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Dimensional reduction of stable bundles, vortices and stable pairs. (English) Zbl 0799.32022
Integrable unitary connections are unitary connections on a hermitian vector bundle over a complex manifold, whose curvature has type (1,1). An integrable unitary connection endows the bundle with a holomorphic structure. Denote by $$F_ A$$ the curvature of the integrable unitary connection $$A$$ and by $$\Lambda$$ the contraction with the Kähler form (in the case where the base manifold is Kähler). $$A$$ is Hermitian- Einstein if $$\Lambda F_ A =$$const $$I$$. Let $$E_ 1$$, $$E_ 2$$ be smooth vector bundles over a compact Kähler manifold $$(X,\omega)$$ equipped with the Hermitian metrics $$h_ 1$$, $$h_ 2$$, respectively. Consider the integrable unitary connections $$A_ 1$$, $$A_ 2$$ on $$(E_ 1,h_ 1)$$, $$(E_ 2,h_ 2)$$, respectively and let $$\varphi$$ be a section in $$\operatorname{Hom} (E_ 2,E_ 1)$$, the Higgs field. Denote by $$A_ 1*A_ 2$$ the induced connection on $$E_ 1\otimes E^*_ 2$$, $$\varphi^*$$ the adjoint of $$\varphi$$ with respect to $$h_ 1$$, $$h_ 2$$ and let $$\tau, \tau'$$ be real parameters.
The author considers the equations $$\overline \partial_{A_ 1*A_ 2} \varphi = 0$$, $$\Lambda F_{A_ 1} - {i \over 2} \varphi \circ \varphi^* + {i \over 2} \tau I_{E_ 1} = 0$$, $$\Lambda F_{A_ 2} + {i \over 2} \varphi^* \circ \varphi + {i\over 2} \tau'I_{E_ 2} = 0$$ and studies the stability conditions governing the existence of the solutions. The triples $$(A_ 1,A_ 2,\varphi)$$ are in one-to-one correspondence with $$SU(2)$$-invariant integrable unitary connections $$A$$ on $$(F,h)$$ where $$F=p^* E_ 1 \oplus p^* E_ 2 \otimes q^*H^{\otimes 2}$$ is a vector bundle over $$X \times P^ 1$$ $$(p:X \times P^ 1\to X$$, $$q:X \times P^ 1 \to P^ 1$$ are the projections and $$H^{\otimes 2}$$ is the line bundle of degree 2 on $$P^ 1)$$ and $$h=p^* h_ 1 \oplus p^*h_ 2\otimes q^*h_ 2'$$ $$(h_ 2'$$ is an $$SU(2)$$-invariant metric on $$H^{\otimes 2})$$. In the case where $$E_ 2$$ is a line bundle, the invariant stability of $${\mathcal F}$$ defined by $$0 \to p^* {\mathcal E}_ 1 \to {\mathcal F} \to p^* {\mathcal E}_ 2 \otimes q^*{\mathcal O} (2) \to 0$$ is equivalent to the S. B. Bradlow stability condition [S. B. Bradlow, J. Differ. Geom. 33, No. 1, 169-213 (1991; Zbl 0706.32013)]. Next, the author gives a construction of the moduli space of stable triplets as the $$SU(2)$$-fixed point set of a certain moduli space of stable bundles over $$X \times P^ 1$$. Then he studies Bradlow’s moduli spaces of stable pairs concluding that the moduli spaces of pairs enjoy the same properties as the moduli spaces of triples.
Reviewer: V.Oproiu (Iaşi)

##### MSC:
 32Q20 Kähler-Einstein manifolds 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 32G08 Deformations of fiber bundles
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