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On orientable real algebraic $$M$$-surfaces. (English. Russian original) Zbl 0936.14039
Math. Notes 62, No. 4, 434-438 (1997); translation from Mat. Zametki 62, No. 4, 520-526 (1997).
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.
##### MSC:
 14P25 Topology of real algebraic varieties 14F25 Classical real and complex (co)homology in algebraic geometry
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