide if and only if x=x′ and (t/t′)⁢(x)∈𝐑>0. We show that if ZG is compact, there are no other orderings of ℳ⁢(Z)G.

Proposition A.9.

Let Z be a G-equivariant complex manifold of dimension 1 such that Z/G is connected and ZG is compact. Then all the orderings of M⁢(Z)G are of the form ≺x,t described above.

Proof.

Fix an ordering ≺ of ℳ⁢(Z)G. With ≺ is associated a valuation ring

A:={f∈ℳ⁢(Z)G∣-r≺f≺r⁢ for some ⁢r∈𝐑}

with maximal ideal

𝔪:={f∈ℳ⁢(Z)G∣-r≺f≺r⁢ for all ⁢r∈𝐑>0}

and residue field isomorphic to 𝐑 [75, Theorems 2 and 3].

Assume for contradiction that for every point x∈ZG, there exists fx∈𝔪 such that fx does not vanish at x. Since ZG is compact, there exist a finite subset Σ⊂ZG and ε∈𝐑>0 such that f:=∑x∈Σfx2∈𝔪 does not take any finite value that is ≤ε on ZG. The field ℳ⁢(Z)G⁢(ε-f) is the field of G-equivariant meromorphic functions of a G-equivariant complex manifold Y of dimension 1, which is a ramified covering of Z (see Proposition A.4). Since f>ε on ZG, we see that YG lies above the poles of ε-f, hence is discrete. But YG is a one-dimensional differentiable manifold as G acts antiholomorphically on Y. It follows that YG=∅. Proposition A.8 implies that ℳ⁢(Z)G⁢(ε-f) cannot be ordered. Since ε-f≻0, this contradicts [74, VIII Basic Lemma 1.4].

We have shown the existence of a point x∈ZG such that all f∈𝔪 vanish at x. All g∈A can be written in a unique way as g=r+f with r∈𝐑 and f∈𝔪, hence have no poles at x. Since for all g∈(ℳ⁢(Z)G)*, one of g and 1/g must belong to A, and since 𝐑⊂A, we deduce that A⊂ℳ⁢(Z)G is the set of functions with no poles at x. It follows that 𝔪={f∈ℳ⁢(Z)G∣f⁢(x)=0}.

Let t∈ℳ⁢(Z)G be a uniformizer at x. After replacing t with -t, assume that t≻0. For f∈(ℳ⁢(Z)G)*, there are unique n∈𝐙, r∈𝐑* and g∈𝔪 such that f=tn⁢(r+g), and f≻0 if and only if r>0. It follows that ≺ and ≺x,t coincide. ∎

A.7 Sections of submersions

Let Z be a G-equivariant complex manifold. The total space E of the holomorphic vector bundle E→Z associated with a G-equivariant locally free coherent sheaf ℰ on Z has a natural structure of G-equivariant complex manifold.

The first part of the following proposition is a G-equivariant variant of a particular case of [42, Proposition 3.2]. We explain how to make the proof work G-equivariantly.

Proposition A.10.

Let f:Z→Y be a G-equivariant holomorphic map of G-equivariant complex manifolds, and let u:Y→Z be a G-equivariant holomorphic section of f. Suppose that Y is Stein.

There exist a G-stable open neighborhood U of u⁢(Y) in Z, a G-stable open neighborhood U′ of the zero section u⁢(Y) in Nu⁢(Y)/Z and a G-equivariant biholomorphism U′→∼U respecting the projections to Y that is the identity on u⁢(Y) and induces the identity Nu⁢(Y)/Z=Nu⁢(Y)/Nu⁢(Y)/Z→∼Nu⁢(Y)/Z between normal bundles.
Suppose that Y has no isolated point. Let K⊂Y be a G-stable compact subset and let S⊂Z be a nowhere dense analytic subset. Choose b1,…,bm∈K and r≥0. Then there exist a G-stable open neighborhood Y′ of K in Y and a sequence un:Y′→Z of G-equivariant holomorphic sections of f above Y′ with the same r-jets as u at the points bi, converging uniformly to u on K, and such that no connected component of un⁢(Y′) is included in S.
If Z′ is a G-equivariant complex manifold and if f′:Z′→Y and g:Z→Z′ are G-equivariant holomorphic maps with f=f′∘g such that g is submersive along u⁢(Y), there exist a G-stable open neighborhood W of g∘u⁢(Y) in Z′ and a G-equivariant holomorphic map w:W→Z with g∘w=IdW and w∘g∘u=u.

