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In this article, we study Hermitian manifolds whose Bismut connection has parallel torsion, which will be called {\em Bismut torsion parallel manifolds,} or {\em BTP} manifolds for brevity. We obtain a necessary and sufficient condition characterizing BTP manifolds in terms of Bismut curvature tensor alone (Theorem 1.1). We also present examples and discuss some general properties for BTP manifolds, as well as give a classification result for non-balanced BTP threefolds (Theorem 1.16)., Comment: 28 pages. This grows out of the first half of the preprint (arXiv: 2208.03071)

An old conjecture in non-K\"ahler geometry states that, if a compact Hermitian manifold has constant holomorphic sectional curvature, then the metric must be K\"ahler (when the constant is non-zero) or Chern flat (when the constant is zero). It is known to be true in complex dimension $2$ by the work of Balas and Gauduchon in 1985 (when the constant is negative or zero) and Apostolov, Davidov and Muskarov in 1996 (when the constant is positive). In dimension $3$ or higher, the conjecture is only known in some special cases, such as the locally conformally K\"ahler case (when the constant is negative or zero) by the work of Chen, Chen and Nie, or for complex nilmanifolds with nilpotent $J$ by the work of Li and the second named author. In this note, we confirm the above conjecture for all non-balanced Bismut torsion parallel (BTP) manifolds. Here the BTP condition means that the Bismut connection has parallel torsion. In particular, the conjecture is valid for all Vaisman manifolds., Comment: 12 pages

In this paper, we confirm the Fino-Vezzoni Conjecture for unimodular Lie algebras which contain abelian ideals of codimension two, a natural generalization to the class of almost abelian Lie algebras. This provides new evidence towards the validity of the conjecture on a very special type of $3$-step solvmanifolds., Comment: 20 pages, updated some reference info and added a couple of appendices

In this article, we investigate the class of Hermitian manifolds whose Bismut connection has parallel torsion ({\rm BTP} for brevity). In particular, we focus on the case where the manifold is (locally) homogeneous with respect to a group of holomorphic isometries and we fully characterize the compact Chern flat {\rm BTP} manifolds. Moreover we show that certain compact flag manifolds are {\rm BTP} if and only if the metric is K\"ahler or induced by the Cartan-Killing form and we then characterize {\rm BTP} invariant metrics on compact semisimple Lie groups which are Hermitian w.r.t. a Samelson structure and are projectable along the Tits fibration. We state a conjecture concerning the question when the Bismut connection of a BTP compact Hermitian locally homogeneous manifold has parallel curvature, giving examples and providing evidence in some special cases.

This paper is a sequel to our studies \cite{ZZ} and \cite{YZZ} on Bismut K\"ahler-like manifolds, or {\em BKL} manifolds for short. We will study the structural theorems for {\em BKL} manifolds, prove a conjecture raised in \cite{YZZ} which states that any {\em BKL} manifold that is Bismut Ricci flat must be Bismut flat, and give complete classifications of {\em BKL} manifolds in dimension $4$ and $5$., Comment: 24 pages

We apply the algebraic consideration of holonomy systems to study Hermitian manifolds whose Chern connection is Ambrose-Singer and prove structure theorems for such manifolds. The main result (Theorem 1.2) asserts that the universal cover of such a Hermitian manifold must be the product of a complex Lie group and Hermitian symmetric spaces, which was previously proved up to complex dimension four by the authors. This in some sense is the Hermitian version of Cartan's classification of Hermitian symmetric spaces. We also obtain results on Hermitian manifolds whose Bismut connection is Ambrose-Singer when the complex dimension $n\le 4$. Furthermore we discuss a project of classifying such manifolds via Alekseevski\u{i} and Kimel\!\'{}\!fel\!\'{}\!d type theorems, which we establish for the family of the Gauduchon connections on a compact Hermitian manifold except for the Bismut connection., Comment: 22 pages

We examine Hermitian metrics on unimodular Lie algebras which contains a $J$-invariant abelian ideal of codimension two, and give a classification for all Bismut K\"ahler-like and all Bismut torsion-parallel metrics on such Lie algebras.

