Solving Optimal Control Problems for Delayed Control-Affine Systems with Quadratic Cost by Numerical Continuation

TL;DR

This paper presents a novel numerical homotopy method for efficiently solving fixed-delay optimal control problems in control-affine systems with quadratic cost, starting from the non-delayed solution.

Contribution

It introduces a homotopy-based approach to handle delays in optimal control problems, overcoming complexities of indirect methods and providing convergence guarantees.

Findings

Method successfully solves delay problems starting from non-delayed solutions.

Numerical efficiency demonstrated through a practical example.

Convergence results support the robustness of the approach.

Abstract

- In this paper we introduce a new method to solve fixed-delay optimal control problems which exploits numerical homotopy procedures. It is known that solving this kind of problems via indirect methods is complex and computationally demanding because their implementation is faced with two difficulties: the extremal equations are of mixed type, and besides, the shooting method has to be carefully initialized. Here, starting from the solution of the non-delayed version of the optimal control problem, the delay is introduced by numerical homotopy methods. Convergence results, which ensure the effectiveness of the whole procedure, are provided. The numerical efficiency is illustrated on an example.

Equations

$${\text{HSC}}_{\omega }(v):=\frac{1}{|v{|}_{\omega }^4}\sum _{i,j,k,\ell =1}^n{R}_{i\overline{j}k\overline{\ell }}{v}_i{\overline{v}}_j{v}_k{\overline{v}}_{\ell }.$$

$${\text{RBC}}_{\omega }:{\mathcal{F}}_M\times {\mathbb{R}}^n\backslash \{0\}⟶\mathbb{R},{\text{RBC}}_{\omega }(v):=\frac{1}{|v{|}^2}\sum _{\alpha ,\gamma }{R}_{\alpha \overline{\alpha }\gamma \overline{\gamma }}{v}_{\alpha }{v}_{\gamma },$$

$${\text{QOBC}}_{\omega }:{\mathcal{F}}_M\times {\mathbb{R}}^n\backslash \{0\}\to \mathbb{R},{\text{QOBC}}_{\omega }(v)=\frac{1}{|v{|}_{\omega }^2}\sum _{\alpha ,\gamma =1}^n{R}_{\alpha \overline{\alpha }\gamma \overline{\gamma }}(v_{\alpha }-v_{\gamma }{)}^2,$$

$$A\succsim B⟺A\text{has a sign}⟹B\text{has the same sign}.$$

$$R_{i\overline{j}k\overline{\ell }}$$

$${\text{HSC}}_{\omega }:{\mathcal{F}}_M\times T^{1,0}M\to \mathbb{R},{\text{HSC}}_{\omega }(v):=\frac{1}{|v{|}_{\omega }^4}\sum _{i,j,k,\ell =1}^n{R}_{i\overline{j}k\overline{\ell }}{v}_i{\overline{v}}_j{v}_k{\overline{v}}_{\ell }.$$

$${\text{HSC}}_{\omega }(v)=\underset{f}{\sup }{K}_{f^{\ast }\omega },$$

$${\text{Ric}}_{\omega }^{(1)}$$

$${\text{Ric}}_{\omega }^{(2)}$$

$${\text{Ric}}_{\omega }^{(3)}={\text{Ric}}_{k\overline{j}}^{(3)}=\sum _{i,\ell }{g}^{i\overline{\ell }}{R}_{i\overline{j}k\overline{\ell }},{\text{Ric}}_{\omega }^{(4)}={\text{Ric}}_{i\overline{\ell }}^{(4)}=\sum _{j,k}{g}^{k\overline{j}}{R}_{i\overline{j}k\overline{\ell }}.$$

$${\text{Scal}}_{\omega }:=\sum _{i,j}{g}^{i\overline{j}}{\text{Ric}}_{i\overline{j}}^{(1)}=\sum _{k,\ell }{g}^{k\overline{\ell }}{\text{Ric}}_{k\overline{\ell }}^{(2)}$$

$${\widetilde{\text{Scal}}}_{\omega }:=\sum _{i,\ell }{g}^{i\overline{\ell }}{\text{Ric}}_{k\overline{j}}^{(3)},$$

$$\underset{M}{\sup }|\partial f{|}^2$$

$$\partial \overline{\partial }|\partial f{|}^2$$

$${\Delta }_{\omega }|\partial f{|}^2$$

$$R_{\alpha \overline{\beta }\gamma \overline{\delta }}^h\left(g^{i\overline{j}}{f}_i^{\alpha }\overline{f_j^{\beta }}\right)\left(g^{k\overline{\ell }}{f}_k^{\gamma }\overline{f_{\ell }^{\delta }}\right).$$