Proof.

(i) Since u is a holomorphic section of f, it follows that the map f is submersive along u⁢(Y), and we may assume that f is submersive. We may moreover assume that Z is Stein by Proposition A.2. The G-stable sub-vector bundle E⊂TZ consisting of those vectors tangent to the fibers of f satisfies E|u⁢(Y)≃Nu⁢(Y)/Z. By Lemma A.1 (i), there exists a G-equivariant surjection p:Z×𝐂N↠E of holomorphic vector bundles on Z, induced by vector fields V1,…,VN on Z. As follows from Lemma A.1 (ii), there is a G-equivariant morphism q:Nu⁢(Y)/Z→u⁢(Y)×𝐂N of holomorphic vector bundles on u⁢(Y) that is a section of p|u⁢(Y)×𝐂N. Let φti be the holomorphic flow of Vi. For (x,t1,…,tN)∈u⁢(Y)×𝐂N in an appropriate neighborhood of u⁢(Y)×{0} in u⁢(Y)×𝐂N, define

F⁢(x,t1,…,tN)=φt11∘…∘φtNN⁢(x)∈Z.

The map x↦F⁢(q⁢(x)) induces the required biholomorphism between neighborhoods of u⁢(Y) in Nu⁢(Y)/Z and Z by the inverse function theorem, as explained in [42, Proof of Proposition 3.2]. Its G-equivariance follows from our choices.

(ii) By (i), we may assume that Z is a neighborhood of the zero section of a G-equivariant vector bundle E on Y and that u is the zero section. Let K′ be a compact G-stable neighborhood of K in Y, and let Y′ be the union of the connected components of the interior of K′ that meet K. The compactness of K implies that Y′ has finitely many connected components Y1′,…,Yl′. The set Σ of x∈Y such that S contains a neighborhood of u⁢(x) in f-1⁢(x) is nowhere dense in Y because S⊂Z is a nowhere dense analytic subset. Since G acts antiholomorphically on Y and Y has no isolated point, the subset YG⊂Y is nowhere dense. It follows that we can choose yj∈Yj′ for 1≤j≤l such that yj∉Σ∪YG and such that yj and σ⁢(yj) are distinct from the bi. Since yj∉Σ, there exists zj∈f-1⁢(yj)⊂Eyj such that t⁢zj∉S for all 0<t≪1. Since Y is Stein, one can find a section ζ∈H0⁢(Y,E) vanishing to order r at the bi and such that ζ⁢(yj)=zj and ζ⁢(σ⁢(yj))=σ⁢(zj). Replacing ζ with (ζ+σ⁢(ζ))/2 ensures that ζ∈H0⁢(Y,E)G. Since K′ is compact, it follows that ζ/n∈H0⁢(Y,E)G induces a section un of f over Y′ as soon as n≫0. The sequence un has the required properties.

(iii) By part (i), we may assume that Z is a neighborhood of the zero section u⁢(Y) of Nu⁢(Y)/Z. Since g is submersive along u⁢(Y), the map g induces a surjection of G-equivariant vector bundles p:Nu⁢(Y)/Z↠Ng∘u⁢(Y)/Z′. By Lemma A.1 (ii), we can find a G-equivariant splitting s:Ng∘u⁢(Y)/Z′→Nu⁢(Y)/Z of p. The composition g∘(s|s-1⁢(Z)):s-1⁢(Z)→Z′ is the identity on g∘u⁢(Y) and a local diffeomorphism along g∘u⁢(Y), hence induces a G-equivariant biholomorphism ψ:W′→∼W between some G-stable open neighborhoods W′ and W of g∘u⁢(Y) in s-1⁢(Z) and Z′. To conclude, define w:=s∘ψ-1:W→Z. ∎

Acknowledgements

Scheiderer’s paper [87] plays a fundamental role in our results on homogeneous spaces of linear algebraic groups. Its influence is gratefully acknowledged. We thank Roland Huber for sending us a copy of [60], Dmitri Pavlov for having made [52] available to us, Jean-Louis Colliot-Thélène for a useful comment on a previous version of this article and the referees for their careful work.

