A nonalgebraic patchwork.

*(English)*Zbl 1142.14037This paper deals with real pseudoholomorphic curves in the \(n\)-th rational geometrically ruled surface, denoted by \(\Sigma_n\). A real pseudoholomorphic curve \(C\) in \(\Sigma_n\) is an immersed Riemann surface which is a \(J\)-holomorphic curve in some tame almost complex structure \(J\) such that the exceptional section is \(J\)-holomorphic, \(\sigma(C)=C\) and \(\sigma_*\circ J_p=-J_p\circ \sigma_*\), where \(\sigma\) is the standard complex conjugation and \(p\) is any point of \(C\).

Since its introduction by M. Gromov [Invent. Math. 82, 307–347 (1985; Zbl 0592.53025)] it has been realized that real pseudoholomorphic curves share a lot of properties with real algebraic ones, as it was shown, e.g. by S. Yu. Orevkov [Topology 38, No. 4, 779–810 (1999; Zbl 0923.14032)]. In the opposite direction, the aim of the paper under review is to construct pseudoholomorphic curves with some kind of nonalgebraic behaviour.

Recall that \(\Sigma_n\) is the surface obtained by taking four copies of \(\mathbb C^2\) with coordinates \((x, y)\), \((x_2, y_2)\), \((x_3, y_3)\) and \((x_4, y_4)\), and gluing them along \((\mathbb C^*)^2\) with the identifications \((x_2, y_2)=(1/x, y/x^n)\), \((x_3, y_3)=(x, 1/y)\) and \((x_4, y_4)=(1/x, x^n/y)\). The projection \(\pi:(x, y)\mapsto x\) on \(\Sigma_n\) defines a \(\mathbb C \mathbb P^1\)-boundle over \(\mathbb C\mathbb P^1\). The surface \(\Sigma_n\) has a natural real structure induced by the complex conjugation in \(\mathbb C^2\), and the real part \(\mathbb R\Sigma_n\) is a torus if \(n\) is even and a Klein bottle if \(n\) is odd. The restriction of \(\pi\) on \(\mathbb R\Sigma_n\) defines a pencil of lines denoted by \({\mathcal L}\).

Denote by \(E\), \(B\) and \(F\) the algebraic curves in \(\Sigma_n\) defined by the equation \(\{y_3=0\}\), \(\{y=0\}\), and \(\{x=0\}\), respectively. The group \(H_2(\Sigma_n, \mathbb Z)\) is isomorphic to \(\mathbb Z\times \mathbb Z\) and it is generated by the homology classes of \(B\) and \(F\). Moreover, one has \(E=B-nF\). The bidegree of an algebraic or pseudoholomorphic curve \(C\) in \(\Sigma_n\) is the pair of integers \((k, \ell)\) if \(C\) realizes the homology class \(kB +\ell F\) in \(H_2(\Sigma_n, \mathbb Z)\). Then its equation in \(\Sigma_n\setminus E\) has the form \(\sum_{i=0}^ka_{k-i}(X, Z)Y^i\) where each \(a_j(X, Z)\) is a homogeneous polynomial of degree \(nj + \ell\).

Two real pseudoholomorphic curves \(C_1\) and \(C_2\) are said to be \({\mathcal L}\)- isotopic if there exists an isotopy \(\phi(t, x)\) of \(\mathbb R \Sigma_n\) mapping \(C_1\) onto \(C_2\) such that for any \(t\in [0, 1]\), for any \(p\in C_1\) and for any fiber \(\xi\) of \(\mathbb R \Sigma_n\), the image \(\phi(t, \xi)\) is a fiber of \(\mathbb R \Sigma_n\) and the intersection multiplicity of \(C_1\) and \(\xi\) at \(p\) equals the intersection multiplicity of \(\phi(t, C_1)\) and \(\phi(t, \xi)\) at the point \(\phi(t, p)\).

A smooth curve \(C\) in \(\mathbb R \Sigma_n\) is said to be \({\mathcal L}\)- nonsingular if \(C\) intersects any fiber transversally, except for a finite number of fibers which have an ordinary tangency point with one of the branches of \(C\) and intersect transversally the other branches of \(C\). A smooth curve \(C\) in \(\mathbb R \Sigma_n\) is said to be smooth \({\mathcal L}\)- singular if \(C\) is \({\mathcal L}\)- singular.

The theorem proved in this paper is stated as follows: For any \(d\geq 3\) there exists a smooth \({\mathcal L}\)- singular real pseudoholomorphic patchworked curve of bidegree \((d, 0)\) in \(\Sigma_2\) which is not \({\mathcal L}\)- isotopic to any real algebraic curve in \(\Sigma_2\) of the same bidegree.

This short paper is very well written, and the interested reader may find interesting to study it in full detail. The construction is rather clever and the proof heavily relies, as expected, on the Pseudoholomorphic Patchworking Theorem by I. Itenberg and E. Shustin [Turk. J. Math. 26, No. 1, 27–51 (2002; Zbl 1047.14047)].

