Given a Hermitian manifold （M-n,g）, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call△=（1-s/2）△c＋s/2△b the s-...Given a Hermitian manifold （M-n,g）, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call△=（1-s/2）△c＋s/2△b the s-Gauduchon connection of M, where △c and △b are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat s-Gauduchon connection. This is related to a question asked by Yau. The cases with s = 0 （a flat Chern connection） or s = 2 （a flat Bismut connection） are classified respectively by Boothby in the 1950s or by the authors in a recent joint work with Q. Wang. In this article, we observe that if either s 〉 4＋2√3 ≈ 7.46 or s 〈 4- 2√3≈ 0.54 and s ≠ 0, then g is Kahler. We also show that, when n = 2, g is always Kahler unless s=2. Therefore non-Kahler compact Gauduchon fiat surfaces are exactly isosceles Hopf surfaces.展开更多
A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kèahler when the constant is non-zero and must be Chern flat when the co...A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kèahler when the constant is non-zero and must be Chern flat when the constant is zero.The conjecture is known in complex dimension 2 by the work of Balas-Gauduchon in 1985(when the constant is zero or negative)and by Apostolov±Davidov±Muskarov in 1996(when the constant is positive).For higher dimensions,the conjecture is still largely unknown.In this article,we restrict ourselves to pluriclosed manifolds,and confirm the conjecture for the special case of Strominger Kèahler-like manifolds,namely,for Hermitian manifolds whose Strominger connection(also known as Bismut connection)obeys all the Kaèhler symmetries.展开更多
文摘Given a Hermitian manifold （M-n,g）, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call△=（1-s/2）△c＋s/2△b the s-Gauduchon connection of M, where △c and △b are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat s-Gauduchon connection. This is related to a question asked by Yau. The cases with s = 0 （a flat Chern connection） or s = 2 （a flat Bismut connection） are classified respectively by Boothby in the 1950s or by the authors in a recent joint work with Q. Wang. In this article, we observe that if either s 〉 4＋2√3 ≈ 7.46 or s 〈 4- 2√3≈ 0.54 and s ≠ 0, then g is Kahler. We also show that, when n = 2, g is always Kahler unless s=2. Therefore non-Kahler compact Gauduchon fiat surfaces are exactly isosceles Hopf surfaces.
基金supported by NSFC(Grant No.12071050)Chongqing Normal University。
文摘A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kèahler when the constant is non-zero and must be Chern flat when the constant is zero.The conjecture is known in complex dimension 2 by the work of Balas-Gauduchon in 1985(when the constant is zero or negative)and by Apostolov±Davidov±Muskarov in 1996(when the constant is positive).For higher dimensions,the conjecture is still largely unknown.In this article,we restrict ourselves to pluriclosed manifolds,and confirm the conjecture for the special case of Strominger Kèahler-like manifolds,namely,for Hermitian manifolds whose Strominger connection(also known as Bismut connection)obeys all the Kaèhler symmetries.