References

[1] D. Abramovich, J. Denef and K. Karu, Weak toroidalization over non-closed fields, Manuscripta Math. 142 (2013), no. 1–2, 257–271. 10.1007/s00229-013-0610-5Search in Google Scholar

[2] D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0, Invent. Math. 139 (2000), no. 2, 241–273. 10.1007/s002229900024Search in Google Scholar

[3] S. Akbulut and H. King, Polynomial equations of immersed surfaces, Pacific J. Math. 131 (1988), no. 2, 209–217. 10.2140/pjm.1988.131.209Search in Google Scholar

[4] C. Araujo and J. Kollár, Rational curves on varieties, Higher dimensional varieties and rational points (Budapest 2001), Bolyai Soc. Math. Stud. 12, Springer, Berlin (2003), 13–68. 10.1007/978-3-662-05123-8_3Search in Google Scholar

[5] E. Artin and O. Schreier, Eine Kennzeichnung der reell abgeschlossenen Körper, Abh. Math. Semin. Univ. Hamburg 5 (1927), no. 1, 225–231. 10.1007/BF02952522Search in Google Scholar

[6] O. Benoist, The period-index problem for real surfaces, Publ. Math. Inst. Hautes Études Sci. 130 (2019), 63–110. 10.1007/s10240-019-00108-7Search in Google Scholar

[7] O. Benoist and O. Wittenberg, On the integral Hodge conjecture for real varieties. I, Invent. Math. 222 (2020), no. 1, 1–77. 10.1007/s00222-020-00965-8Search in Google Scholar

[8] O. Benoist and O. Wittenberg, On the integral Hodge conjecture for real varieties. II, J. Éc. Polytech. Math. 7 (2020), 373–429. 10.5802/jep.120Search in Google Scholar

[9] V. G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Math. Surveys Monogr. 33, American Mathematical Society, Providence 1990. Search in Google Scholar

[10] J. Bochnak, M. Coste and M.-F. Roy, Real algebraic geometry, Ergeb. Math. Grenzgeb. (3) 36, Springer, Berlin 1998. 10.1007/978-3-662-03718-8Search in Google Scholar

[11] J. Bochnak and W. Kucharz, The Weierstrass approximation theorem for maps between real algebraic varieties, Math. Ann. 314 (1999), no. 4, 601–612. 10.1007/s002080050309Search in Google Scholar

[12] J. Bochnak and W. Kucharz, Approximation of holomorphic maps by algebraic morphisms, Ann. Pol. Math. 80 (2003), 85–92. 10.4064/ap80-0-5Search in Google Scholar

[13] J. Bochnak and W. Kucharz, Algebraic approximation of smooth maps, Univ. Iagel. Acta Math. 48 (2010), 9–40. Search in Google Scholar

[14] A. Borel, Linear algebraic groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York 1991. 10.1007/978-1-4612-0941-6Search in Google Scholar

[15] A. Borel and A. Haefliger, La classe d’homologie fondamentale d’un espace analytique, Bull. Soc. Math. France 89 (1961), 461–513. 10.24033/bsmf.1571Search in Google Scholar

[16] S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models, Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin 1990. 10.1007/978-3-642-51438-8Search in Google Scholar

[17] G. E. Bredon, Sheaf theory, 2nd ed., Grad. Texts in Math. 170, Springer, New York 1997. 10.1007/978-1-4612-0647-7Search in Google Scholar

[18] G. W. Brumfiel, Quotient spaces for semialgebraic equivalence relations, Math. Z. 195 (1987), no. 1, 69–78. 10.1007/BF01161599Search in Google Scholar

[19] V. I. Chernousov, The group of similarity ratios of a canonical quadratic form, and the stable rationality of the variety PSO, Mat. Zametki 55 (1994), no. 4, 114–119, 144. 10.1007/BF02112482Search in Google Scholar

[20] C. Ciliberto and C. Pedrini, Real abelian varieties and real algebraic curves, Lectures in real geometry (Madrid 1994), De Gruyter Exp. Math. 23, De Gruyter, Berlin (1996), 167–256. 10.1515/9783110811117.167Search in Google Scholar

