We prove that any topological real line bundle on a compact

real algebraic curve X is isomorphic to an algebraic line bundle. The

result is then generalized to vector bundles of an arbitrary constant rank.

As a consequence we prove that any continuous map from X into a real

Grassmannian can be approximated by regular maps

Maciejewski, Łukasz

Acta Mathematica

English

We prove that any topological real line bundle on a compact real algebraic curve $X$ is isomorphic to an algebraic line bundle. The result is then generalized to vector bundles of an arbitrary constant rank. As a consequence we prove that any continuous map from $X$ into a real Grassmannian can be approximated by regular maps

Biblioteka Nauki - repozytorium artykuÅÃ³w

Vector bundles on real algebraic curves

Jagiellonian Univeristy Repository

UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICAdoi: 10.4467/20843828AM.12.005.1123 FASCICULUS L, 2012VECTOR BUNDLES ON REAL ALGEBRAIC CURVESby  Lukasz MaciejewskiAbstract. We prove that any topological real line bundle on a compactreal algebraic curve X is isomorphic to an algebraic line bundle. Theresult is then generalized to vector bundles of an arbitrary constant rank.As a consequence we prove that any continuous map from X into a realGrassmannian can be approximated by regular maps.1. Introduction. Throughout this paper X denotes a compact real alge-braic curve, that is, a compact 1-dimensional algebraic subset of Rd for somed ∈ N. We refer to [1] for terminology and background material on real alge-braic geometry. In this paper all vector bundles are real vector bundles. Recallthat algebraic vector bundles on X correspond to finitely generated projectivemodules over the ring of real-valued regular functions on X, cf. [1, p. 302]. Ourmain goal is the following:Theorem 1.1. Any topological line bundle on X is isomorphic to an alge-braic line bundle.Theorem 1.1 is proved in section 2. It can be easily generalized.Corollary 1.2. Any topological constant rank vector bundle on X is iso-morphic to an algebraic vector bundle.Proof. Any topological vector bundle on X of constant rank r ≥ 1 splitsoff a trivial vector bundle of rank r− 1, since dim(X) = 1. Hence it suffices toapply Theorem 1.1.2000 Mathematics Subject Classification. 14P05, 14P25.Key words and phrases. Real algebraic curve, algebraic vector bundle, approximation ofcontinuous maps by regular maps.64As a consequence of Corollary 1.2, we obtain a counterpart of the classi-cal Weierstrass approximation theorem for maps from X into the Grassmannvariety Gn,k of k-dimensional vector subspaces of Rn.Corollary 1.3. Let f : X −→Gn,k be a continuous map. Each neighbor-hood of f in the compact-open topology contains a regular map.Proof. It suffices to show that the pullback vector bundle f∗γn,k on X,where γn,k is the tautological vector bundle on Gn,k, is isomorphic to an al-gebraic vector bundle, cf. [1, Theorem 13.3.1]. This however follows fromCorollary 1.2.Since the real variety G2,1 is biregularly isomorphic to the unit circleS1 = {(x, y) ∈ R2 : x2 + y2 = 1},we immediately get:Corollary 1.4. Let f : X −→ S1 be a continuous map. Each neighborhoodof f in the compact-open topology contains a regular map.All the results above are proved in [1] under the assumption that the curveX is nonsingular. The arguments presented in [1] do not directly generalize toyield Theorem 1.1.Corollary 1.5. For every cohomology class u in H1(X;Z/2), there existsa regular map f : X −→ S1 such that f∗(s1) = u, where s1 is the uniquegenerator of the cohomology group H1(S1;Z/2) ∼= Z/2.Proof. There is a one-to-one correspondence between the homotopyclasses of continuous maps from X into S1 and the cohomology classes inH1(X;Z), cf., [2, p. 300]. Since the reduction modulo 2 homomorphismH1(X;Z)−→H1(X;Z/2) is surjective, it