On the Second Cohomology Group of a Kaehler Manifold of Positive Curvature

Bishop, R. L.; Goldberg, Samuel I.

doi:10.2307/2034011

Cited by 36 publications

(65 citation statements)

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“…It is well-known (p. 527, [3]) that a complete Kahler manifold M of positive holomorphic curvature is compact and is simplyconnected; moreover, its second Betti number is 1 [2] and its EulerPoincare characteristic E is positive (Theorem 2, [9]). Thus we may assume that M = K/L is the quotient of a compact semi-simple Lie group by a closed subgroup by a well-known theorem of Montgomery.…”

Section: Theorem a Homogeneous Kahler N-manifold M Of Positive Holommentioning

confidence: 99%

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Topology of some Kähler manifolds

Srinivasacharyulu¹

1967

Pacific J. Math.

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Section: Theorem a Homogeneous Kahler N-manifold M Of Positive Holommentioning

confidence: 99%

“…negative) [5]. The positively curved compact Kahler manifolds are simplycopnected (cf p. 528, [3]) and their second Betti number b 2 is equal to one [2]. In §2, we prove that the first Betti number b x of a negatively curved compact Kahler surface is always zero.…”

mentioning

confidence: 98%

Topology of some Kähler manifolds

Srinivasacharyulu¹

1967

Pacific J. Math.

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“…For instance, we have 4) and the curvature form 0 = (OJ) of its curvature R = DoD is given by OJ = 8w}, which we can write…”

Section: Negative Vector Bundles and Complex Finsler Metricsmentioning

confidence: 99%

Harmonic Maps in Complex Finsler Geometry

Nishikawa

2004

Variational Problems in Riemannian Geometry

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Abstract. Given a smooth map from a compact Riemann surface to a complex manifold M equipped with a strongly pseudoconvex Finsler metric F, we define a natural notion of the a-energy of the map. A harmonic map is then defined to be a critical point of the a-energy functional. Under the condition that F is weakly Kahler, we obtain the second variation formula of the functional, and prove that any a-energy minimizing harmonic map from a Riemann sphere to a weakly Kahler Finsler manifold M of positive curvature is either holomorphic or antiholomorphic. Significance of complex Finsler metrics in the study of holomorphic vector bundles, in particular its relation with the Hartshorne conjecture in complex algebraic geometry, is represented. A brief overview of complex Finsler geometry is also provided.

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“…(For convenience, we write Φ*(di, * '>d k ) and Φ ft _i(<Zi, ,d*-i) simply as Φ fc and 0 k _ x respectively.) = {pr* (aO-flr^'-^pr- 1 Step 5. By Steps 2,3, and 4, we obtain:…”

Section: L{e*)~ι) Corresponding To S E H\m E) (Cf (21)) the Firsmentioning

confidence: 99%