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Morphisms, line bundles and moduli spaces in real algebraic geometry. (English) Zbl 0938.14033

From the introduction: Given two nonsingular real algebraic varieties \(X\) and \(Y\), with \(X\) always assumed to be compact, we regard the set \({\mathcal R}(X,Y)\) of all regular maps from \(X\) into \(Y\) as a subset of the space \({\mathcal C}^\infty (X,Y)\) of all \({\mathcal C}^\infty\)-maps from \(X\) into \(Y\), endowed with the \({\mathcal C}^\infty\) topology. The main object of our interest is the set \({\mathcal C}^\infty_{\mathcal R}(X,Y)=\) the closure of \({\mathcal R}(X,Y)\) in \({\mathcal C}^\infty(X,Y)\). In other words, we investigate which \({\mathcal C}^\infty\)-maps from \(X\) into \(Y\) can be approximated by regular maps. Of course, a precursor of this problem is the classical Stone-Weierstrass approximation theorem, where \(Y=\mathbb{R}\).
We denote by \(VB^1_\mathbb{C}(X)\) the group of isomorphism classes of topological \(\mathbb{C}\)-line bundles on \(X\), with group operation induced by tensor product of \(\mathbb{C}\)-bundles. Since \(X\) is compact, the subgroup \(VB^1_{\mathbb{C} \text{-alg}} (X)\) of \(VB^1_\mathbb{C}(X)\) that consists of the isomorphism classes of topological \(\mathbb{C}\)-line bundles on \(X\) admitting an algebraic structure is canonically isomorphic to the Picard group \(\text{Pic}({\mathcal R} (X,\mathbb{C}))\) of isomorphism classes of invertible \({\mathcal R}(X, \mathbb{C})\)-modules.
The importance of the group \(H^2_{\mathbb{C}\text{-alg}}(X,\mathbb{Z})\) stems from the following, already known result:
Theorem 1.0. Let \(X\) be a compact nonsingular real algebraic variety. Then the canonical isomorphism \(c_1:VB^1_\mathbb{C}(X)\to H^2(X,\mathbb{Z})\), induced by the first Chern class, maps \(VB^1_{\mathbb{C} \text{-alg}}(X)\) onto \(H^2_{\mathbb{C}\text{-alg}}(X, \mathbb{Z})\). Furthermore, given a \({\mathcal C}^\infty\)-map \(f:X\to S^2\), the following conditions are equivalent:
(a) \(f\) is in \({\mathcal C}^\infty_{\mathcal R} (X,S^2)\);
(b) \(f\) is homotopic to a regular map from \(X\) into \(S^2\);
(c) \(H^2(f) (\kappa)\) is in \(H^2_{\mathbb{C} \text{-alg}} (X,\mathbb{Z})\), where \(\kappa\) is a generator of \(H^2(S^2,\mathbb{Z})\cong \mathbb{Z}\).
Denote by \(\pi^2(X)\) the set of homotopy classes \([f]\) of \({\mathcal C}^\infty\)-maps \(f:X\to S^2\).
It follows from theorem 1.0 that the image of \(\pi^2_{\mathcal R}(X)= \{[f]\in \pi^2(X)|f\in {\mathcal R} (X,S^2)\}\), under \(h_X\) is precisely \(H^2_{\mathbb{C}\text{-alg}} (X,\mathbb{Z})\), where \(h_X([f])= H^2(f)(\kappa)\). In particular, \(\pi^2_{\mathcal R}(X)\) is a subgroup of \(\pi^2(X)\) that determines completely \({\mathcal C}^\infty_{\mathcal R} (X,S^2)\). If \(X\) is connected and orientable, then \(\pi^2(X)\) is isomorphic to \(\mathbb{Z}\) and, in turn, the subgroup \(\pi^2_{\mathcal R}(X)\) is determined completely by a single numerical invariant \(b(X)\), wich is a unique nonnegative integer satisfying \(b(X)\pi^2(X)= \pi^2_{\mathcal R} (X)\).
Clearly, \(b(X)=1\) if and only if the set \({\mathcal R}(X,S^2)\) is dense in \({\mathcal C}^\infty(X,S^2)\). Similarly, \(b(X)=0\) if and only if every regular map from \(X\) into \(S^2\) is null homotopic. More generally, a \({\mathcal C}^\infty\)-map \(f:X\to S^2\) belongs to \({\mathcal C}^\infty_{\mathcal R}(X,S^2)\) if and only if the topological degree \(\deg(f)\) of \(f\), computed with respect to some fixed orientations on \(X\) and \(S^2\), is a multiple of \(b(X)\).
