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.
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 |