[21] J.-L. Colliot-Thélène, Real rational surfaces without a real point, Arch. Math. (Basel) 58 (1992), no. 4, 392–396. 10.1007/BF01189931Search in Google Scholar

[22] J.-L. Colliot-Thélène, Groupes linéaires sur les corps de fonctions de courbes réelles, J. reine angew. Math. 474 (1996), 139–167. 10.1515/crll.1996.474.139Search in Google Scholar

[23] J.-L. Colliot-Thélène and P. Gille, Remarques sur l’approximation faible sur un corps de fonctions d’une variable, Arithmetic of higher-dimensional algebraic varieties (Palo Alto 2002), Progr. Math. 226, Birkhäuser, Boston (2004), 121–134. 10.1007/978-0-8176-8170-8_7Search in Google Scholar

[24] J.-L. Colliot-Thélène and J.-J. Sansuc, La R-équivalence sur les tores, Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), no. 2, 175–229. 10.24033/asens.1325Search in Google Scholar

[25] J.-L. Colliot-Thélène and J.-J. Sansuc, The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group), Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math. 19, Tata Institute of Fundamental Research, Mumbai (2007), 113–186. Search in Google Scholar

[26] J.-L. Colliot-Thélène, J.-J. Sansuc and P. Swinnerton-Dyer, Intersections of two quadrics and Châtelet surfaces. I, J. reine angew. Math. 373 (1987), 37–107. 10.1515/crll.1987.373.37Search in Google Scholar

[27] A. Comessatti, Fondamenti per la geometria sopra le suerficie razionali dal punto di vista reale, Math. Ann. 73 (1912), no. 1, 1–72. 10.1007/BF01456659Search in Google Scholar

[28] M. Coste and M.-F. Roy, La topologie du spectre réel, Ordered fields and real algebraic geometry (San Francisco 1981), Contemp. Math. 8, American Mathematical Society, Providence (1982), 27–59. 10.1090/conm/008/653174Search in Google Scholar

[29] D. A. Cox, J. B. Little and H. K. Schenck, Toric varieties, Grad. Stud. Math. 124, American Mathematical Society, Providence 2011. 10.1090/gsm/124Search in Google Scholar

[30] H. Delfs and M. Knebusch, Semialgebraic topology over a real closed field. II. Basic theory of semialgebraic spaces, Math. Z. 178 (1981), no. 2, 175–213. 10.1007/BF01262039Search in Google Scholar

[31] H. Delfs and M. Knebusch, On the homology of algebraic varieties over real closed fields, J. reine angew. Math. 335 (1982), 122–163. 10.1515/crll.1982.335.122Search in Google Scholar

[32] H. Delfs and M. Knebusch, An introduction to locally semialgebraic spaces, Rocky Mountain J. Math.14 (1984), 945–963. 10.1216/RMJ-1984-14-4-945Search in Google Scholar

[33] H. Delfs and M. Knebusch, Locally semialgebraic spaces, Lecture Notes in Math. 1173, Springer, Berlin 1985. 10.1007/BFb0074551Search in Google Scholar

[34] J.-P. Demailly, L. Lempert and B. Shiffman, Algebraic approximations of holomorphic maps from Stein domains to projective manifolds, Duke Math. J. 76 (1994), no. 2, 333–363. 10.1215/S0012-7094-94-07612-6Search in Google Scholar

[35] M. Demazure and A. Grothendieck, Schémas en groupes. I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Math. 151, Springer, Berlin 1970. 10.1007/BFb0058993Search in Google Scholar

[36] M. Demazure and A. Grothendieck, Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Math. 152, Springer, Berlin 1970. 10.1007/BFb0059027Search in Google Scholar

[37] F. R. Demeyer and M. A. Knus, The Brauer group of a real curve, Proc. Amer. Math. Soc. 57 (1976), no. 2, 227–232. 10.1090/S0002-9939-1976-0412193-7Search in Google Scholar

[38] A. Ducros, Principe de Hasse pour les espaces principaux homogènes sous les groupes classiques sur un corps de dimension cohomologique virtuelle au plus 1, Manuscripta Math. 89 (1996), no. 3, 335–354. 10.1007/BF02567522Search in Google Scholar