$$R_{\alpha \overline{\beta }\gamma \overline{\delta }}^h\left(g^{i\overline{j}}{f}_i^{\alpha }\overline{f_j^{\beta }}\right)\left(g^{k\overline{\ell }}{f}_k^{\gamma }\overline{f_{\ell }^{\delta }}\right)$$

$${\text{RBC}}_{\omega }:{\mathcal{F}}_M\times {\mathbb{R}}^n\backslash \{0\}\to \mathbb{R},{\text{RBC}}_{\omega }(v):=\frac{1}{|v{|}_{\omega }^2}\sum _{\alpha ,\gamma =1}^n{R}_{\alpha \overline{\alpha }\gamma \overline{\gamma }}{v}_{\alpha }{v}_{\gamma },$$

$${\int }_{{\mathbb{P}}^{n-1}}\frac{w_i{\overline{w}}_j{w}_k{\overline{w}}_{\ell }}{|w{|}^4}{\omega }_{\text{FS}}^{n-1}$$

$$\sum _{i,j,k,\ell =1}^n{\int }_{{\mathbb{P}}^{n-1}}\frac{1}{|v{|}^4}{R}_{i\overline{j}k\overline{\ell }}{v}_i{\overline{v}}_j{v}_k{\overline{v}}_{\ell }{\omega }_{\text{FS}}^{n-1}$$

$${\text{RBC}}_{\omega }(v)$$

$${\lambda }_{\min }(\hat{\mathcal{R}})\le {\text{RBC}}_{\omega }\le {\lambda }_{\max }(\hat{\mathcal{R}})$$

$${\Gamma }_{\circ }:=\{(x_1,...,x_n)\in {\mathbb{R}}_{+}^n:x_1^2+\cdots +x_n^2\ne 0\}$$

$${\text{RBC}}_{\omega }^{\Gamma }:{\mathcal{F}}_M\times \Gamma \to \mathbb{R},{\text{RBC}}_{\omega }(v):=\frac{1}{|v{|}_{\omega }^2}\sum _{\alpha ,\gamma =1}^n{R}_{\alpha \overline{\alpha }\gamma \overline{\gamma }}{v}_{\alpha }{v}_{\gamma },$$

$$\Gamma :=\{(x_1,...,x_n)\in {\mathbb{R}}_{+}^n:x_1\ge \cdots \ge x_n\}\cap {\Gamma }_{\circ }.$$

$${\widetilde{\text{HSC}}}_{\omega }^{\Gamma }:{\mathcal{F}}_M\times \Gamma \to \mathbb{R},{\widetilde{\text{HSC}}}_{\omega }^{\Gamma }(v):=\frac{1}{|v{|}^2}\sum _{\alpha ,\gamma =1}^n(R_{\alpha \overline{\alpha }\gamma \overline{\gamma }}+R_{\alpha \overline{\gamma }\gamma \overline{\alpha }})v_{\alpha }{v}_{\gamma },$$

$${\text{HSC}}_{\omega }\succsim {\widetilde{\text{HSC}}}_{\omega }^{\circ }.$$

$${\widetilde{\text{HSC}}}_{\omega }^{\circ }\succsim {\text{HSC}}_{\omega }\succsim {\widetilde{\text{HSC}}}_{\omega }^{\circ }.$$

$${\widetilde{\text{HSC}}}_{\omega }(v)$$

Full text

Introduction

The curvature properties of a Hermitian metric are far from being understood. Even within the confines of the God-fearing Kähler world, many problems remain open. One of the most curious objects in the subject is the holomorphic sectional curvature

[TABLE]

The sign of the holomorphic sectional curvature is known to impact the complex structure of the underlying manifold: A compact Hermitian manifold $(M,\omega)$ with $\text{HSC}_{\omega}<0$ is hyperbolic in the sense of Kobayashi. The relationship between the better understood Ricci curvature is also mysterious, with the best indication of their relationship coming from the Wu–Yau theorem [56, 57, 54, 43, 62, 51, 9]: If $(M,\omega)$ is a compact Kähler manifold with $\text{HSC}_{\omega}<0$, then there exists a Kähler metric $\eta$ such that $\text{Ric}_{\eta}<0$. Hitchin’s examples [27] of Hodge metrics on the Hirzebruch surfaces $\mathbb{F}_{n}$ for $n\geq 2$, however, indicate that $\text{HSC}_{\omega}>0$ does not imply $\text{Ric}_{\eta}>0$, in general.