follows that each cohomology classin H1(X;Z/2) is of the form f∗(s1) for some continuous map f : X −→ S1.According to Corollary 1.4, the map f can be assumed to be regular.Let us note that Corollary 1.5 implies Theorem 1.1. Indeed, let ξ be a topo-logical line bundle on X. The first Stiefel–Whitney class w1(γ2,1) of the tauto-logical line bundle γ2,1 on G2,1 generates the cohomology group H1(G2,1;Z/2).According to Corollary 1.5, there exists a regular map f : X −→G2,1 satisfyingw1(ξ) = f∗(w1(γ2,1)) = w1(f∗γ2,1). Since topological line bundles are classifiedby the first Stiefel-Whitney class (cf. [3, Proposition 3.10]), it follows that ξ isisomorphic to the algebraic line bundle f∗γ2,1. However, we do not know howto prove Corollary 1.5 without making use of Theorem 1.1.652. Line bundles on real algebraic curves. We first recall a useful con-struction of algebraic line bundles on an arbitrary affine real algebraic varietyV . Lemma 2.1 below is a special case of [1, Theorem 12.1.11].Lemma 2.1. Let {U1, . . . , Ur} be a Zariski open cover of V and lethij : Uj −→R be a regular function satisfying hij(Ui ∩ Uj) ⊂ R \{0} for1 ≤ i, j ≤ r. Assume that hij · hjk = hik on Uj ∩ Uk for all i, j, k, andhii(x) = 1 for all i and x in Ui. LetE = {(x, (v1, . . . , vr)) ∈ V × Rr : vi = hij(x)vj for x ∈ Uj , 1 ≤ i, j ≤ r}and let p : E−→V be defined by p(x, (v1, . . . , vr)) = x. Then ξ = (E, p, V ) isan algebraic line subbundle of the product vector bundle on V with total spaceV × Rr, and the mapUi × R−→ p−1(Ui), (x, v) 7→ (h1i(x)v, . . . , hri(x)v))is an algebraic trivialization of ξ over Ui for 1 ≤ i ≤ r.For any vector bundle η and any global section s of η, let Z(s) denote thezero locus of s.The set Reg(X) of nonsingular points of X in dimension 1 is a Zariskiopen subset of X, cf. [1, p. 69]. Furthermore, Reg(X) is a 1-dimensional C∞manifold.Lemma 2.2. Let x0 be a point in Reg(X). There exists an algebraic linebundle ξ = (E, p,X) on X which admits an algebraic section s : X −→E suchthat Z(s) = {x0} and the restriction of s to Reg(X) is transverse to the zerosection of ξ.Proof. Let RX be the sheaf of real-valued regular functions on X. Forany point x on X, we identify the stalk RX,x with the localization of the ringRX(X) at the maximal idealmx = {f ∈ RX(X) : f(x) = 0},cf. [1, Proposition 3.2.3]. Since the point x0 is in Reg(X), the stalk RX,x0is a regular local ring of dimension 1 and thus a principal ideal domain. Inparticular, the ideal mx0 RX,x0 of the ring RX,x0 is principal. Thus we canfind a regular function f1 in mx0 and a Zariski open neighborhood U1 of x0 inReg(X) such thatmx0 RX(U1) = (f1)RX(U1).In particular, f1|U1 : U1−→R is a C∞ function for which 0 in R is a regularvalue and (f1|U1)−1(0) = {x0}.Let f2 be any regular function in mx0 with f−12 (0) = {x0}, e.g., a polyno-mial given by the formula ‖x− x0‖2, where ‖ · ‖ denotes the euclidean metric66in Rd. We havef2|U1 = h21f1|U1for some regular function h21 : U1−→R. If U2 = X \ {x0}, thenh12 =f1f2: U2−→Ris a regular function on U2. By construction, the sets h21(U1∩U2) and h12(U1∩U2) are contained in R \{0}. Define h11 : U1−→R and h22 : U2−→R to beconstant functions identically equal to 1. Let ξ = (E, p,X) be the algebraicline bundle on X determined, as in Lemma 2.1, by the Zariski open cover{U1, U2} of X and the regular functions hij . Note thats : X −→E, s(x) = (x, (f1(x), f2(x)))is an algebraic section of ξ with Z(s) = {x0}. On the set U1, the section s isrepresented by the mapU1−→U1 × R , x 7→ (x, f1(x)),and hence the restriction of s to Reg(X) is transverse to the zero sectionof ξ.We will now give a convenient description of the first cohomology groupH1(X;Z/2) of the curve X. The subset X \ Reg(X) of X is finite. If X hasnonsingular connected components, we choose one arbitrary point in each ofthose and denote