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Seiberg Witten and Gromov invariants for symplectic \(4\)-manifolds. (English) Zbl 0967.57001

First International Press Lecture Series. 2. Somerville, MA: International Press. vi, 401 p. (2000).
The book under review is a collection of four of the author’s articles containing the complete proof of his remarkable result relating the Seiberg-Witten and Gromov-Witten invariants of symplectic 4-manifolds. These four papers first appeared in print as follows:
[1] SW \(\Rightarrow\) Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves, J. Am. Math. Soc. 9, No. 3, 845-918 (1996; Zbl 0867.53025);
[2] Counting pseudo-holomorphic submanifolds in dimension 4, J. Differ. Geom. 44, No. 4, 818-893 (1996; Zbl 0883.57020);
[3] Gr \(\Rightarrow\) SW: from pseudo-holomorphic curves to Seiberg-Witten solutions, J. Differ. Geom. 51, No. 2, 203-334 (1999);
[4] GR = SW: counting curves and connections, J. Differ. Geom. 52, No. 3, 453-609 (1999; Zbl 1040.53096).
Seiberg-Witten invariants of a compact, oriented, 4-dimensional manifold \(X\) constitute a map from the set Spin of equivalence classes of \(\text{Spin}^C\) structures on \(X\) to the integers. They are defined when the number \(b_2^+=(b_2+ \text{signature})/2\) is greater than 1 by making a suitable count of solutions to a certain nonlinear system of differential equations on the manifold, known as the Seiberg-Witten equations. (There is a more complicated structure in the case when \(b_2^+=1\).)
In the case when \(X\) is a symplectic manifold, the set Spin is identified with the second integral cohomology of \(X\), so Seiberg-Witten invariants define a map \(\text{SW}:H^2(X;\mathbb Z) \to \mathbb Z\).
On the other hand, there is another map \(\text{Gr}: H^2(X;\mathbb Z) \to \mathbb Z\) (called the Gromov invariant) which assigns an integer to each 2-dimensional cohomology class of a compact symplectic 4-manifold. Rougly speaking, the invariant counts, with suitable weights, compact, pseudo-holomorphic submanifolds whose fundamental class is Poincaré dual to the cohomology class in question.
The fundamental theorem, proved in the four papers constituting the book, states that Gr = SW (i.e. the Gromov invariants are equal to the Seiberg-Witten invariants) for any compact, symplectic 4-manifold \(X\) with \(b_+^2>1\).
Although the book is just four separate papers brought together under the same volume, it can be considered as an excellent survey monograph on the subject, appropriate for both experts and beginners.

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57R57 Applications of global analysis to structures on manifolds
53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
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