an:01257137
Zbl 0936.14039
Krasnov, V. A.
On orientable real algebraic \(M\)-surfaces
EN
Math. Notes 62, No. 4, 434-438 (1997); translation from Mat. Zametki 62, No. 4, 520-526 (1997).
00054228
1997
j
14P25 14F25
real algebraic \(M\)-surface; homology classes; Euler characteristic; fundamental classes
From the text: Let \(X\) be a nonsingular projective real algebraic \(M\)-surface. Suppose that the set of real points \(X(\mathbb{R})\) is orientable, and the homology group \(H_* (X(\mathbb{C}),\mathbb{Z})\) is free. Let \(X_1,\dots, X_s\) be the connected components of \(X(\mathbb{R})\); they determine homology classes
\[
x_1,\dots, x_s\in H_2 (X(\mathbb{C}), \mathbb{F}_2).
\]
Theorem 1. Let \(q\) be the irregularity of \(X\). Then there exists \(q+1\) linearly independent relations between the classes \(x_1,\dots, x_s\).
Note that the number of linearly independent relations between \(x_1,\dots, x_s\) does not exceed \(q+1\). In what follows, \(\chi_k\) denotes the Euler characteristic of \(X_k\), \(k=1,\dots, s\).
Theorem 2. Suppose that \(\chi_k\equiv 0\pmod {2^\mu}\), where \(\mu\geq 1\) and \(k=1,\dots, s\). If \(x_{i_1}+\cdots+ x_{i_n}=0\), then the congruence \(\chi_{i_1}+\cdots+ \chi_{i_n}\equiv 0\pmod {2^{\mu+2}}\) holds.
If \(H_1(X(\mathbb{C}),\mathbb{Z})=0\), then there is only one relation between \(x_1,\dots, x_s\), namely \(x_1+\cdots+ x_s=0\). In the general case, the description of possible relations is based on the Albanese mapping \(\alpha:X\to A\). The set of complex points of the Albanese variety \(A(\mathbb{C})\) is a complex torus with an antiholomorphic involution \(\theta: A(\mathbb{C})\to A(\mathbb{C})\). Since \(X\) is an \(M\)-surface, \(A\) is an \(M\)-variety and the pair \((A(\mathbb{C}),\theta)\) is topologically identified with the complex torus \(\mathbb{C}^q/ \mathbb{Z}^q\) endowed with the complex conjugation involution. The set of real points \(A(\mathbb{R})\) is the union of \(2^q\) real tori. Let \(A(\mathbb{R})^{(i)}\), \(i=1,\dots, q\), be the union of the connected components of \(A(\mathbb{R})\) that are contained in the preimage of the curve \(\mathbb{R}/ \mathbb{Z}\subset \mathbb{C}/\mathbb{Z}\) under the projection \(\pi_i: \mathbb{C}^q/ \mathbb{Z}^q\to \mathbb{C}/\mathbb{Z}\), where \(\pi_i (z_1,\dots, z_q)= z_i\). By \(X(\mathbb{R})^{(i)}\) denote the union of the connected components of \(X(\mathbb{R})\) that are contained in the preimge of \(A(\mathbb{R})^{(i)}\) under the Albanese mapping \(\alpha:X\to A\).
Theorem 3. The fundamental classes \([X(\mathbb{R})]\) and \([X(\mathbb{R})^{(i)}]\), \(i= 1,\dots, q\), equal 0 in \(H_2(X(\mathbb{C}), \mathbb{F}_2)\); the corresponding relations between the classes \(x_1,\dots, x_s\) form a basis of the system of relations.
Now suppose that the basis of the Albanese mapping \(\alpha:X\to A\) is a curve. Then there is another method for obtaining the basis of the system of relations between \(x_1,\dots, x_s\). Let \(C= \alpha(X)\) be a curve; then \(C\) has no singular points. The homomorphism \(\alpha_*: H_1(X(\mathbb{C}), \mathbb{F}_2)\to H_1(C(\mathbb{C}), \mathbb{F}_2)\) is an isomorphism, and the compex conjugation involution acts trivially on the group \(H_1(X(\mathbb{C}), \mathbb{F}_2)\). Therefore, \(C\) is an \(M\)-curve, and the number of the connected components of \(C(\mathbb{R})\) is \(q+1\); denote these components by \(C_1,\dots, C_{q+1}\). Let \(X(\mathbb{R})_i\) be the union of the connected components of \(X(\mathbb{R})\) that are contained in the preimage of \(C_i\), where \(i=1,\dots, q+1\).
Theorem 4. The fundamental classes \([X(\mathbb{R})_i]\), \(i= 1,\dots, q+1\), equal 0 in \(H_2(X(\mathbb{C}), \mathbb{F}_2)\); the corresponding relations between the classes \(x_1,\dots, x_s\) form a basis of the system of relations.