[39] A. Ducros, Fibrations en variétés de Severi–Brauer au-dessus de la droite projective sur le corps des fonctions d’une courbe réelle, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 1, 71–75. 10.1016/S0764-4442(98)80105-5Search in Google Scholar

[40] A. Ducros, L’obstruction de réciprocité à l’existence de points rationnels pour certaines variétés sur le corps des fonctions d’une courbe réelle, J. reine angew. Math. 504 (1998), 73–114. 10.1515/crll.1998.113Search in Google Scholar

[41] S. Finashin and V. Kharlamov, First homology of a real cubic is generated by lines, preprint (2019), https://arxiv.org/abs/1911.07008. 10.1090/conm/772/15485Search in Google Scholar

[42] F. Forstnerič, The Oka principle for sections of stratified fiber bundles, Pure Appl. Math. Q. 6 (2010), no. 3, 843–874. 10.4310/PAMQ.2010.v6.n3.a11Search in Google Scholar

[43] W. Fulton, Introduction to toric varieties, Ann. of Math. Stud. 131, Princeton University, Princeton 1993. 10.1515/9781400882526Search in Google Scholar

[44] S. Gitler, The cohomology of blow ups, Bol. Soc. Mat. Mex. (2) 37 (1992), 167–175. Search in Google Scholar

[45] T. Graber, J. Harris and J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, 57–67. 10.1090/S0894-0347-02-00402-2Search in Google Scholar

[46] H. Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. 10.2307/1970257Search in Google Scholar

[47] H. Grauert and R. Remmert, Theory of Stein spaces, Class. Math., Springer, Berlin 2004. 10.1007/978-3-642-18921-0Search in Google Scholar

[48] M. J. Greenberg, Rational points in Henselian discrete valuation rings, Publ. Math. Inst. Hautes Études Sci. 31 (1966), 59–64. 10.1007/BF02684802Search in Google Scholar

[49] A. Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221. 10.2748/tmj/1178244839Search in Google Scholar

[50] A. Grothendieck and M. Raynaud, Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois Marie 1960/61 (SGA 1). Dirigé par A. Grothendieck, Lecture Notes in Math. 224, Springer, Berlin 1971. 10.1007/BFb0058657Search in Google Scholar

[51] F. Guaraldo, P. Macrì and A. Tancredi, Topics on real analytic spaces, Adv. Lect. Math. (ALM), Friedrich Vieweg & Sohn, Braunschweig 1986. 10.1007/978-3-322-84243-5Search in Google Scholar

[52] R. M. Guralnick, Matrices and representations over rings of analytic functions and other one-dimensional rings, Visiting scholars’ lectures – 1987, Texas Tech Univ. Math. Ser. 15, Texas Tech University, Lubbock (1988), 15–35. Search in Google Scholar

[53] Y. Harpaz and O. Wittenberg, On the fibration method for zero-cycles and rational points, Ann. of Math. (2) 183 (2016), no. 1, 229–295. 10.4007/annals.2016.183.1.5Search in Google Scholar

[54] B. Hassett, Weak approximation and rationally connected varieties over function fields of curves, Variétés rationnellement connexes: aspects géométriques et arithmétiques, Panor. Synthèses 31, Société Mathématique de France, Paris (2010), 115–153. Search in Google Scholar

[55] B. Hassett and Y. Tschinkel, Weak approximation over function fields, Invent. Math. 163 (2006), no. 1, 171–190. 10.1007/s00222-005-0458-8Search in Google Scholar

[56] B. Hassett and Y. Tschinkel, Weak approximation for hypersurfaces of low degree, Algebraic geometry – Seattle 2005. Part 2, Proc. Sympos. Pure Math. 80, American Mathematical Society, Providence (2009), 937–955. 10.1090/pspum/080.2/2483957Search in Google Scholar

[57] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, Ann. of Math. (2) 79 (1964), 109–203. 10.2307/1970486Search in Google Scholar

[58] M. W. Hirsch, Differential topology, Grad. Texts in Math. 33, Springer, New York 1994. Search in Google Scholar

[59] L. Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Math. Libr. 7, North-Holland, Amsterdam 1990. Search in Google Scholar

[60] R. Huber, Isoalgebraische Räume, Ph.D. thesis, Regensburg, 1984. Search in Google Scholar

