zbMATH — the first resource for mathematics

A nonalgebraic patchwork. (English) Zbl 1142.14037
This 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)].
14P25 Topology of real algebraic varieties
14J26 Rational and ruled surfaces
14H50 Plane and space curves
32Q65 Pseudoholomorphic curves
Full Text: DOI
[1] Bertrand, B., Brugallé, E.: A Viro theorem without convexity hypothesis for trigonal curves. International Mathematics Research Notices. Article ID 87604, 33 pp (2006). doi:10.1155/IMRN/2006/87604 · Zbl 1109.14039
[2] Bertrand B. (2006). Asymptotically maximal families of hypersurfaces in toric varieties. Geom. Dedicata. 118(1): 49–70 · Zbl 1103.14033 · doi:10.1007/s10711-005-9016-1
[3] Bihan, F.: Asymptotiques de nombres de Betti d’hypersurfaces projectives réelles (in French). Preprint. Available at: arXiv:math.AG/0312259
[4] Brugallé, E.: Symmetric plane curves of degree 7: pseudo-holomorphic and algebraic classifications. Crelle’s J. (2007, in press). Available at: arXiv:math.GT/0404030
[5] Brugallé E. (2006). Real plane algebraic curves with asymptotically maximal number of even ovals. Duke Math. J. 131(3): 575–587 · Zbl 1109.14040 · doi:10.1215/S0012-7094-06-13136-8
[6] Fiedler-Le Touzé S., Orevkov S.Yu. (2002). A flexible affine M-sextic which is algebraically unrealizable. J. Algebraic Geom. 11(2): 293–310 · Zbl 1054.14071
[7] Gromov M. (1985). Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82(2): 307–347 · Zbl 0592.53025 · doi:10.1007/BF01388806
[8] Haas, B.: Les multilucarnes: nouveaux contre-exemples à la conjecture de Ragsdale (in French). C. R. Acad. Sci. Paris Sér. I Math. 320(12), 1507–1512 (1995) · Zbl 0856.14020
[9] Itenberg I., Shustin E. (2002). Combinatorial patchworking of real pseudo-holomorphic curves. Turkish J. Math. 26(1): 27–51 · Zbl 1047.14047
[10] Itenberg I. (1993). Contre-exemples à la conjecture de Ragsdale. C. R. Acad. Sci. Paris Sér. I Math. 317(3): 277–282 (French) · Zbl 0787.14040
[11] Itenberg, I.: On the number of even ovals of a nonsingular curve of even degree in \({\mathbb{R}}P^2\) . In: Topology, Ergodic Theory, Real Algebraic Geometry. Amer. Math. Soc. Transl. Ser. 2, vol. 202, pp. 121–129. Amer. Math. Soc., Providence (2001) · Zbl 1002.14011
[12] Itenberg, I., Viro, O.Ya.: Maximal real algebraic hypersurfaces of projective spaces (in preparation) · Zbl 1180.14055
[13] Lopez de Medrano, L.: Courbure totale des variétés algébriques réelles projectives (in French). Thèse doctorale (2006)
[14] Mikhalkin G. (2005). Enumerative tropical algebraic geometry in \(\mathbb R^2\) . J. Amer. Math. Soc. 18(2): 313–377 · Zbl 1092.14068 · doi:10.1090/S0894-0347-05-00477-7
[15] Orevkov, S.Yu.: Arrangements of an M-quintic with respect to a conic which maximally intersects its odd branch. St. Petersbourg Math. J. (in press). http://picard.ups-tlse.fr/\(\sim\)orevkov · Zbl 1206.14082
[16] Orevkov S.Yu. (1999). Link theory and oval arrangements of real algebraic curves. Topology 38(4): 779–810 · Zbl 0923.14032 · doi:10.1016/S0040-9383(98)00021-4
[17] Orevkov S.Yu., Shustin E.I. (2002). Flexible, algebraically unrealizable curves: rehabilitation of Hilbert–Rohn–Gudkov approach. J. Reine Angew. Math. 551: 145–172 · Zbl 1014.14028 · doi:10.1515/crll.2002.080
[18] Orevkov, S.Yu., Shustin, E.I.: Pseudoholomorphic algebraically unrealizable curves. Mosc. Math. J. 3(3), 1053–1083, 1200–1201 (2003) · Zbl 1049.14044
[19] Shustin E. (1998). Gluing of singular and critical points. Topology 37(1): 195–217 · Zbl 0905.14008 · doi:10.1016/S0040-9383(97)00008-6
[20] Shustin E. (1999). Lower deformations of isolated hypersurface singularities. Algebra i Analiz 11(5): 221–249 · Zbl 0967.14002
[21] Shustin E. (2005). Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry. Algebra i Analiz 17: 170–214
[22] Viro, O.Ya.: Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7. In: Topology (Leningrad, 1982). Lecture Notes in Mathematics, vol. 1060, pp. 187–200. Springer, Berlin (1984)
[23] Viro O.Ya. (1989). Real plane algebraic curves: constructions with controlled topology. Leningrad Math. J. 1(5): 1059–1134 · Zbl 0732.14026
[24] Welschinger, J.-Y.: Courbes algébriques réelles et courbes flexibles sur les surfaces réglées de base \({\mathbb{C}}P^1\) (in French). Proc. London Math. Soc. (3) 85(2), 367–392 (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.