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A new approach to the symplectic isotopy problem. (English) Zbl 1454.53072

Summary: The symplectic isotopy conjecture states that every smooth symplectic surface in \(\mathbb{C}\text{P}^2\) is symplectically isotopic to a complex algebraic curve. Progress began with M. Gromov’s pseudoholomorphic curves [Invent. Math. 82, 307–347 (1985; Zbl 0592.53025)], and progressed further culminating in B. Siebert and G. Tian’s proof of the conjecture up to degree 17 [Ann. Math. (2) 161, No. 2, 959–1020 (2005; Zbl 1090.53072)], but further progress has stalled. In this article we provide a new direction of attack on this problem. Using a solution to a nodal symplectic isotopy problem we guide model symplectic isotopies of smooth surfaces. This results in an equivalence between the smooth symplectic isotopy problem and an existence problem of certain embedded Lagrangian disks. This redirects study of this problem from the realm of pseudoholomorphic curves of high genus to the realm of Lagrangians and Floer theory. Because the main theorem is an equivalence going both directions, it could theoretically be used to either prove or disprove the symplectic isotopy conjecture.

MSC:

53D35 Global theory of symplectic and contact manifolds
53D05 Symplectic manifolds (general theory)
57R17 Symplectic and contact topology in high or arbitrary dimension
14J42 Holomorphic symplectic varieties, hyper-Kähler varieties