[61] R. Huber and M. Knebusch, A glimpse at isoalgebraic spaces, Note Mat. 10 (1990), no. 2, 315–336. Search in Google Scholar

[62] J. Huisman, The exponential sequence in real algebraic geometry and Harnack’s inequality for proper reduced real schemes, Comm. Algebra 30 (2002), no. 10, 4711–4730. 10.1081/AGB-120014664Search in Google Scholar

[63] D. Husemoller, Fibre bundles, 3rd ed., Grad. Texts in Math. 20, Springer, New York 1994. 10.1007/978-1-4757-2261-1Search in Google Scholar

[64] B. Kahn, Construction de classes de Chern équivariantes pour un fibré vectoriel réel, Comm. Algebra 15 (1987), no. 4, 695–711. 10.1080/00927872.1987.12088443Search in Google Scholar

[65] M. Kneser, Schwache Approximation in algebraischen Gruppen, Colloque sur la théorie des groupes algébriques (Bruxelles 1962), Librairie Universitaire, Louvain (1962), 41–52. Search in Google Scholar

[66] H. Köditz and S. Timmann, Randschlichte meromorphe Funktionen auf endlichen Riemannschen Flächen, Math. Ann. 217 (1975), no. 2, 157–159. 10.1007/BF01351295Search in Google Scholar

[67] J. Kollár, Rational curves on algebraic varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin 1996. 10.1007/978-3-662-03276-3Search in Google Scholar

[68] J. Kollár, Rationally connected varieties over local fields, Ann. of Math. (2) 150 (1999), no. 1, 357–367. 10.2307/121107Search in Google Scholar

[69] J. Kollár, Specialization of zero cycles, Publ. Res. Inst. Math. Sci. 40 (2004), no. 3, 689–708. 10.2977/prims/1145475489Search in Google Scholar

[70] J. Kollár and F. Mangolte, Approximating curves on real rational surfaces, J. Algebraic Geom. 25 (2016), no. 3, 549–570. 10.1090/jag/658Search in Google Scholar

[71] V. A. Krasnov, Characteristic classes of vector bundles on a real algebraic variety, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 4, 716–746. 10.1070/IM1992v039n01ABEH002223Search in Google Scholar

[72] V. A. Krasnov, On the equivariant Grothendieck cohomology of a real algebraic variety and its application, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 3, 36–52. 10.1070/IM1995v044n03ABEH001608Search in Google Scholar

[73] B. Kripke, Finitely generated coherent analytic sheaves, Proc. Amer. Math. Soc. 21 (1969), 530–534. 10.1090/S0002-9939-1969-0240332-6Search in Google Scholar

[74] T. Y. Lam, Introduction to quadratic forms over fields, Grad. Stud. Math. 67, American Mathematical Society, Providence 2005. Search in Google Scholar

[75] S. Lang, The theory of real places, Ann. of Math. (2) 57 (1953), 378–391. 10.2307/1969865Search in Google Scholar

[76] Y. Liang, Arithmetic of 0-cycles on varieties defined over number fields, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 1, 35–36. 10.24033/asens.2184Search in Google Scholar

[77] P. W. Michor, Manifolds of differentiable mappings, Shiva Math. Ser. 3, Shiva Publishing, Nantwich 1980. Search in Google Scholar

[78] J. S. Milne, Étale cohomology, Princeton Math. Ser. 33, Princeton University, Princeton 1980. 10.1515/9781400883981Search in Google Scholar

[79] S. Mori and S. Mukai, The uniruledness of the moduli space of curves of genus 11, Algebraic geometry (Tokyo/Kyoto 1982), Lecture Notes in Math. 1016, Springer, Berlin (1983), 334–353. 10.1007/BFb0099970Search in Google Scholar

[80] M. Nagata, Local rings, Interscience Tracts Pure Appl. Math. 13, Interscience, New York 1962. Search in Google Scholar

[81] A. Pál and E. Szabó, The fibration method over real function fields, Math. Ann. 378 (2020), no. 3–4, 993–1019. 10.1007/s00208-020-02053-xSearch in Google Scholar

[82] Y. Peterzil and S. Starchenko, Mild manifolds and a non-standard Riemann existence theorem, Selecta Math. (N. S.) 14 (2009), no. 2, 275–298. 10.1007/s00029-008-0064-xSearch in Google Scholar

