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Symplectic and Kähler structures on \(\mathbb{C}P^1\)-bundles over \(\mathbb{C}P^2\). (English) Zbl 1487.53107

The authors focus on Tolman’s manifolds to study symplectic manifolds which does not admit Kähler metric. Tolman’s manifolds are compact symplectic manifolds of dimension 6 with Hamiltonian torus actions constructed by S. Tolman [Invent. Math. 131, no. 2, 299–310 (1998; Zbl 0901.58018)]. The authors discuss when the symplectic manifolds does not admit Kähler structures compatible with them. As a result, they describe some conditions for such a Kähler metric to exist.
The paper consists of five sections. Section 1 is a commentary on the background of the study. Main theorems (Theorem 1.3, 1.5 and 1.6) are also provided in the section. Section 2 is the quick review of the construction of Tolman’s manifolds. The authors explain simply some results on Tolman’s manifolds including Hamiltonian circle actions. Section 3 treats topological aspects of Tolman’s manifold needed for proving Theorem 1.3. The authors describe the intersection form on the second cohomology group of Tolman’s manifold \(M_{\mathcal{T}}\) with the Chern classes to prove that \(M_\mathcal{T}\) is diffeomorphic to the projectivisation \(\mathbb{P}(E)\) of a complex two bundle \(E\) over the complex projective space \(\mathbb{C}P^2\). The proofs of main theorems begin from Section 4. In Section 4, the authors prove the main theorem 1.3 and 1.6. Theorem 1.6 is shown first by using basic properties of holomorphic rank two bundles over \(\mathbb{C}P^2\). The proof of Theorem 1.3 uses Mori theory together with classical results on holomorphic rank two bundles over \(\mathbb{C}P^2\). Section 5 is devoted to the proof of Theorem 1.5. The basic results on projective manifolds with circle actions is reviewed plain and the theorem is shown by using them.

MSC:

53D20 Momentum maps; symplectic reduction
53D05 Symplectic manifolds (general theory)
32Q15 Kähler manifolds

Citations:

Zbl 0901.58018
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References:

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