The present state of affairs is much worse if one enters the wilderness of the Hermitian category. The presence of torsion in the Chern connection splits the Ricci curvature into four variants, and it is not even clear whether the holomorphic sectional curvature controls either of the two scalar curvatures, let alone the Ricci curvatures. Many open problems in complex geometry are a symptom of this gap in our understanding.

Curvatures as quadratic form-valued functions on the unitary frame bundle

In [9, 10], the first named author initiated a program to understand the relationship between these curvatures by interpreting them as quadratic form-valued functions on the unitary frame bundle. This was first applied to the Yang–Zheng real bisectional curvature [62]:

[TABLE]

where it was observed in [9, 10] that (in each frame) we can write the real bisectional curvature as a Rayleigh quotient for a certain matrix of curvatures. Realizing the real bisectional curvature as a Rayleigh quotient makes it immediate that (in each frame) the maximum (respectively, minimum) is always reached and occurs when $v$ is the eigenvector corresponding to the largest (respectively, smallest) eigenvalue of the symmetric part $\frac{1}{2}(\mathcal{R}+\mathcal{R}^{t})$. Further, the computation of the real bisectional curvature, and construction of examples is made substantially more transparent.

In [11, 12, 13] this program was extended to the quadratic orthogonal bisectional curvature:

[TABLE]

which has been important in the study nef classes on Kähler manifolds which admit semi-positive representatives [58]. Despite the similarity in appearance with the real bisectional curvature, the interpretations of these curvatures as quadratic forms illuminates the fact that the QOBC is, in a sense, ‘opposite’ to the real bisectional curvature. We will discuss the precise meaning of this in $\S 1$.

Purpose and structure of this manuscript

The purpose of the present manuscript is to give a systematic treatment of the amazon of curvatures that appear in the Hermitian category. Particular focus is placed on the holomorphic sectional curvature, the real bisectional curvature, and the quadratic orthogonal bisectional curvature.

In the Hermitian category, we show that it is necessary to consider certain $\Gamma$–variants of these curvatures, i.e., curvatures associated with certain cones $\Gamma\subseteq\mathbb{R}^{n}\backslash\{0\}$.

We exhibit examples that illustrate both frame dependence and frame-independence of these curvatures. This is the first instance in where the variation of the frame has been investigated for the real bisectional curvature or quadratic orthogonal bisectional curvature.

In more detail, here is the structure of the manuscript:

• ($\dagger$)

In $\S 1$ we remind ourselves of some basic Hermitian geometry. The real bisectional curvature is recalled and its $\Gamma$–variant is introduced. The relationship between the $\Gamma$–real bisectional curvature and the holomorphic sectional curvature is studied. We also compute the real bisectional curvature and the $\Gamma$–real bisectional curvature of the standard metric on the Hopf surface $\mathbb{S}^{3}\times\mathbb{S}^{1}$. For this metric, the curvatures are shown to be independent of the choice of unitary frame.

• ($\dagger$)

In $\S 2$ we address the question of constant holomorphic sectional curvature, producing a theorem of Wu–Yau-type. The altered real bisectional curvature (and its $\Gamma$–variant) is introduced here. This curvature is a cousin of the real bisectional curvature and is shown to exhibit curious behavior.

• ($\dagger$)

In $\S 3$ we introduce the altered holomorphic bisectional curvature. This curvature is invariantly defined and is much weaker than the holomorphic bisectional curvature. It is strong enough to control the real bisectional curvature, however, and therefore produces a natural curvature constraint for the Schwarz lemma.

• ($\dagger$)

In $\S 4$ we study the quadratic orthogonal bisectional curvature of a Hermitian metric, and define its altered variant. The relationship between these curvatures and the scalar curvatures is exhibited in this section.

• ($\dagger$)

In $\S 5$ we show that these curvatures are frame-dependent, explicitly computing the curvature of the Tricerri metric on the Inoue surface.

Acknowledgements

The first named author would like to thank his supervisors Ben Andrews and Gang Tian for their support and encouragement. He would also like to thank Fangyang Zheng for valuable conversations. The second author is grateful to Professor Fangyang Zheng for constant encouragement and support.