[83] V. P. Platonov, Birational properties of spinor varieties, Trudy Mat. Inst. Steklov. 157 (1981), 161–169. Search in Google Scholar

[84] M. Rapoport, N. Schappacher and P. Schneider, Beilinson’s conjectures on special values of L-functions, Perspect. Math. 4, Academic Press, Boston 1988. Search in Google Scholar

[85] D. J. Saltman, Noether’s problem over an algebraically closed field, Invent. Math. 77 (1984), no. 1, 71–84. 10.1007/BF01389135Search in Google Scholar

[86] C. Scheiderer, Real and étale cohomology, Lecture Notes in Math. 1588, Springer, Berlin 1994. 10.1007/BFb0074269Search in Google Scholar

[87] C. Scheiderer, Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one, Invent. Math. 125 (1996), no. 2, 307–365. 10.1007/s002220050077Search in Google Scholar

[88] J.-P. Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413–420. 10.1007/978-3-642-37726-6_66Search in Google Scholar

[89] J.-P. Serre, Corps locaux, Hermann, Paris 1968. Search in Google Scholar

[90] J.-P. Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Math. 5, Springer, Berlin 1994. 10.1007/BFb0108758Search in Google Scholar

[91] M. Shen, Rationality, universal generation and the integral Hodge conjecture, Geom. Topol. 23 (2019), no. 6, 2861–2898. 10.2140/gt.2019.23.2861Search in Google Scholar

[92] V. V. Shokurov, Riemann surfaces and algebraic curves, Algebraic geometry. I, Encyclopaedia Math. Sci. 23, Springer, Berlin, (1994) 1–166. Search in Google Scholar

[93] R. Silhol, Real algebraic surfaces, Lecture Notes in Math. 1392, Springer, Berlin 1989. 10.1007/BFb0088815Search in Google Scholar

[94] R. R. Simha, The Behnke–Stein theorem for open Riemann surfaces, Proc. Amer. Math. Soc. 105 (1989), no. 4, 876–880. 10.1090/S0002-9939-1989-0953748-XSearch in Google Scholar

[95] Y. T. Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 (1976/77), no. 1, 89–100. 10.1007/BF01390170Search in Google Scholar

[96] A. Skorobogatov, Torsors and rational points, Cambridge Tracts in Math. 144, Cambridge University, Cambridge 2001. 10.1017/CBO9780511549588Search in Google Scholar

[97] Z. Tian, Weak approximation for cubic hypersurfaces, Duke Math. J. 164 (2015), no. 7, 1401–1435. 10.1215/00127094-2916322Search in Google Scholar

[98] Z. Tian and H. R. Zong, One-cycles on rationally connected varieties, Compos. Math. 150 (2014), no. 3, 396–408. 10.1112/S0010437X13007549Search in Google Scholar

[99] A. Tognoli, Approximation theorems in real analytic and algebraic geometry, Lectures in real geometry (Madrid 1994), De Gruyter Exp. Math. 23, de Gruyter, Berlin (1996), 113–166. 10.1515/9783110811117.113Search in Google Scholar

[100] C. Voisin, On integral Hodge classes on uniruled or Calabi–Yau threefolds, Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math. 45, Mathematical Society of Japan, Tokyo (2006), 43–73. 10.2969/aspm/04510043Search in Google Scholar

[101] C. Voisin, Some aspects of the Hodge conjecture, Jpn. J. Math. 2 (2007), no. 2, 261–296. 10.1007/s11537-007-0639-xSearch in Google Scholar

[102] V. E. Voskresenskiĭ, Algebraic groups and their birational invariants, Transl. Math. Monogr. 179, American Mathematical Society, Providence 1998. Search in Google Scholar

[103] H. Whitney, Differentiable manifolds, Ann. of Math. (2) 37 (1936), no. 3, 645–680. 10.1007/978-1-4612-2974-2_1Search in Google Scholar

[104] E. Witt, Theorie der quadratischen Formen in beliebigen Körpern, J. reine angew. Math. 176 (1937), 31–44. 10.1007/978-3-642-41970-6_1Search in Google Scholar

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Published Online: 2021-03-11

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