Asymptotically holomorphic theory for symplectic orbifolds. (English) Zbl 1519.53068
A landmark contribution by S. K. Donaldson [J. Differ. Geom. 44, No. 4, 666–705 (1996; Zbl 0883.53032)] aimed at constructing codimension-2 symplectic submanifolds with prescribed cohomology class pointed out the concept of asymptotically holomorphic sections of a complex line bundle, subject to a certain transversality condition. The article under review explores Donaldson’s technique in the context of symplectic orbifolds.
On a compact symplectic orbifold such that the symplectic form defines an integer cohomology class, the authors prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. Analogs of the Lefschetz hyperplane theorem for these suborbifolds are derived in mid-dimension. Also the hard Lefschetz and formality properties of these submanifolds are studied.
The contribution aligns with a vibrant array of recent discoveries, some due to the authors, in symplectic geometry.
On a compact symplectic orbifold such that the symplectic form defines an integer cohomology class, the authors prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. Analogs of the Lefschetz hyperplane theorem for these suborbifolds are derived in mid-dimension. Also the hard Lefschetz and formality properties of these submanifolds are studied.
The contribution aligns with a vibrant array of recent discoveries, some due to the authors, in symplectic geometry.
Reviewer: Mihai Putinar (Santa Barbara)
MSC:
| 53D35 | Global theory of symplectic and contact manifolds |
| 57R17 | Symplectic and contact topology in high or arbitrary dimension |
| 32L05 | Holomorphic bundles and generalizations |
Keywords:
asymptotically holomorphic sections; symplectic orbifolds; line bundles; Lefschetz hyperplane theorem; symplectic submanifoldsCitations:
Zbl 0883.53032References:
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