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On the total curvature and Betti numbers of complex projective manifolds. (English) Zbl 1494.53083

The author establishes an inequality between \(\beta_j\), the Betti numbers of \(M^m\), an \(m\)-dimensional compact complex projective manifold and its total curvature \(\mathcal{T}(M^m)\) (Definition 1.4). To be precise, the main result states that \(\displaystyle\sum_{j=1}^{2m}\beta_j\le \frac{m+1}{2}\mathcal{T}(M^m).\) Let us describe some sections. In the second section, the author reviews known results on the total curvature in the complex projective setting and states a formulation of the Chern-Lashof theorems for submanifolds of spheres. The third and the fourth sections are reserved for the proof of the main result. In the fifth section, the author states a result on the geometric meaning of the total curvature and another one on the extrinsic formulation of the Gauss-Bonnet-Chern.

MSC:

53C56 Other complex differential geometry
53C55 Global differential geometry of Hermitian and Kählerian manifolds
51N35 Questions of classical algebraic geometry
53C65 Integral geometry
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