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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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