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Tim Perutz
Tim Perutz
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There is a proof in section 2.2 of The geometry of four-manifolds, by Donaldson and Kronheimer. They point out the similarities between the proofs of the integrability theorems for Dolbeault operators and for flat connections.

(While it's possible to prove this integrability theorem by quoting Newlander-Nirenberg, the PDE problem that underlies it is considerably easier to solve than the N-N problem, because the bundle is decoupled from the coordinates on the base. Similarly, it's easier to show that a flat connection is locally trivial than to show that a flat Riemannian manifold is locally euclidean.)

There is a proof in section 2.2 of The geometry of four-manifolds, by Donaldson and Kronheimer. They point out the similarities between the proofs of the integrability theorems for Dolbeault operators and for flat connections.

(While it's possible to prove this integrability theorem by quoting Newlander-Nirenberg, the PDE problem that underlies it is considerably easier to solve than the N-N problem, because the bundle is decoupled from the coordinates on the base. Similarly, it's easier to show that a flat connection is locally trivial than to show that a flat Riemannian manifold is locally euclidean.)

There is a proof in section 2.2 of The geometry of four-manifolds, by Donaldson and Kronheimer. They point out the similarities between the proofs of the integrability theorems for Dolbeault operators and for flat connections.

(While it's possible to prove this integrability theorem by quoting Newlander-Nirenberg, the PDE problem that underlies it is considerably easier to solve than the N-N problem, because the bundle is decoupled from the coordinates on the base.)

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Tim Perutz
Tim Perutz
  • 13.2k
  • 1
  • 53
  • 79

There is a proof in section 2.2 of The geometry of four-manifolds, by Donaldson and Kronheimer. They point out the similarities between the proofs of the integrability theorems for Dolbeault operators and for flat connections.

(While it's possible to prove this integrability theorem by quoting Newlander-Nirenberg, the PDE problem that underlies it is considerably easier to solve than the N-N problem, because the bundle is decoupled from the coordinates on the base. Similarly, it's easier to show that a flat connection is locally trivial than to show that a flat Riemannian manifold is locally euclidean.)

There is a proof in section 2.2 of The geometry of four-manifolds, by Donaldson and Kronheimer. They point out the similarities between the proofs of the integrability theorems for Dolbeault operators and for flat connections.

There is a proof in section 2.2 of The geometry of four-manifolds, by Donaldson and Kronheimer. They point out the similarities between the proofs of the integrability theorems for Dolbeault operators and for flat connections.

(While it's possible to prove this integrability theorem by quoting Newlander-Nirenberg, the PDE problem that underlies it is considerably easier to solve than the N-N problem, because the bundle is decoupled from the coordinates on the base. Similarly, it's easier to show that a flat connection is locally trivial than to show that a flat Riemannian manifold is locally euclidean.)

Source Link
Tim Perutz
Tim Perutz
  • 13.2k
  • 1
  • 53
  • 79

There is a proof in section 2.2 of The geometry of four-manifolds, by Donaldson and Kronheimer. They point out the similarities between the proofs of the integrability theorems for Dolbeault operators and for flat connections.