$\S 1$. The Holomorphic Sectional Curvature and its variants

Central to understanding questions of curvature is when the sign of one curvature determines the sign of another curvature. Let us introduce a partial ordering $\succsim$ on the set of curvature notions of a Hermitian metric defined by the rule:

[TABLE]

If $A\succsim B$ and $B\succsim A$, then we say that $A$ and $B$ are comparable. If a curvature $A$ is constant, equal to some $c\in\mathbb{R}$, we write $A\equiv c$.

Reminder 1.1

Let $(M^{n},\omega)$ be a Hermitian manifold. The Chern connection is the unique connection on $T^{1,0}M$ which is compatible with the metric and the complex structure, and whose torsion has vanishing $(1,1)$–part.

Let $(z_{1},...,z_{n})$ denote local holomorphic coordinates near a point $p\in M$. The components of the Chern curvature tensor $R_{i\overline{j}k\overline{\ell}}$ in a coordinate frame $\partial_{z_{1}},...,\partial_{z_{n}}$ afford the description:

[TABLE]

Reminder 1.2

Let $v=(v_{1},...,v_{n})\in T^{1,0}M$ be a $(1,0)$–tangent vector. The holomorphic sectional curvature is the function

[TABLE]

Remark 1.3

A more enlightening description of the holomorphic sectional curvature is the following observation of Wu [59]: Fix a point $p\in M$ and let $v\in T_{p}^{1,0}M$. Let $f:\mathbb{D}\hookrightarrow M$ be an embedded holomorphic disk with $f(0)=p$ and $f^{\prime}(0)=v$. The pullback metric $f^{\ast}\omega$ yields a non-degenerate Hermitian metric on the unit disk, from which we may calculate the Gauss curvature $K_{f^{\ast}\omega}$. The holomorphic sectional curvature of $\omega$ is then

[TABLE]

where the supremum is taken over all embedded holomorphic disks $f:\mathbb{D}\to M$ with $f(0)=p$ and $f^{\prime}(0)=v$.

Reminder 1.4

The Chern curvature is an $\text{End}(T^{1,0}M)$–valued $(1,1)$–form. The first Chern–Ricci curvature is the contraction over the endomorphism part:

[TABLE]

and is a $(1,1)$–form representing the first Chern class the anti-canonical bundle $c_{1}(K_{M}^{-1})$. The second Chern–Ricci curvature is a contraction over the $(1,1)$–part:

[TABLE]

Similarly, the third and fourth Chern–Ricci curvature are defined:

[TABLE]

Note that $\text{Ric}_{\omega}^{(4)}$ is the conjugate of $\text{Ric}_{\omega}^{(3)}$.

Reminder 1.5

The contraction

[TABLE]

is the Chern scalar curvature. In general, this differs from the contraction

[TABLE]

which we call the altered Chern scalar curvature.

Reminder 1.6

Recall that a Hermitian metric $\omega$ is said to be balanced if (after identifying $\omega$ with its $(1,1)$–form) $d\omega^{n-1}=0$. It is well-known that this is equivalent to $\text{Scal}_{\omega}\equiv\widetilde{\text{Scal}}_{\omega}$ (see, e.g., [25]).

Remark 1.7

If the metric is Kähler (or more generally, Kähler-like [60]), then all Chern–Ricci curvatures coincide. Recall that a Hermitian metric $\omega$ is said to be Kähler-like (in the sense of [60]) if the Chern curvature tensor has the symmetries of the Kähler curvature tensor. The standard example of a non-Kähler Kähler-like metric is the standard metric on the Iwasawa threefold. This metric is Chern-flat, i.e., all components of the Chern curvature tensor vanish, and hence the Kähler-like condition is trivially satisfied.

Remark 1.8

To the best of the authors’ knowledge, there are no known examples of compact Hermitian manifolds with non-Kähler Kähler-like metrics that are not Chern flat. It is generally suspected, however, that many such metrics exist.

Remark 1.9

The Wu–Yau theorem [56, 57, 54, 21, 62, 9] indicates an important relationship between the holomorphic sectional curvature $\text{HSC}_{\omega}$ and the Ricci curvature(s). In the compact Kähler case, it states that if $(M,\omega)$ is compact Kähler with $\text{HSC}_{\omega}<0$, then $\text{Ric}_{\eta}<0$ for some (a priori different) Kähler metric $\eta$.

It remains unknown whether there is a Kähler metric on a compact Kähler manifold with $\text{HSC}_{\omega}<0$, but $\text{Ric}_{\omega}$ does not have a sign (c.f., [20]). In the ‘positive case’, Hitichin’s examples of Hodge metrics on the Hirzebruch surfaces $\mathbb{F}_{n}$ show that $\text{HSC}_{\omega}>0$, but there is no metric of positive Ricci curvature on $\mathbb{F}_{n}$ for $n\geq 2$.

