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A characterization of seven compact Kähler submanifolds by holomorphic pinching. (English) Zbl 0589.53065
Let $$P_ m(C)$$ be an m-dimensional complex projective space with the Fubini-Study metric of constant holomorphic sectional curvature 1, and let M be an n-dimensional compact Kähler submanifold immersed in $$P_ m(C)$$. By applying an ingenious but simple method, the author proves that if the holomorphic sectional curvature of M is $$\geq$$, then M is a totally geodesic $$P_ n(C)$$, Veronese $$P_ n(C)$$, a Segre submanifold $$P_ k(C)$$, $$P_{n-k}(C)$$ or an Hermitian symmetric space of rank 2 with parallel second fundamental form.
Reviewer: K.Ogiue

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C40 Global submanifolds
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