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| Format: | Preprint |
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2019
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| Online Access: | https://arxiv.org/abs/1903.02701 |
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| _version_ | 1866912275083821056 |
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| author | Ni, Lei Zheng, Fangyang |
| author_facet | Ni, Lei Zheng, Fangyang |
| contents | In this article we continue the study of the two curvature notions for Kähler manifolds introduced by the first named author earlier: the so-called cross quadratic bisectional curvature (CQB) and its dual ($^d$CQB). We first show that compact Kähler manifolds with CQB$_1>0$ or $\mbox{}^d$CQB$_1>0$ are Fano, while nonnegative CQB$_1$ or $\mbox{}^d$CQB$_1$ leads to a Fano manifold as well, provided that the universal cover does not contain a flat de Rham factor. For the latter statement we employ the Kähler-Ricci flow to deform the metric. We conjecture that all Kähler C-spaces will have nonnegative CQB and positive $^d$CQB. By giving irreducible such examples with arbitrarily large second Betti numbers we show that the positivity of these two curvature put no restriction on the Betti number. A strengthened conjecture is that any Kähler C-space will actually have positive CQB unless it is a ${\mathbb P}^1$ bundle. Finally we give an example of non-symmetric, irreducible Kähler C-space with $b_2>1$ and positive CQB, as well as compact non-locally symmetric Kähler manifolds with CQB$<0$ and $^d$CQB$<0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1903_02701 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Kähler manifolds and cross quadratic bisectional curvature Ni, Lei Zheng, Fangyang Differential Geometry In this article we continue the study of the two curvature notions for Kähler manifolds introduced by the first named author earlier: the so-called cross quadratic bisectional curvature (CQB) and its dual ($^d$CQB). We first show that compact Kähler manifolds with CQB$_1>0$ or $\mbox{}^d$CQB$_1>0$ are Fano, while nonnegative CQB$_1$ or $\mbox{}^d$CQB$_1$ leads to a Fano manifold as well, provided that the universal cover does not contain a flat de Rham factor. For the latter statement we employ the Kähler-Ricci flow to deform the metric. We conjecture that all Kähler C-spaces will have nonnegative CQB and positive $^d$CQB. By giving irreducible such examples with arbitrarily large second Betti numbers we show that the positivity of these two curvature put no restriction on the Betti number. A strengthened conjecture is that any Kähler C-space will actually have positive CQB unless it is a ${\mathbb P}^1$ bundle. Finally we give an example of non-symmetric, irreducible Kähler C-space with $b_2>1$ and positive CQB, as well as compact non-locally symmetric Kähler manifolds with CQB$<0$ and $^d$CQB$<0$. |
| title | Kähler manifolds and cross quadratic bisectional curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/1903.02701 |