the set of such points by Z. The curve X can be regardedas a graph (1-dimensional CW complex) with (X \ Reg(X)) ∪ Z as the set ofvertices. This assertion is a straightforward consequence of the triangulationtheorem for semi-algebraic sets, cf. [1, Theorem 9.2.1].Lemma 2.3. There exist subgraphs X1, . . . , Xn of X such that each Xi ishomeomorphic to the unit circle S1, and the inclusion maps Xi ↪→ X inducean isomorphismϕ : H1(X;Z/2)−→n⊕i=1H1(Xi;Z/2)Proof. Let K be a connected 1-dimensional component of X and let Tbe a maximal tree of the graph K. The quotient map q : K −→K/T is ahomotopy equivalence and the quotient space K/T is homeomorphic to thewedge sum of a finite number of pointed circles, [2, p. 153]. Each such pointedcircle corresponds to a subset of K/T of the form q(C), where C is a subgraphof K homeomorphic to the unit circle. The inclusion maps q(C) ↪→ K/Tinduce an isomorphismψ : H1(K/T ;Z/2)−→⊕CH1(q(C);Z/2)67If qC : C −→ q(C) is the restriction of the map q, then the homomorphismα =⊕Cq∗C :⊕CH1(q(C);Z/2)−→⊕CH1(C;Z/2)is an isomorphism. The homomorphismq∗ : H1(K/T ;Z/2)−→H1(K;Z/2)is an isomorphism, the quotient map being a homotopy equivalence. Finally,the inclusion maps C ↪→ K induce a homomorphismϕK : H1(K;Z/2)−→⊕CH1(C;Z/2)satisfying ϕK ◦ q∗ = α ◦ ψ. Consequently, ϕK is an isomorphism.The assertion of the lemma follows, because X has finitely many connectedcomponents.Proof of Theorem 1.1. The isomorphism classes of topological linebundles on X form a group, denoted Vect1(X), with tensor product as thegroup operation. The first Stiefel–Whitney class gives a group isomorphismbetween Vect1(X) and the first cohomology group H1(X;Z/2), cf. [3, Proposi-tion 3.10]. Also, note that the isomorphism classes of algebraic vector bundlesform a subgroup of Vect1(X). Hence, in view of Lemma 2.3, it remains to con-struct for each i = 1, . . . , n an algebraic line bundle ξi on X with w1(ξi|Xi) 6= 0and w1(ξi|Xj ) = 0 for all j 6= i (note that H1(Xi;Z/2) ∼= Z/2). Such a linebundle ξi can be obtained as follows.Let xi be a point in(Xi ∩ Reg(X)) \⋃j 6=iXjand let ξ = (E, p,X) be an algebraic line bundle on X as in Lemma 2.2 withx0 = xi. There exists an algebraic section s : X −→E such that Z(s) = {xi}and the restriction of s to Reg(X) is transverse to the zero section of ξ. Itfollows that the line bundle ξ|Xj is trivial and w1(ξ|Xj ) = 0 for j 6= i.Suppose for a moment that the line bundle ξ|Xi is trivial, and letθ : p−1(Xi)−→Xi × Rbe a topological trivialization of ξ|Xi . Then θ(s(x)) = (x, f(x)) for each x inXi,where f : Xi−→R is a continuous function. By construction, f−1(0) = {xi}.The function f does not change sign on Xi \ {xi}, the set Xi \ {xi} beinghomeomorphic to R. This however is impossible since s is transverse to thezero section of ξ in a neighborhood of xi. Consequently, the line bundle ξ|Xiis nontrivial and w1(ξ|Xi) 6= 0.We complete the proof by setting ξi = ξ.68References1. Bochnak J., Coste M., Roy M.-F., Real Algebraic Geometry, Springer-Verlag, Berlin–Heidelberg–New York 1998.2. Bredon G. E., Topology and geometry, Springer-Verlag, New York, 1993.3. Hatcher A., Vector Bundles and K-theoryhttp://www.math.cornell.edu/~hatcher/VBKT/VBpage.htmlReceived August 7, 2012Institute of MathematicsJagiellonian University Lojasiewicza 630-348 Kraków, Poland

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Vector bundles on real algebraic curves

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Biblioteka Nauki - repozytorium artykuÅÃ³w

Jagiellonian Univeristy Repository

Vector bundles on real algebraic curves

Abstract

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Full text

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Biblioteka Nauki - repozytorium artykuÅÃ³w

Jagiellonian Univeristy Repository

Biblioteka Nauki - repozytorium artykuÅÃ³w