We consider the class of compact Hermitian manifolds whose Chern connection is Ambrose-Singer, namely, it has parallel torsion and curvature. We prove structure theorems for such manifolds.

In this article, we study Hermitian manifolds whose Bismut-Strominger connection has parallel torsion tensor, which will be called {\em Bismut torsion parallel manifolds,} or {\em BTP} manifolds for short. We obtain a necessary and sufficient condition characterizing this class in terms of the curvature tensor. In particular, Bismut flat or Bismut K\"ahler-like manifolds are {\em BTP} manifolds, known by our earlier results. We also obtain structural results for non-balanced {\em BTP} manifolds, and classification theorems for non-balanced {\em BTP} threefolds and balanced {\em BTP} threefolds., Comment: 46 pages. All comments are welcome

In 1984, Gauduchon considered the functional of $L^2$-norm of his torsion $1$-form on a compact Hermitian manifold. He obtained the Euler-Lagrange equation for this functional, and showed that in dimension $2$ the critical metrics must be balanced (namely with vanishing torsion $1$-form). In this note we extend his result to higher dimensions, and show that critical metrics are balanced in all dimensions. We also consider the $L^2$-norm of the full Chern torsion, and show by examples that there are critical points of this functional that are not K\"ahler.

In this article, we propose the following conjecture: if the Strominger connection of a compact Hermitian manifold has constant non-zero holomorphic sectional curvature, then the Hermitian metric must be K\"ahler. The main result of this article is to confirm the conjecture in dimension $2$. We also verify the conjecture in higher dimensions in a couple of special situations., Comment: Added the missing term in formula (32) (the ninth parenthesis on the right hand side), which resulted in changes in formula (33), (34) and (37, and a couple of sentences after (37) to make the argument go through. The paper will appear in Journal of Geometric Analysis

In a paper by Angella, Otal, Ugarte, and Villacampa, the authors conjectured that on a compact Hermitian manifold, if a Gauduchon connection other than Chern or Strominger is K\"ahler-like, then the Hermitian metric must be K\"ahler. They also conjectured that if two Gauduchon connections are both K\"ahler-like, then the metric must be K\"ahler. In this paper, we discuss some partial answers to the first conjecture, and give a proof to the second conjecture. In the process, we discovered an interesting `duality' phenomenon amongst Gauduchon connections, which seems to be intimately tied to the question, though we do not know if there is any underlying reason for that from physics.

A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be K\"ahler 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 K\"ahler-like manifolds, namely, for Hermitian manifolds whose Strominger connection (also known as Bismut connection) obeys all the K\"ahler symmetries.

A well known conjecture in complex geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature must be K\"ahler if the constant is non-zero and must be Chern flat if the constant is zero. The conjecture is confirmed 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 the class of complex nilmanifolds and confirm the conjecture in that case.

We examine the class of compact Hermitian manifolds with constant holomorphic sectional curvature. Such manifolds are conjectured to be K\"ahler (hence a complex space form) when the constant is non-zero and Chern flat (hence a quotient of a complex Lie group) when the constant is zero. The conjecture is known in complex dimension two but open in higher dimensions. In this paper, we establish a partial solution in complex dimension three by proving that any compact Hermitian threefold with zero real bisectional curvature must be Chern flat. Real bisectional curvature is a curvature notion introduced by Xiaokui Yang and the second named author in 2019, generalizing holomorphic sectional curvature. It is equivalent to the latter in the K\"ahler case and is slightly stronger in general.

In recent work (Pure Appl. Anal. 2 (2020), 397-426), the first named author and J. Zhang found a connection between the regularity theory of optimal transport and the curvature of K\"ahler manifolds. In particular, we showed that the MTW tensor for a cost function $c(x,y)=\Psi(x-y)$ can be understood as the anti-bisectional curvature of an associated K\"ahler metric defined on a tube domain. Here, the anti-bisectional curvature is defined as $R(\mathcal{X}, \overline{ \mathcal{Y}},\mathcal{X}, \overline{ \mathcal{Y}}) $ where $\mathcal{X}$ and $\mathcal{Y}$ are polarized $(1,0)$ vectors and $R$ is the curvature tensor. The correspondence between the anti-bisectional curvature and the MTW tensor provides a meaningful sense in which the anti-bisectional curvature can have a sign (i.e., be positive or negative). In this paper, we study the behavior of the anti-bisectional curvature under K\"ahler-Ricci flow. We find that non-positive anti-bisectional curvature is preserved under the flow. In complex dimension two, we also show that non-negative orthogonal anti-bisectional curvature (i.e., the MTW(0) condition) is preserved under the flow. We provide several applications of these results -- in complex geometry, optimal transport, and affine geometry., Comment: 41 pages. Version changes: A stronger version of Corollary 6 and minor edits

