The symplectic isotopy conjecture states that every smooth symplectic surface in $CP^2$ is symplectically isotopic to a complex algebraic curve. Progress began with Gromov's pseudoholomorphic curves [Gro85], and progressed further culminating in Siebert and Tian's proof of the conjecture up to degree 17 [ST05], 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.

中文翻译：

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辛同位素问题的一种新方法

辛同位素猜想指出，$CP^2$ 中的每个光滑辛表面都是复杂代数曲线的辛同位素。进展始于 Gromov 的伪全纯曲线 [Gro85]，并进一步发展到 Siebert 和 Tian 对猜想的证明达到 17 次 [ST05]，但进一步的进展停滞不前。在本文中，我们针对这个问题提供了一个新的攻击方向。使用节点辛同位素问题的解决方案，我们指导平滑表面的辛同位素模型。这导致平滑辛同位素问题与某些嵌入的拉格朗日圆盘的存在性问题之间具有等价性。这将对该问题的研究从高属伪全纯曲线领域转向拉格朗日和弗洛尔理论领域。