Remark 1.10

The relationship between the holomorphic sectional curvature and the Ricci curvature(s) is of particular interest because of the Schwarz lemma (more precisely, because of the Chern–Lu formula). Indeed, let $f:(M,\omega)\to(N,\eta)$ be a holomorphic map between complete Kähler manifolds. If $\text{Ric}_{\omega}\geq-C_{1}\omega+C_{2}\eta$, with $C_{2}\geq 0$, and $\text{HSC}_{\omega}\leq-C_{3}<0$, then

[TABLE]

The above statement has a rich history [1, 34, 63, 29, 49, 39, 62, 9] which we will not discuss here. For a survey of these developments, we invite the reader to consult [10, 49].

For the purpose of properly motivating some considerations, however, let us recall: The Bochner formula, applied to the section $\partial f$ of the twisted cotangent bundle $T^{\ast}M\otimes f^{\ast}TN$, yields

[TABLE]

where $\nabla$ denotes the connection on $T^{\ast}M\otimes f^{\ast}TN$ induced by the Chern connection on $T^{1,0}M$ and $T^{1,0}N$, and $R^{T^{\ast}M\otimes f^{\ast}TN}$ denotes the curvature of this connection. By standard theory, the curvature of the tensor product of bundles splits additively, giving

[TABLE]

Taking the trace with respect to the metric $\omega$, produces the following formula for the Laplacian of the energy density:

[TABLE]

The target curvature term $(\omega^{\#}\otimes\omega^{\#}\otimes\text{Rm}_{\eta})(\partial f,\overline{\partial f},\partial f,\overline{\partial f})$ is the object of interest. If the metric is Kähler-like, then Royden showed that this is controlled by the holomorphic sectional curvature. This is not the case in general: With respect to a local unitary frame, this target curvature term reads:

[TABLE]

Fix a point $p\in M$, and choose a unitary frame at $p$ and $f(p)$ such that $f_{i}^{\alpha}=\lambda_{i}\delta_{i}^{\alpha}$, where $\lambda_{1}\geq\cdots\lambda_{r}>\lambda_{r+1}=\cdots=\lambda_{n}=0$ are the principal values of $\partial f$. Then

[TABLE]

Definition 1.11

Let $(M^{n},\omega)$ be a Hermitian manifold. The real bisectional curvature is the function

[TABLE]

where $v=(v_{1},...,v_{n})\in\mathbb{R}^{n}\backslash\{0\}$ and $R_{i\overline{j}k\overline{\ell}}$ denote the components of the Chern curvature tensor with respect to a fixed unitary frame (i.e., a section of the unitary frame bundle $\mathcal{F}_{M}$).

We declare the real bisectional curvature to be negative and write $\text{RBC}_{\omega}<0$ if this holds for all vectors and all unitary frames. The notions of positive real bisectional curvature, vanishing real bisectional curvature, etc., are defined with the obvious modifications. The same is to be said for all subsequent definitions of frame-dependent curvatures.

Remark 1.12

The real bisectional curvature is an unfortunately deceptive name, since it is much weaker than the holomorphic bisectional curvature $\text{HBC}_{\omega}$. Indeed, as shown in [62], the (sign of the) real bisectional curvature does not control the sign of the Ricci curvatures. Moreover, if the metric is Kähler-like, then the real bisectional curvature is comparable to the holomorphic sectional curvature. Indeed, let $\omega_{\text{FS}}$ denote the Fubini–Study metric (of unit volume) on $\mathbb{P}^{n-1}$. Write $[w_{1}:\cdots:w_{n}]$ for the standard homogeneous coordinates on $\mathbb{P}^{n-1}$. We have the identity

[TABLE]

Hence, at a fixed point $p\in M$, write $v=(t_{1}w_{1},...,t_{n}w_{n})=(t_{1},...,t_{n})\circ(w_{1},...,w_{n})$, where $\circ$ denotes the Hadamard product, and $t\in\mathbb{R}^{n}\backslash\{0\},w\in T_{p}^{1,0}M\simeq\mathbb{C}^{n}\backslash\{0\}$. Then

[TABLE]

If the metric is Kähler-like, then $R_{i\overline{i}k\overline{k}}=R_{i\overline{k}k\overline{i}}$, and hence, the holomorphic sectional curvature dominates the real bisectional curvature. In general, however, the real bisectional curvature dominates the holomorphic sectional curvature. To see this, fix a $(1,0)$–tangent vector $u$, and choose a unitary frame $\{e_{1},...,e_{n}\}$ such that $e_{1}$ is parallel to $u$. Then, for $v=(1,0,...,0)\in\mathbb{R}^{n}\backslash\{0\}$, we have $\text{RBC}_{\omega}(v)=\text{HSC}_{\omega}(u)$. That is, the holomorphic sectional curvature is a component of the real bisectional curvature.