Let \({\mathcal X}\) be a \(g\)-dimensional abelian variety over \(\mathbb{R}\). Then \(X={\mathcal X}(\mathbb{R})\) is a commutative real algebraic group with \(2^r\) connected components, \(0\leq r\leq g\) each of them diffeomorphic to \(\mathbb{R}^g/ \mathbb{Z}^g\). Given a point \(x\) in \(X\), let \(t_x: X\to X\) denote the translation by \(x\).
Proposition 1.1. Let \(VB^1_\mathbb{C}(X)^{\text{inv}}\) and \(H^2(X,\mathbb{Z})^{\text{inv}}\) be the \(t_x\)-invariant subgroups of \(VB^1_\mathbb{C}(X)\) and \(H^2(X,\mathbb{Z})\). They are free abelian groups of rank \((g-1)g/2\), which satisfy \[ c_1(VB^1_\mathbb{C} (X)^{\text{inv}})=H^2(X,\mathbb{Z})^{\text{inv}},\quad VB^1_{\mathbb{C} \text{-alg}} (X)\subseteq VB^1_\mathbb{C} (X)^{\text{inv}}, \quad H^2_{\mathbb{C} \text{-alg}} (X,\mathbb{Z})\subseteq H^2(X, \mathbb{Z})^{\text{inv}}. \] Proposition 1.1 provides a natural “upper bound” for the size of the groups \(VB^1_{\mathbb{C} \text{-alg}} (X)\) and \(H^2_{\mathbb{C}\text{-alg}} (X,\mathbb{Z})\). Clearly, \(H^2(X, \mathbb{Z})^{\text{inv}}= H^2(X, \mathbb{Z})\) is equivalent to the connectedness of \(X\), and hence, in view of proposition 1.1, \(X\) is connected if \(VB^1_{\mathbb{C}\text{-alg}}(X)= VB^1_\mathbb{C} (X)\). Interjecting into this argument theorem 1.0 and the fact that the group \(H^2(X,\mathbb{Z})\) is generated by the elements of the form \(H^2(f) (\kappa)\), where \(f:X\to S^2\) is a \({\mathcal C}^\infty\)-map, we also conclude that density of \({\mathcal R}(X,S^2)\) in \({\mathcal C}^\infty(X,S^2)\) implies connectedness of \(X\).
It is known that \({\mathcal X}\) admits a period matrix of the form \((Z,I_g)\), where \(Z\) is a complex \(g\times g\) matrix and \(I_g\) is the identity \(g\times g\) matrix. Denote by \(\text{Mat}_g(\mathbb{Z})\) the \(\mathbb{Z}\)-module of all \(g\times g\) matrices with entries in \(\mathbb{Z}\). Let \(\text{Alt}_g(\mathbb{Z})\) denote all antisymmetric matrices, \(A=-^tA\).
Theorem 1.3. Let \({\mathcal X}\) be a \(g\)-dimensional abelian variety over \(\mathbb{R}\) and let \(X= {\mathcal X}(\mathbb{R})\). If \(\Omega=(Z,I_g)\) is a period matrix of \({\mathcal X}\), then every \(\text{Gal} (\mathbb{C}/\mathbb{R})\)-equivariant isomorphism of complex Lie groups \(\varphi: \mathbb{C}^g/[\Omega] \to{\mathcal X} (\mathbb{C})\) gives rise to a group isomorphism \(\tau_\varphi:H^2(X,\mathbb{Z})^{\text{inv}}\to\text{Alt}_g(\mathbb{Z})\).
Theorem 1.6. Let \({\mathcal A}^g_\mathbb{R}\) be the moduli space of \(g\)-dimensional principally polarized abelian varieties over \(\mathbb{R}\).
(i) The set \(\{[{\mathcal Y}]\in{\mathcal A}^g_\mathbb{R}\mid VB^1_{\mathbb{C} \text{-alg}} ({\mathcal Y}(\mathbb{R}))=0\}\) is the intersection of a countable family of open and dense subsets of \({\mathcal A}^g_\mathbb{R}\).
(ii) The set \(\{[{\mathcal Y}] \in {\mathcal A}^g_\mathbb{R} \mid\text{rank} VB^1_{\mathbb{C} \text{-alg}} ({\mathcal Y}(\mathbb{R}))= (g-1)g/2\}\) is uncountable and dense in \({\mathcal A}^g_\mathbb{R}\).
