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The Seiberg-Witten equations on Hermitian surfaces. (English) Zbl 0993.57016
This paper studies the Seiberg-Witten equations on an arbitrary compact complex surface endowed with a Hermitian metric. The Seiberg-Witten equations on \(X\) are \[ D_{A} \varphi+\sigma \cdot \varphi= 0,\qquad (F_{A}+2\pi i\beta)^{+} = 2(\varphi\otimes \varphi^{*})_{0}=0, \] where \(\varphi\) is a section of the \(\text{spin}^c\) bundle, \(A\) is a connection on the determinant line bundle, \(D_{A}\) is the Dirac operator and \(\sigma\) and \(\beta\) are perturbative one- and two-forms, respectively.
Let \(X\) be a complex surface with Hermitian metric \(g\). This gives rise to a one-form \(\theta_{g}\) such that \(d \omega_{g}= \omega_{g}\wedge \theta_{g}\). Then setting \(\sigma=-\theta_{g}/4\), the Seiberg-Witten equations decouple into two vortex-type equations for \(\varphi=(\alpha,\beta)\), as in the Kähler case [E. Witten, Math. Res. Lett. 1, No. 6, 769-796 (1994; Zbl 0867.57029)].
The results of [P. Lupascu, The abelian vertex equations on Hermitian manifolds, Math. Nachr. 230, 99-113 (2001; Zbl 1018.32014)] relate the moduli space of vortex equations to the subset \(V(c_{1}(L))\) of the Douady space of effective divisors on \(X\) of fixed homology class \(c_{1}(L)\), satisfying also a bound in their volume depending on \(\beta\). Using this, the Seiberg-Witten moduli space is \(V(c_{1}(L)) \sqcup V(c_{1}(K)-c_{1}(L))\).
When \(X\) is Kähler, one of the two sets is empty. The paper gives examples of non-Kähler surfaces where both sets are non-empty. Unfortunately, these examples correspond to surfaces with \(b_{2}=0\), so there are no associated Seiberg-Witten invariants.
MSC:
57R57 Applications of global analysis to structures on manifolds
58D27 Moduli problems for differential geometric structures
32J15 Compact complex surfaces
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