In this article we give a necessary and sufficient condition to characterize projective submanifolds in ${\mathbb P}^N$ with codimensions 2 and 3. The conditions involve the Chern classes of the manifold and a very ample line bundle on the manifold. This generalizes our earlier characterization for hypersurfaces. The higher codimensional cases are proposed as a general question.

Let the pair $(\mathfrak{g},J)$ be a nilpotent Lie algebra $\mathfrak{g}$ (NLA for short) endowed with a nilpotent complex structure $J$. In this paper, motivated by a question in the work of Cordero, Fern\'andez, Gray and Ugarte, we prove that $2\leq \nu(J) \leq 3$ for $(\mathfrak{g},J)$ when $\nu(\mathfrak{g})=2$, where $\nu(\mathfrak{g})$ is the step of $\mathfrak{g}$ and $\nu(J)$ is the unique smallest integer such that $\mathfrak{a}(J)_{\nu(J)}=\mathfrak{g}$ as in Definition 1 and 8 of the paper by Cordero, Fern\'andez, Gray and Ugarte. When $\nu(\mathfrak{g})=3$, for arbitrary $n \geq 3$, there exists a pair $(\mathfrak{g},J)$ such that $\nu(J)=\dim_{\mathbb{C}}\mathfrak{g}=n$, for which we call the $J$ in the pair $(\mathfrak{g},J)$, satisfying $\nu(J)=\dim_{\mathbb{C}}\mathfrak{g}=n$, a maximal nilpotent (MaxN for short) complex structure. The algebraic dimension of a nilmanifold endowed with a left invariant MaxN complex structure is discussed. Furthermore, a structure theorem is proved for the pair $(\mathfrak{g},J)$, where $\nu(\mathfrak{g})=3$ and $J$ is a MaxN complex structure.

This article is concerned with Chern class and Chern number inequalities on polarized manifolds and nef vector bundles. For a polarized pair $(M,L)$ with $L$ very ample, our first main result is a family of sharp Chern class inequalities. Among them the first one is a variant of a classical result and the equality case of the second one is a characterization of hypersurfaces. The second main result is a Chern number inequality on it, which includes a reverse Miyaoka-Yau type inequality. The third main result is that the Chern numbers of a nef vector bundle over a compact K\"{a}hler manifold are bounded below by the Euler number. As an application, we classify compact K\"{a}hler manifolds with nonnegative bisectional curvature whose Chern numbers are all positive. A conjecture related to the Euler number of compact K\"{a}hler manifolds with nonpositive bisectional curvature is proposed, which can be regarded as a complex analogue to the Hopf conjecture., Comment: 19 pages, comments very welcome!

In this article, we study a class of K\"ahler manifolds defined on tube domains in $\mathbb{C}^n$, and in particular those which have $O(n) \times \mathbb{R}^n$ symmetry. For these, we prove a uniqueness result showing that any such manifold which is complete and has non-negative orthogonal bisectional curvature ($n \geq 3$) or non-negative bisectional curvature ($n \geq 2$) is biholomorphically isometric to $\mathbb{C}^n$. We also consider another curvature tensor called the \emph{orthogonal anti-bisectional curvature}. We find necessary and sufficient conditions for a complete $O(n)$-symmetric tube domain to have non-negative orthogonal anti-bisectional curvature and provide several examples of complete metrics which satisfy this condition. Finally, we discuss some applications of these spaces within optimal transport. In particular, we study "synthetic" curvature bounds for non-smooth geometries and how they can be applied to the rough geometry induced by the Monge cost $c(x,y)=\|x-y\|$., Comment: 23 pages. This version replaces the previous version which was titled "The geometry of positively curved K\"ahler metrics on tube domains" (arXiv:2001.06155v1). Section 6 is changed substantially in this new version and Appendix B is omitted

