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SW \(\Rightarrow\) Gr: From the Seiberg-Witten equations to pseudo-holomorphic curves. (English) Zbl 0867.53025

A symplectic four-manifold \(X\) is a compact oriented four-manifold endowed with a closed 2-form \(\omega\) such that \(\omega\wedge\omega\neq 0\) everywhere. A submanifold \(\Sigma\) of \(X\) is called symplectic if the restriction of \(\omega\) to the tangent bundle \(T(\Sigma)\) is nondegenerate. Symplectic submanifolds are closely related to pseudo-holomorphic submanifolds. Suppose that \(J\) is an almost complex structure on \(X\), i.e., an endomorphism of \(T(X)\) whose square is minus the identity. A submanifold \(\Sigma\) is called pseudo-holomorphic if \(J\) maps \(T(\Sigma)\) to itself. An almost complex structure is said to be \(\omega\)-compatible if the bilinear form \(\omega(.,J(.))\) defines a Riemannian metric on \(X\). If \(\Sigma\) is pseudo-holomophic for an \(\omega\)-compatible almost complex structure, then \(\Sigma\) is also symplectic. On the other hand, if \(\Sigma\) is symplectic, then there is an \(\omega\)-compatible almost complex structure which makes \(\Sigma\) pseudo-holomorphic.
In this fundamental paper, the author proves beautiful results on the existence of certain types of pseudo-holomorphic submanifolds, and shows how pseudo-holomorphic curves in \(X\) can be constructed from solutions to the Seiberg-Witten equations (for definitions see: N. Seiberg and E. Witten [Nucl. Phys. B 426, No. 1, 19-52 (1994) (corrected in ibid. 430, No. 2, 485-486 (1994)) and Nucl. Phys. B 431, No. 3, 485-550 (1994); the author, Math. Res. Lett. 1, No. 6, 809-822 (1994; Zbl 0853.57019) and ibid. 2, No. 2, 221-238 (1995; Zbl 0854.57020)]).

MSC:

53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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