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Symplectic Calabi-Yau surfaces. (English) Zbl 1216.32016

Ji, Lizhen (ed.) et al., Handbook of geometric analysis. No. 3. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-205-3/hbk). Advanced Lectures in Mathematics (ALM) 14, 231-356 (2010).
The research of Calabi-Yau manifolds is an extremely active research field both in mathematics and in mathematical physics. Calabi-Yau manifolds are defined as compact, complex Kähler manifolds that have trivial first Chern classes. It was conjectured by Calabi in 1957 and proved by Yau in 1977 that Calabi-Yau manifolds admit Kähler metrics with vanishing Ricci curvatures which are solutions of the Einstein field equation. The theory of motions of loops inside a Calabi-Yau manifold provides a model of a conformal field theory, called the sigma model in physics. Many authors studied them from different points of view: Arthur L. Besse, Brian Greene, Nigel Hitchin, Tristan Hübsch, Gang Tian, etc.
The present work gives an overview of some general aspects of Kähler manifolds, underlying certain topological properties of them. In the first chapter of the paper, the author presents basic notions and properties of linear symplectic geometry: linear symplectic and compatible complex structures, Hermitian vector spaces, Clifford algebras and some elements of spin geometry. Further, he develops a brief study of topological aspects of closed symplectic manifolds, giving the Moser stability theorem, the Lagrangian neighborhood theorem and the Thurston theorem. Concerning several aspects of Kähler geometry, characterization of integrability, properties of Hermitian scalar curvature and Weitzenböck formulas regarding a Hermitian connection are mentioned. General aspects of Seiberg-Witten and Taubes’ symplectic Seiberg-Witten theory from the point of view of almost Kähler geometry are also reviewed, presenting Taubes’ fundamental nonvanishing result of symplectic \(4\)-manifolds. Some remarks on Donaldson’s conjecture on almost Kähler Calabi-Yau equations are also made.
The main chapter of the paper deals with symplectic Calabi-Yau surfaces. The author proves that symplectic Calabi-Yau surfaces are minimal symplectic \(4\)-manifolds with Kodaira dimension equal to zero. A homological classification of symplectic Calabi-Yau surfaces is given, including all the known Calabi-Yau surfaces. Some facts about the moduli space of symplectic Calabi-Yau surfaces are indicated and the existence of canonical metrics is postulated.
The paper ends with some questions relating symplectic Calabi-Yau surfaces with complex Calabi-Yau surfaces and affine \(4\)-manifolds, namely: \(Q_1\): {Is a smooth \(4\)-manifold which admits affine structures and symplectic structures a torus bundle over a torus?} \(Q_2\): {Is a symplectic affine 4-manifold a symplectic Calabi-Yau surface?} \(Q_3\): {Is there a “mirror symmetry” between symplectic and complex Calabi-Yau surfaces?}
For the entire collection see [Zbl 1193.53002].

MSC:

32Q25 Calabi-Yau theory (complex-analytic aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
57R57 Applications of global analysis to structures on manifolds
53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
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