In this paper, we study a special type of compact Hermitian manifolds that are Strominger K\"ahler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is K\"ahler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a K\"ahler manifold. Previously, we have shown that any SKL manifold $(M^n,g)$ is always pluriclosed, and when the manifold is compact and $g$ is not K\"ahler, it can not admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when $n=2$, the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-K\"ahler SKL manifolds in dimension $3$ and those with degenerate torsion in higher dimensions. We also present some properties about SKL manifolds in general dimensions, for instance, given any compact non-K\"ahler SKL manifold, its K\"ahler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal K\"ahler when $n\geq 3$, and the manifold does not admit any Hermitian symplectic metric.

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.

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. What we actually showed is a bit more: for any given Hermitian manifold, the Strominger K\"ahler-like condition is equivalent to the pluriclosedness of the metric plus the parallelness of the torsion.

In this article we continue the study of the two curvature notions for K\"ahler 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 K\"ahler 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 K\"ahler-Ricci flow to deform the metric. We conjecture that all K\"ahler 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 K\"ahler C-space will actually have positive CQB unless it is a ${\mathbb P}^1$ bundle. Finally we give an example of non-symmetric, irreducible K\"ahler C-space with $b_2>1$ and positive CQB, as well as compact non-locally symmetric K\"ahler manifolds with CQB$<0$ and $^d$CQB$<0$., Comment: One page was added in Section 3

In this paper we study the class of compact K\"ahler manifolds with positive orthogonal Ricci curvature: $Ric^\perp>0$. First we illustrate examples of K\"ahler manifolds with $Ric^\perp>0$ on K\"ahler C-spaces, and construct ones on certain projectivized vector bundles. These examples show the abundance of K\"ahler manifolds which admit metrics of $Ric^\perp>0$. Secondly we prove some (algebraic) geometric consequences of the condition $Ric^\perp>0$ to illustrate that the condition is also quite restrictive. Finally this last point is made evident with a classification result in dimension three and a partial classification in dimension four.

We pursuit the research line proposed in \cite{YZ-Gflat} about the classification of Hermitian manifolds whose $s$-Gauduchon connection $\nabla^s =(1-\frac{s}{2})\nabla^c + \frac{s}{2}\nabla^b$ is flat, where $s \in \mathbb{R}$ and $\nabla^c$ and $\nabla^b$ are the Chern and the Bismut connections, respectively. We focus on Lie groups equipped with a left invariant Hermitian structure. Such spaces provide an important class of Hermitian manifolds in various contexts and are often a valuable vehicle for testing new phenomena in complex and Hermitian geometry. More precisely, we consider a connected $2n$-dimensional Lie group $G$ equipped with a left-invariant complex structure $J$ and a left-invariant compatible metric $g$ and we assume that its connection $\nabla^s$ is flat. Our main result states that if either $n$=2 or there exits a $\nabla^s$-parallel left invariant frame on $G$, then $g$ must be K\"ahler. This result demonstrates rigidity properties of some complete Hermitian manifolds with $\nabla^s$-flat Hermitian metrics., Comment: 10 pages, In this new version, we add Cor 1.6 in the introduction and also an appendix on Kahler flat Lie groups

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.

In this paper, we consider orthogonal Ricci curvature $Ric^{\perp}$ for K\"ahler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate their subtle relationship.

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 Lu

We consider a family of K\"ahler structures on products of 2-spheres, arising from complex Bott manifolds. These are obtained via iterated $\mathbb P^1$-bundle constructions, generalizing the classical Hirzebruch surfaces. We show that the resulting K\"ahler structures all have identical Chern classes. We construct Bott diagrams, which are rooted forests with an edge labelling by positive integers, and show that these classify these K\"ahler structures up to biholomorphism., Comment: 12 pages

In this note, we continue the investigation of a projective K\"ahler manifold $M$ of semi-negative holomorphic sectional curvature $H$. We introduce a new differential geometric numerical rank invariant which measures the number of linearly independent {\it truly flat} directions of $H$ in the tangent spaces. We prove that this invariant is bounded above by the nef dimension and bounded below by the numerical Kodaira dimension of $M$. We also prove a splitting theorem for $M$ in terms of the nef dimension and, under some additional hypotheses, in terms of the new rank invariant.