In light of the Wu–Yau theorem, however, it is natural to ask:

Question 1.13

Let $(M^{n},\omega)$ be a compact Hermitian manifold with $\text{RBC}_{\omega}<0$. Does there exist a Hermitian metric $\eta$ on $M$ with $\text{Ric}^{(k)}_{\eta}<0$ for $k=1,...,4$?

Remark 1.14

An important remark must be made concerning the definition of the real bisectional curvature. In Definition 1.11, we permit the vector $v$ to be an arbitrary non-zero vector in $\mathbb{R}^{n}$. To see why this is natural, fix a unitary frame, and let $\mathcal{R}\in\mathbb{R}^{n\times n}$ be the matrix with entries $\mathcal{R}_{\alpha\gamma}:=R_{\alpha\overline{\alpha}\gamma\overline{\gamma}}$. The real bisectional curvature (as defined in Definition 1.11) is then the Rayleigh quotient

[TABLE]

in each frame. Let $\widehat{\mathcal{R}}:=\frac{1}{2}(\mathcal{R}+\mathcal{R}^{t})$ denote the symmetric part of $\mathcal{R}$. Then from the variational characterization of the eigenvalues, we have the sharp bounds:

[TABLE]

in each frame. In particular, $\text{RBC}_{\omega}>0$ coincides with the positive-definiteness of $\mathcal{R}$ in each frame.111More precisely, $\text{RBC}_{\omega}>0$ coincides with the positive-definiteness of $\mathcal{R}$ in the unitary frame which minimizes the smallest eigenvalue of $\mathcal{R}$.

There is a necessary refinement one must make, however, of the real bisectional curvature: Let $\mathbb{R}^{n}_{+}:=\{(x_{1},...,x_{n})\in\mathbb{R}^{n}:x_{k}\geq 0\ \forall k\}$ denote the non-negative orthant. In comparing the various curvatures, the cone

[TABLE]

will play an important role. We introduce the following:

Definition 1.15

Let $(M^{n},\omega)$ be a Hermitian manifold. Let $\Gamma\subseteq\mathbb{R}^{n}\backslash\{0\}$ be a cone. The $\Gamma$–real bisectional curvature is the function

[TABLE]

where $v=(v_{1},...,v_{n})\in\Gamma$. When $\Gamma=\Gamma_{\circ}$, we write $\text{RBC}_{\omega}^{\circ}$ for the $\Gamma_{\circ}$–real bisectional curvature.

Remark 1.16

We saw in Remark 1.14 that the real bisectional curvature can be viewed as the Rayleigh quotient of $\widehat{\mathcal{R}}$. Hence, in each frame, the extrema of the real bisectional curvature are realized by the eigenvalues of $\widehat{\mathcal{R}}$. This is no longer true for the $\Gamma$–real bisectional curvature, where the notion of positive-definiteness is replaced with $\Gamma$–copositivity: A real symmetric matrix $A$ is said to be $\Gamma$–copositive (for some cone $\Gamma$) if $v^{t}Av>0$ for all $v\in\Gamma$.

Remark 1.17

Note that, the natural cone to consider for the Hermitian Chern–Lu inequality is given by

[TABLE]

For this cone, the $\Gamma$–real bisectional curvature is called the second Schwarz bisectional curvature, written $\text{SBC}_{\omega}^{(2)}$ (see, e.g., [9, 10]).