Theorem 1.7. The intersection of the set \(\{[{\mathcal Y}]\in{\mathcal A}^2_\mathbb{R}\mid b({\mathcal Y} (\mathbb{R})) =1\}\), with each connected component of \({\mathcal A}^2_\mathbb{R}\) is uncountable.
We restrict our attention to the case \(n=2\). Then the set \({\mathcal C}^\infty_{\mathcal R} (X_1\times X_2,S^2)\) is completely determined by the group \(\pi^2_{\mathcal R}(X_1 \times X_2)\), which is used as a main device in our presentation. Let \({\mathcal M}^g_\mathbb{R}\) be the moduli space of algebraic curves over \(\mathbb{R}\) of genus \(g\). It is well known that the family \(\{{\mathcal M}_\mathbb{R}^{(g,s, \varepsilon)} \mid (s, \varepsilon)\to\Lambda_g \cup\{(0,2)\}\}\), where \(\Lambda_g = \Lambda^1_g \cup\Lambda_g^2\), \(\Lambda^1_g=\bigl\{(s,1)\mid s\in\mathbb{Z}\), \(1\leq s\leq g\}\), \(\Lambda^2_g =\bigl\{(s,2) \mid s\in\mathbb{Z}\), \(1\leq s\leq g+1,\;s\equiv g+1 \mod 2\bigr\}\), \({\mathcal M}_\mathbb{R}^{(g,s,\varepsilon)} =\bigl\{[X]\in {\mathcal M}^g_\mathbb{R} \mid\bigl(s( {\mathcal X},\varepsilon({\mathcal X}))=(s, \varepsilon) \bigr\}\), is the set of connected components of \({\mathcal M}^g_\mathbb{R}\). Furthermore, \[ \dim {\mathcal M}_\mathbb{R}^{(g,s, \varepsilon)}= \begin{cases} g\quad & \text{ for }0\leq g\leq 1\\ 3g-3 \quad & \text{for } g\geq 2 \end{cases} \] for all \((s, \varepsilon)\) in \(\Lambda_g\cup \{(0,2).\}\).
Proposition 1.8. With the notation as above, \[ \text{rank} \pi^2_{\mathfrak R}(X_1\times X_2)\leq\bigl( ({\mathcal X}_1)-\varepsilon ({\mathcal X}_1)+ 1\bigr)\leq g({\mathcal X}_1)g({\mathcal X}_2). \] Theorem 1.10. Let \({\mathcal X}_k\) be an algebraic curve over \(\mathbb{R}\) of genus \(g_k\) with \(X_k={\mathcal X}_k(\mathbb{R})\) nonempty for \(k=1,2\). Let \(Z(Z_k,I_{g_k})\) be a period matrix of the Jacobian variety of \({\mathcal X}_k\). Then there exists a homomorphism \(\tau:\text{Mat}(g_1\times g_2,\mathbb{Z})\to\pi^2(X_1\times X_2)\).

MSC:

14P25 Topology of real algebraic varieties
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14D20 Algebraic moduli problems, moduli of vector bundles
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References:

[1] J. Bochnak, M. Coste andM.-F. Roy,Géométrie Algébrique Réelle, Ergeb. Math., vol.12, Springer-Verlag, 1987.