This paper is the first step in a systematic project to study examples of K\"ahler manifolds with positive holomorphic sectional curvature ($H > 0$). Previously Hitchin proved that any compact K\"ahler surface with $H>0$ must be rational and he constructed such examples on Hirzebruch surfaces $M_{2, k}=\mathbb{P}(H^{k}\oplus 1_{\mathbb{CP}^1})$. We generalize Hitchin's construction and prove that any Hirzebruch manifold $M_{n, k}=\mathbb{P}(H^{k}\oplus 1_{\mathbb{CP}^{n-1}})$ admits a K\"ahler metric of $H>0$ in each of its K\"ahler classes. We demonstrate that the pinching behaviors of holomorphic sectional curvatures of new examples differ from those of Hitchin's which were studied in the recent work of Alvarez-Chaturvedi-Heier. Some connections to recent works on the K\"ahler-Ricci flow on Hirzebruch manifolds are also discussed. It seems interesting to study the space of all K\"ahler metrics of $H>0$ on a given K\"ahler manifold. We give higher dimensional examples such that some K\"ahler classes admit K\"ahler metrics with $H>0$ and some do not., Comment: In this 2nd version. we add a corollary (Corollary 1.8) and update some typos and inaccuaricies pointed by Gordon Heier

Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact K\"ahler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called $\textbf{real bisectional curvature}$ for Hermitian manifolds. When the metric is K\"ahler, this is just the holomorphic sectional curvature $H$, and when the metric is non-K\"ahler, 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.

We generalize a construction of Hitchin to prove that, given any compact K\"ahler manifold $M$ with positive holomorphic sectional curvature and any holomorphic vector bundle $E$ over $M$, the projectivized vector bundle ${\mathbb P}(E)$ admits a K\"ahler metric with positive holomorphic sectional curvature.

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 pages

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 corrected

In this article, we examine the behavior of the Riemannian and Hermitian curvature tensors of a Hermitian metric, when one of the curvature tensors obeys all the symmetry conditions of the curvature tensor of a K\"ahler metric. We will call such metrics G-K\"ahler-like or K\"ahler-like, for lack of better terminologies. Such metrics are always balanced when the manifold is compact, so in a way they are more special than balanced metrics, which drew a lot of attention in the study of non-K\"ahler Calabi-Yau manifolds. In particular we derive various formulas on the difference between the Riemannian and Hermitian curvature tensors in terms of the torsion of the Hermitian connection. We believe that these formulas could lead to further applications in the study of Hermitian geometry with curvature assumptions., Comment: This is the final version, accepted by CAG

In this article, we prove a Kahler extension theorem for real Kahler submanifolds of codimension 4 and rank at least 5. Our main theorem states that such a manifold is a holomorphic hypersurface in another real Kahler submanifold of codimension 2. This generalizes a result of Dajczer and Gromoll in 1997 which states that any real Kahler submanifolds of codimension 3 and rank at least 4 admits a Kahler extension., Comment: 23 pages

In 1991, Dajczer and Rodriguez proved in [10] that a complete minimal real Kahler submanifold of codimension 2, if with complex dimension > 2, would be either holomorphic, or a cylinder, or complex ruled. In this article, we generalize their result to real analytic complete real Kahler submanifolds of codimension 4. The conclusion is that such the submanifold, if with complex dimension > 4, would be either partially holomorphic, or a cylinder, or a twisted cylinder in the sense that the complex relative nullity foliation is contained in a strictly larger holomorphic foliation, whose leaves are cylinders. We also examine the question of when such a submanifold is complex ruled., Comment: 12 pages

We construct a compact K\"ahler manifold of nonnegative quadratic bisectional curvature, which does not admit any K\"ahler metric of nonnegative orthogonal bisectional curvature. The manifold is a 7-dimensional K\"ahler C-space with second Betti number equal to 1, and its canonical metric is a K\"ahler-Einstein metric of positive scalar curvature, Comment: 11 pages