# zbMATH — the first resource for mathematics

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
##### Keywords:
Seiberg-Witten moduli space; Hermitian surface
Full Text: