Courbes symplectiques de haute auto-intersection dans les surfaces symplectiques

TL;DR

This paper investigates symplectically embedded curves with high self-intersection in symplectic surfaces, showing they determine the surface's topology and embedding uniquely, and explores symplectic sections in ruled surfaces over elliptic curves.

Contribution

It proves that high self-intersection symplectic curves uniquely determine the surface and embedding, using Seiberg-Witten and pseudoholomorphic techniques, and shows symplectic sections are isotopic to complex sections.

Findings

High self-intersection curves determine surface topology and embedding.

Unique strong symplectic fillings of associated contact 3-manifolds.

Symplectic sections are isotopic to complex sections.

Abstract

We first study symplectically embedded curves in symplectic surfaces with high self-intersection numbers compared to their genus. We prove in two different ways that such a curve completely determines both the diffeomorphism type of the surface in which it is embedded and the embedding itself. The first proof uses Seiberg-Witten theory whereas the second one only involves pseudoholomorphic techniques. We deduce from this result that the contact $3$-manifolds naturally associated with those curves admit a unique strong symplectic filling up to diffeomorphism. We next examine symplectic sections of geometrically ruled complex surfaces over elliptic curves. We show that such a section is symplectically isotopic to a complex section. -- On \'etudie dans un premier temps les courbes symplectiquement plong\'ees dans les surfaces symplectiques dont les nombres d'auto-intersection sont suffisamment grands par rapport leurs genres. On montre de deux mani\`eres diff\'erentes qu'une telle courbe d\'etermine \`a la fois la classe de diff\'eomorphisme de la surface symplectique qui la contient et la mani\`ere dont elle est plong\'ee dans cette surface. La premi\`ere d\'emonstration fait appel \`a la th\'eorie de Seiberg-Witten, alors que la seconde se restreint aux techniques pseudoholomorphes. On d\'eduit de ce r\'esultat l'unicit\'e \`a diff\'eomorphisme pr\`es des remplissages symplectiques forts des vari\'et\'es de contact de dimension $3$ naturellement associ\'ees \`a ce type de courbes. Dans un second temps, on s'int\'eresse aux sections symplectiques des surfaces complexes g\'eom\'etriquement r\'egl\'ees au-dessus de courbes elliptiques. On montre qu'une telle section est symplectiquement isotope \`a une section complexe.