[2] J. Bochnak andW. Kucharz, Algebraic approximation of mappings into spheres,Michigan Math. J.,34 (1987), 119–125. · Zbl 0631.14019 · doi:10.1307/mmj/1029003489
[3] J. Bochnak andW. Kucharz, Representation of homotopy classes by algebraic mappings,J. Reine Angew. Math. 377 (1987), 159–169. · Zbl 0619.14014
[4] J. Bochnak andW. Kucharz, On real algebraic morphisms into even-dimensional spheres,Ann. of Math. 128 (1988), 415–433. · Zbl 0674.14013 · doi:10.2307/1971431
[5] J. Bochnak andW. Kucharz, Algebraic models of smooth manifolds,Invent. Math. 97 (1989), 585–611. · Zbl 0687.14023 · doi:10.1007/BF01388891
[6] J. Bochnak andW. Kucharz, Nonisomorphic algebraic models of a smooth manifold,Math. Ann. 290 (1991), 1–2. · Zbl 0714.14012 · doi:10.1007/BF01459234
[7] J. Bochnak andW. Kucharz, Vector bundles on a product of real cubic curves,K-Theory (1992) 487–497. · Zbl 0782.14014
[8] J. Bochnak andW. Kucharz, Complex cycles on real algebraic models of a smooth manifold,Proc. Amer. Math. Soc. 114 (1992), 1097–1104. · Zbl 0760.57011 · doi:10.1090/S0002-9939-1992-1093594-2
[9] J. Bochnak andW. Kucharz, Elliptic curves and real algebraic morphisms,J. Algebraic Geometry 2 (1993), 635–666. · Zbl 0807.14046
[10] J. Bochnak, M. Buchner andW. Kucharz, Vector bundles over real algebraic varieties,K-Theory 3 (1989), 271–298. Erratum,K-Theory 4 (1990), p. 103. · Zbl 0761.14020 · doi:10.1007/BF00533373
[11] P. Deligne, Le théorème de Noether, in: SGA 7II, p. 328–340,Lecture Notes in Math. 340, Springer, 1973. · doi:10.1007/BFb0060515
[12] A. Dold,Lectures on Algebraic Topology, Springer-Verlag, 1972. · Zbl 0234.55001
[13] T. Ekedahl etJ.-P. Serre, Exemples de courbes algébriques à Jacobienne complètement décomposable,C.R. Acad. Sci. Paris 317, Série I (1993), 509–513. · Zbl 0789.14026
[14] B. H. Gross andJ. Harris, Real algebraic curves,Ann. Scient. Éc. Norm. Sup. 14 (1981), 157–182. · Zbl 0533.14011
[15] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero,Ann. of Math. 79 (1964), 109–326. · Zbl 0122.38603 · doi:10.2307/1970486
[16] M. Hirsch,Differential Topology, New York and Berlin, Springer, 1976.
[17] S. T. Hu,Homotopy Theory, New York, Academic Press, 1959.
[18] S. Lang,Abelian Varieties, Interscience, New York, 1959.
[19] H. Lange, Produkte elliptischer Kurven,Nachr. Akad. Wiss. Göttingen 8 (1975), 95–108. · Zbl 0317.14022
[20] H. Lange and Ch.Birkenhake,Complex Abelian Varieties, Grundlehren der Math. Wissenschaften, vol.302, Springer-Verlag, 1992. · Zbl 0779.14012
[21] J.-L. Loday, Applications algébriques du tore dans la sphère et du S p {\(\times\)}S q dans S p+q , Algebraic K-Theory II,Lecture Notes in Math., vol.342, 79–91, Springer-Verlag, 1973, p. 79–91. · doi:10.1007/BFb0073720
[22] J. S. Milne, Jacobian varieties, inArithmetic Geometry, edited by G. Cornell and J. H. Silverman, Springer-Verlag, 1986, p. 167–212.
[23] Y. Namikawa,Toroidal Compactification of Siegel Spaces, Lecture Notes in Math. vol.812, Springer-Verlag, 1980. · Zbl 0466.14011
[24] S. M. Natanzon, Moduli spaces of real curves,Trans. Moscow Math. Soc. Issue 1 (1980), 233–272. · Zbl 0452.14006
[25] M. Seppälä, Moduli space of stable real algebraic curves,Ann. Scient. Éc. Norm. Sup. 24 (1991), 519–544. · Zbl 0757.32011
[26] M. Seppälä andR. Silhol, Moduli spaces for real algebraic curves and real abelian varieties,Math. Zeitschrift 201 (1989), 151–165. · Zbl 0645.14012 · doi:10.1007/BF01160673
[27] J.-P. Serre, Faisceaux algébriques cohérents,Ann. of Math. 61 (1955), 197–278. · Zbl 0067.16201 · doi:10.2307/1969915
[28] I. R. Shafarevich,Basic Algebraic Geometry, Springer-Verlag, 1977. · Zbl 0362.14001
[29] R. Silhol,Real Algebraic Surfaces, Lecture Notes in Math. vol.1392, Springer-Verlag, 1989. · Zbl 0691.14010
[30] R. Silhol, Compactifications of moduli spaces in real algebraic geometry,Invent. Math. 107 (1992), 151–201. · Zbl 0777.14014 · doi:10.1007/BF01231886
[31] R. G. Swan, Vector bundles and projective modules,Trans. Amer. Math. Soc. 105 (1962), 264–277. · Zbl 0109.41601 · doi:10.1090/S0002-9947-1962-0143225-6
[32] R. G. Swan, Topological examples of projective modules,Trans. Amer. Math. Soc. 230 (1977), 201–234. · Zbl 0443.13005 · doi:10.1090/S0002-9947-1977-0448350-9
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