Form-type equations on Kähler manifolds of nonnegative orthogonal bisectional curvature

Fu, Jixiang; Wang, Zhizhang; Wu, Damin

doi:10.1007/s00526-014-0714-0

Cited by 68 publications

(52 citation statements)

References 10 publications

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“…Remark 2.1. Note that the definition of real bisectional curvature is somewhat analogous to the notion of quadratic orthogonal bisectional curvature defined in [21], see also [3,4,8,12,27]. However, the two curvature notions are actually quite different, in the sense that the former is a slight generalization of holomorphic sectional curvature H (and is actually equivalent to H when the metric is Kähler) while the latter is closer to orthogonal bisectional curvature.…”

Section: The Real Bisectional Curvature Of Hermitian Manifoldsmentioning

confidence: 99%

On real bisectional curvature for Hermitian manifolds

Yang

Zheng

2018

Trans. Amer. Math. Soc.

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Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact Kähler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called real bisectional curvature for Hermitian manifolds. When the metric is Kähler, this is just the holomorphic sectional curvature H, and when the metric is non-Kähler, it is slightly stronger than H. We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case. The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature.It is a natural question to ask when will a Hermitian manifold have constant real bisectional curvature. To this end, we have the following:Theorem 1.4. Let (M n , g) be a compact Hermitian manifold whose real bisectional curvature is constantly equal to c. Then c ≤ 0. Moreover, when c = 0, then (M, g) is a balanced manifold with vanishing first, second, and third Ricci tensors, and its Chern curvature satisfies the property R xyuv = −R uvxy for any type (1, 0) complex tangent vectors x, y, u, v.We would like to propose the following conjecture:Conjecture 1.5. Let M n (n ≥ 3) be a compact Hermitian manifold with vanishing real bisectional curvature c. Then c = 0, and its Chern curvature tensor R = 0.By Boothby's theorem, compact Hermitian manifolds with vanishing Chern curvature are precisely the compact quotients of complex Lie groups equipped with left invariant metrics.Besides the constant real bisectional curvature cases, more generally, it would certainly be very interesting to try to understand the class of all compact Hermitian

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Section: The Real Bisectional Curvature Of Hermitian Manifoldsmentioning

confidence: 99%

On real bisectional curvature for Hermitian manifolds

Yang

Zheng

2018

Trans. Amer. Math. Soc.

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“…A different recent extension on complex manifolds [41], confirming a conjecture of Gauduchon [21], is that one can prescribe the volume form of a Gauduchon metric (satisfying ∂∂(ω n−1 ) = 0). In this case, (1.1) is replaced by a Monge-Ampère type equation for (n − 1)-plurisubharmonic functions [26,17,18,38,50,51].…”

Section: Introductionmentioning

confidence: 99%

The Monge–Ampère equation for non-integrable almost complex structures

2019

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“…In this case the Calabi-Yau theorem is replaced by its Hermitian counterpart, proved by Weinkove and the author [41] (see also [2,7,9,17,24,26,27,34,44,42] for earlier results and later developments, [19,20,21,22,30,36,45,46] for other Monge-Ampère type equations on non-Kähler manifolds, and [40] for a very recent Calabi-Yau theorem for Gauduchon metrics on Hermitian manifolds). The key new difficulty is that now in general we have to modify the function F in (1.1) by adding a constant to it, namely we obtain…”

Section: Introductionmentioning

confidence: 99%