Taming the pseudoholomorphic beasts in R x (S1 x S2)
Taming the pseudoholomorphic beasts in R x (S1 x S2)
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DOI：
10.2140/gt.2020.24.1791
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发表时间：
2020-01-01
影响因子：
2
通讯作者：
Gerig, Chris
中科院分区：
文献类型：
--
作者：
Gerig, Chris
对于具有b(+)(2)(X)>；0的闭定向光滑4-流形X，Seiberg-Witten不变量是定义良好的。陶布斯的“Sw=Gr”定理断言，如果X具有辛形式，则这些不变量等于定义良好的伪全纯曲线的计数，即陶布斯的Gromov不变量。在没有辛形式的情况下，仍然存在非平凡的闭自对偶2-形式，它们沿着不相交的圆并消失，并且在其他地方是辛的。本文及其续篇描述了拟全纯曲线在这类近辛2-形式的零集补中的定义良好的积分计数，并证明了它们恢复了Z/2Z上的Seiberg-Witten不变量。这是“Sw=Gr”到非辛4-流形的推广。本文的主要结果证明了以下结论。给出其零点集的一个合适的近辛形式omega和管状邻域N-，在辛余边(X-N，omega)的完备化中存在定义良好的伪全纯曲线的计数，它们在末端渐近于某些Reeb轨道。它们可以包装在一起，形成“近辛”的格罗莫夫不变量，作为X上自旋c结构的函数。
For a closed oriented smooth 4-manifold X with b(+)(2) (X) > 0, the Seiberg-Witten invariants are well-defined. Taubes' "SW = Gr" theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. In the absence of a symplectic form, there are still nontrivial closed self-dual 2-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This paper and its sequel describe welldefined integral counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic 2-forms, and it is shown that they recover the Seiberg- Witten invariants over Z/2Z. This is an extension of "SW = Gr" to nonsymplectic 4-manifolds.The main result of this paper asserts the following. Given a suitable near-symplectic form omega and tubular neighborhood N - of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism (X -N , omega) which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form "near-symplectic" Gromov invariants as a function of spin-c structures on X.