55 research outputs found

### On compact Hermitian manifolds with flat Gauduchon connections

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 $\nabla^s = (1-\frac{s}{2})\nabla^c +
\frac{s}{2}\nabla^b$ the $s$-Gauduchon connection of $M$, where $\nabla^c$ and
$\nabla^b$ 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 \cite{Yau}. The cases
with $s=0$ (a flat Chern connection) or $s=2$ (a flat Bismut connection) are
classified respectively by Boothby \cite{Boothby} in the 1950s or by Q. Wang
and the authors recently \cite{WYZ}. In this article, we observe that if either
$s\geq 4+2\sqrt{3} \approx 7.46$ or $s\leq 4-2\sqrt{3}\approx 0.54$ and $s\neq
0$, then $g$ is K\"ahler. We also show that, when $n=2$, $g$ is always K\"ahler
unless $s=2$. Note that non-K\"ahler compact Bismut flat surfaces are exactly
those isosceles Hopf surfaces by \cite{WYZ}.Comment: 9 pages. This preprint was submitted to Acta Mathematica Sinica, a
special issue dedicated to Professor Qikeng L

### Strominger connection and pluriclosed metrics

In this paper, we prove a conjecture raised by Angella, Otal, Ugarte, and
Villacampa recently, which states that if the Strominger connection (also known
as Bismut connection) of a compact Hermitian manifold is K\"ahler-like, in the
sense that its curvature tensor obeys all the symmetries of the curvature of a
K\"ahler manifold, then the metric must be pluriclosed. Actually, we show that
Strominger K\"ahler-like is equivalent to the pluriclosedness of the Hermitian
metric plus the parallelness of the torsion, even without the compactness
assumption

### Complex product manifolds cannot be negatively curved

We show that if $M = X \times Y$ is the product of two complex manifolds (of
positive dimensions), then $M$ does not admit any complete K\"ahler metric with
bisectional curvature bounded between two negative constants. More generally, a
locally-trivial holomorphic fibre-bundle does not admit such a metric.Comment: 6 Pages. To appear in The Asian Journal of Mathematic

### Positivity and Kodaira embedding theorem

Kodaira embedding theorem provides an effective characterization of
projectivity of a K\"ahler manifold in terms the second cohomology. Recently X.
Yang [21] proved that any compact K\"ahler manifold with positive holomorphic
sectional curvature must be projective. This gives a metric criterion of the
projectivity in terms of its curvature. In this note, we prove that any compact
K\"ahler manifold with positive 2nd scalar curvature (which is the average of
holomorphic sectional curvature over 2-dimensional subspaces of the tangent
space) must be projective. In view of generic 2-tori being non-abelian, this
new curvature characterization is sharp in certain sense

### Complex nilmanifolds and K\"ahler-like connections

In this note, we analyze the question of when will a complex nilmanifold have
K\"ahler-like Strominger (also known as Bismut), Chern, or Riemannian
connection, in the sense that the curvature of the connection obeys all the
symmetries of that of a K\"ahler metric. We give a classification in the first
two cases and a partial description in the third case. It would be interesting
to understand these questions for all Lie-Hermitian manifolds, namely, Lie
groups equipped with a left invariant complex structure and a compatible left
invariant metric

### On Bismut Flat Manifolds

In this paper, we give a classification of all compact Hermitian manifolds
with flat Bismut connection. We show that the torsion tensor of such a manifold
must be parallel, thus the universal cover of such a manifold is a Lie group
equipped with a bi-invariant metric and a compatible left invariant complex
structure. In particular, isosceles Hopf surfaces are the only Bismut flat
compact non-K\"ahler surfaces, while central Calabi-Eckmann threefolds are the
only simply-connected compact Bismut flat threefolds.Comment: In this 3rd version, we add a lemma on Hermitian surfaces with flat
Riemannian connection. References are updated and typos correcte

### The set of all orthogonal complex structures on the flat $6$-tori

In \cite{BSV}, Borisov, Salamon and Viaclovsky constructed non-standard
orthogonal complex structures on flat tori $T^{2n}_{\mathbb R}$ for any $n\geq
3$. We will call these examples BSV-tori. In this note, we show that on a flat
$6$-torus, all the orthogonal complex structures are either the complex tori or
the BSV-tori. This solves the classification problem for compact Hermitian
manifolds with flat Riemannian connection in the case of complex dimension
three.Comment: 14 page

- …