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 \(\tau:\text{Mat}(g_1\times g_2,\mathbb{Z})\to\pi^2(X_1\times X_2)\).


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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