Compact Kähler manifolds with nonpositive bisectional curvature

Gang Liu

doi:10.1007/s00039-014-0290-7

Compact Kähler manifolds with nonpositive bisectional curvature

Gang Liu^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

Let (Mⁿ, g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold N^k with c₁ < 0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.

Original language	English (US)
Pages (from-to)	1591-1607
Number of pages	17
Journal	Geometric and Functional Analysis
Volume	24
Issue number	5
DOIs	https://doi.org/10.1007/s00039-014-0290-7
State	Published - Sep 1 2014

ASJC Scopus subject areas

Analysis
Geometry and Topology

Access to Document

10.1007/s00039-014-0290-7

Cite this

@article{0bc590dc2bc44c7a8ca215ef000a9e97,

title = "Compact K{\"a}hler manifolds with nonpositive bisectional curvature",

abstract = "Let (Mn, g) be a compact K{\"a}hler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact K{\"a}hler manifold Nk with c1 < 0. This confirms a conjecture of Yau. As a corollary, for any compact K{\"a}hler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.",

author = "Gang Liu",

note = "Publisher Copyright: {\textcopyright} 2014, Springer Basel.",

year = "2014",

month = sep,

day = "1",

doi = "10.1007/s00039-014-0290-7",

language = "English (US)",

volume = "24",

pages = "1591--1607",

journal = "Geometric and Functional Analysis",

issn = "1016-443X",

publisher = "Birkhauser Verlag Basel",

number = "5",

}

TY - JOUR

T1 - Compact Kähler manifolds with nonpositive bisectional curvature

AU - Liu, Gang

PY - 2014/9/1

Y1 - 2014/9/1

N2 - Let (Mn, g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold Nk with c1 < 0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.

AB - Let (Mn, g) be a compact Kähler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold Nk with c1 < 0. This confirms a conjecture of Yau. As a corollary, for any compact Kähler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.

UR - http://www.scopus.com/inward/record.url?scp=84907701064&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907701064&partnerID=8YFLogxK

U2 - 10.1007/s00039-014-0290-7

DO - 10.1007/s00039-014-0290-7

M3 - Article

AN - SCOPUS:84907701064

SN - 1016-443X

VL - 24

SP - 1591

EP - 1607

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 5

ER -

Compact Kähler manifolds with nonpositive bisectional curvature

Abstract

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this