Example 1.18

As the example of the standard metric on the Hopf surface $\mathbb{S}^{3}\times\mathbb{S}^{1}$ illustrates, the real bisectional curvature is strictly stronger than the $\Gamma_{\circ}$–real bisectional curvature. The cone $\Gamma_{\circ}$ is also important for the quadratic form realization of the holomorphic sectional curvature. To expound upon this, let us formalize the following, which appeared implicitly in [62]:

Definition 1.19

Let $\Gamma\subseteq\mathbb{R}^{n}\backslash\{0\}$ be a cone. The $\Gamma$–altered holomorphic sectional curvature $\widetilde{\text{HSC}}_{\omega}^{\Gamma}$ is defined to be the function

[TABLE]

where $v=(v_{1},...,v_{n})\in\Gamma$. In the case that $\Gamma=\mathbb{R}^{n}\backslash\{0\}$, we simply write $\widetilde{\text{HSC}}_{\omega}$, and refer to $\widetilde{\text{HSC}}_{\omega}$ as the altered holomorphic sectional curvature. When $\Gamma=\Gamma_{\circ}$, we write $\widetilde{\text{HSC}}_{\omega}^{\circ}$ for the $\Gamma_{\circ}$–altered holomorphic sectional curvature.

Remark 1.20

It follows from (0.10) that the sign of the holomorphic sectional curvature controls the sign of the $\Gamma$–altered holomorphic sectional curvature:

[TABLE]

The converse is true by the same argument that was used to show that $\text{RBC}_{\omega}\succsim\text{HSC}_{\omega}$. Hence, the holomorphic sectional curvature is comparable to the $\Gamma_{\circ}$–altered holomorphic sectional curvature:

[TABLE]

The $\Gamma_{\circ}$–altered holomorphic sectional curvature, therefore, provides us with a quadratic form-valued function on the unitary frame bundle, analogous to the $\Gamma_{\circ}$–real bisectional curvature. Given the success in utilizing the real bisectional curvature in [62, 64], the $\Gamma$–altered holomorphic sectional curvature offers a bridge to the holomorphic sectional curvature.

Remark 1.21

Let us note that, like the real bisectional curvature, the altered holomorphic sectional curvature is naturally identified (in each frame) as the Rayleigh quotient

[TABLE]

where $\mathcal{P}\in\mathbb{R}^{n\times n}$ is the real matrix with entries $\mathcal{P}_{\alpha\gamma}:=R_{\alpha\overline{\gamma}\gamma\overline{\alpha}}$, and $\mathcal{R}$ is the matrix in the definition of the real bisectional curvature. In particular, the extrema of the altered holomorphic sectional curvature $\widetilde{\text{HSC}}_{\omega}$ are realized by the eigenvalues of the symmetric part of $\mathcal{R+P}$; and $\widetilde{\text{HSC}}_{\omega}^{\Gamma}>0$ coincides with the $\Gamma$–copositivity of $\mathcal{R+P}$ (in each frame).

Example 1.22

It is not true, however, that $\text{HSC}_{\omega}\succsim\widetilde{\text{HSC}}_{\omega}$. Indeed, consider the standard metric

[TABLE]

on the Hopf surface $\mathbb{S}^{3}\times\mathbb{S}^{1}$. The curvature of the Chern connection has components

[TABLE]

Hence,

[TABLE]

and we may form the matrices

[TABLE]

The extrema of the altered holomorphic sectional curvature $\widetilde{\text{HSC}}_{\omega}$ are realized by the eigenvalues of the symmetric part of

[TABLE]

Hence,

[TABLE]

and we note that the altered holomorphic sectional curvature does not have a sign. The $\Gamma_{\circ}$–altered holomorphic sectional curvature, on the other hand, is given by

[TABLE]

for $v\in\Gamma_{\circ}$. Of course, to deduce that the holomorphic sectional curvature is non-negative, we must have $\widetilde{\text{HSC}}_{\omega}^{\circ}\geq 0$ in all frames, in the special case considered here, we can establish this to be the case:

Theorem 1.23

Let

[TABLE]

denote the standard metric on the Hopf surface $\mathbb{S}^{3}\times\mathbb{S}^{1}$. For any cone $\Gamma\subseteq\mathbb{R}^{n}\backslash\{0\}$, the $\Gamma$–real bisectional curvature $\text{RBC}_{\omega}^{\Gamma}$, the $\Gamma$–altered real bisectional curvature $\widetilde{\text{RBC}}_{\omega}^{\Gamma}$, and the $\Gamma$–altered holomorphic sectional curvature $\widetilde{\text{HSC}}_{\omega}^{\Gamma}$ are independent of the choice of unitary frame.

Proof.