Since its introduction by M. Gromov [Invent. Math. 82, 307–347 (1985; Zbl 0592.53025)] it has been realized that real pseudoholomorphic curves share a lot of properties with real algebraic ones, as it was shown, e.g. by S. Yu. Orevkov [Topology 38, No. 4, 779–810 (1999; Zbl 0923.14032)]. In the opposite direction, the aim of the paper under review is to construct pseudoholomorphic curves with some kind of nonalgebraic behaviour.

Recall that \(\Sigma_n\) is the surface obtained by taking four copies of \(\mathbb C^2\) with coordinates \((x, y)\), \((x_2, y_2)\), \((x_3, y_3)\) and \((x_4, y_4)\), and gluing them along \((\mathbb C^*)^2\) with the identifications \((x_2, y_2)=(1/x, y/x^n)\), \((x_3, y_3)=(x, 1/y)\) and \((x_4, y_4)=(1/x, x^n/y)\). The projection \(\pi:(x, y)\mapsto x\) on \(\Sigma_n\) defines a \(\mathbb C \mathbb P^1\)-boundle over \(\mathbb C\mathbb P^1\). The surface \(\Sigma_n\) has a natural real structure induced by the complex conjugation in \(\mathbb C^2\), and the real part \(\mathbb R\Sigma_n\) is a torus if \(n\) is even and a Klein bottle if \(n\) is odd. The restriction of \(\pi\) on \(\mathbb R\Sigma_n\) defines a pencil of lines denoted by \({\mathcal L}\).

Denote by \(E\), \(B\) and \(F\) the algebraic curves in \(\Sigma_n\) defined by the equation \(\{y_3=0\}\), \(\{y=0\}\), and \(\{x=0\}\), respectively. The group \(H_2(\Sigma_n, \mathbb Z)\) is isomorphic to \(\mathbb Z\times \mathbb Z\) and it is generated by the homology classes of \(B\) and \(F\). Moreover, one has \(E=B-nF\). The bidegree of an algebraic or pseudoholomorphic curve \(C\) in \(\Sigma_n\) is the pair of integers \((k, \ell)\) if \(C\) realizes the homology class \(kB +\ell F\) in \(H_2(\Sigma_n, \mathbb Z)\). Then its equation in \(\Sigma_n\setminus E\) has the form \(\sum_{i=0}^ka_{k-i}(X, Z)Y^i\) where each \(a_j(X, Z)\) is a homogeneous polynomial of degree \(nj + \ell\).

Two real pseudoholomorphic curves \(C_1\) and \(C_2\) are said to be \({\mathcal L}\)- isotopic if there exists an isotopy \(\phi(t, x)\) of \(\mathbb R \Sigma_n\) mapping \(C_1\) onto \(C_2\) such that for any \(t\in [0, 1]\), for any \(p\in C_1\) and for any fiber \(\xi\) of \(\mathbb R \Sigma_n\), the image \(\phi(t, \xi)\) is a fiber of \(\mathbb R \Sigma_n\) and the intersection multiplicity of \(C_1\) and \(\xi\) at \(p\) equals the intersection multiplicity of \(\phi(t, C_1)\) and \(\phi(t, \xi)\) at the point \(\phi(t, p)\).

A smooth curve \(C\) in \(\mathbb R \Sigma_n\) is said to be \({\mathcal L}\)- nonsingular if \(C\) intersects any fiber transversally, except for a finite number of fibers which have an ordinary tangency point with one of the branches of \(C\) and intersect transversally the other branches of \(C\). A smooth curve \(C\) in \(\mathbb R \Sigma_n\) is said to be smooth \({\mathcal L}\)- singular if \(C\) is \({\mathcal L}\)- singular.

The theorem proved in this paper is stated as follows: For any \(d\geq 3\) there exists a smooth \({\mathcal L}\)- singular real pseudoholomorphic patchworked curve of bidegree \((d, 0)\) in \(\Sigma_2\) which is not \({\mathcal L}\)- isotopic to any real algebraic curve in \(\Sigma_2\) of the same bidegree.

This short paper is very well written, and the interested reader may find interesting to study it in full detail. The construction is rather clever and the proof heavily relies, as expected, on the Pseudoholomorphic Patchworking Theorem by I. Itenberg and E. Shustin [Turk. J. Math. 26, No. 1, 27–51 (2002; Zbl 1047.14047)].

Reviewer: Jose Manuel Gamboa (Madrid)

##### MSC:

14P25 | Topology of real algebraic varieties |

14J26 | Rational and ruled surfaces |

14H50 | Plane and space curves |

32Q65 | Pseudoholomorphic curves |

##### Keywords:

topology of real algebraic curves; Viro method; patchworking; rational ruled surfaces; pseudoholomorphic curves
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\textit{B. Bertrand} and \textit{E. Brugallé}, Math. Z. 259, No. 3, 481--486 (2008; Zbl 1142.14037)

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##### References:

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