The proof is elementary, albeit surprising: Fix a local unitary frame and view $R_{i\overline{j}k\overline{\ell}}$ as a matrix of $(1,1)$–forms. That is, for each $i,j$ fixed, we have

[TABLE]

Each entry of the matrix is invariant under a change of unitary frame, but the matrix varies according to the adjoint action $R\mapsto\widetilde{R}:=URU^{-1}$ of the unitary group, $U\in\text{U}(2)$. Let $U=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ be a unitary matrix. Then

[TABLE]

The components which appear in the $\Gamma$–real bisectional curvature are

[TABLE]

hence the $\Gamma$–real bisectional curvature and real bisectional curvature of the standard metric on the Hopf surface are invariant under the choice of frame. For the components involved in the $\Gamma$–altered real bisectional curvature, we have

[TABLE]

∎

Remark 1.24

The above theorem is the first result in the direction of analyzing the frame dependence of these curvatures. Moreover, it is the first explicit example of a Hermitian metric on a compact Hermitian manifold where precise information regarding the real bisectional curvature (and variants) is precisely understood.

Remark 1.25

The above example of the standard metric on the Hopf surface also illustrates that $\text{RBC}_{\omega}^{\circ}\not\succsim\widetilde{\text{HSC}}_{\omega}$. Indeed, the $\Gamma_{\circ}$–real bisectional curvature is non-negative:

[TABLE]

but $\widetilde{\text{HSC}}_{\omega}$ does not have a sign. Moreover, by Theorem 1.23, $\text{RBC}_{\omega}^{\circ}$ is independent of the choice of frame. Hence,

[TABLE]

but

[TABLE]

This again illustrates the necessity of considering the $\Gamma$–variants. One cannot avoid these considerations by passing to a different frame.

$\S 2$. Constant holomorphic sectional curvature

A long-standing conjecture in complex geometry predicts that a compact Hermitian manifold with constant holomorphic sectional curvature $\text{HSC}_{\omega}\equiv c$ is Chern-flat if $c=0$ and Kähler otherwise. This is known for complex surfaces by Balas–Gauduchon [5] in the $c\leq 0$ case and Apostolov–Davidov–Muskarov [2] in the $c>0$ case. For locally conformally Kähler metrics, the conjecture was settled by Chen–Chen–Nie [17] and for Kähler-like metrics, the conjecture was settled by the second named author [52]. For compact Hermitian threefolds with constant vanishing real bisectional curvature, the conjecture was verified by Zhou–Zheng [64]. In [32], Li–Zheng showed that if a Lie–Hermitian manifold supports a metric with $\text{HSC}_{\omega}\equiv c$, then $c=0$ and the metric is Chern-flat. In [47], it was shown that if $\omega$ is Strominger Kähler-like with $\text{HSC}_{\omega}\equiv c$, then $\omega$ is a Kähler metric.

With this conjecture as motivation, we have the following:

Theorem 2.1

Let $(M^{n},\omega)$ be a compact Hermitian manifold with constant holomorphic sectional curvature $\text{HSC}_{\omega}\equiv c$. Then

[TABLE]

Proof.

If $(M^{n},\omega)$ is a compact Hermitian manifold with $\text{HSC}_{\omega}\equiv c$, the Balas lemma [3] gives

[TABLE]

Hence, from (0.5), we have

[TABLE]

From (0.2), we see that

[TABLE]

Therefore,

[TABLE]

From (0.6), it follows that

[TABLE]

or equivalently,

[TABLE]

∎

Proposition 2.2

Let $(M^{n},\omega)$ be a compact Hermitian manifold with $\text{HSC}_{\omega}\equiv 0$. Assume the total Chern scalar curvature of $\omega$ vanishes. Then $\omega$ is balanced, and there are three distinguished cases:

• (i)

$\kappa(M)=-\infty$ and $K_{M}$ is unitary flat.

• (ii)

$\kappa(M)=-\infty$ and neither $K_{M}$ or $K_{M}^{-1}$ are pseudoeffective.

• (iii)

$\kappa(M)=0$ and $K_{M}$ is holomorphically torsion.

Proof.

If the holomorphic sectional curvature of $\omega$ vanishes identically, then Theorem 2.1 implies that the total Chern scalar curvature is non-negative. Assuming the total Chern scalar curvature vanishes, we see that $\omega$ is balanced and, in particular, Gauduchon. From [61, Theorem 1.4], there are the three cases (i)–(iii) in the statement. ∎

Corollary 2.3

Let $(M^{n},\omega)$ be a compact Hermitian manifold with $\text{HSC}_{\omega}\equiv 0$. Assume the total Chern scalar curvature of $\omega$ vanishes. If $\omega$ is $k$–Gauduchon for some $1\leq k\leq n-1$, then the metric $\omega$ is Chern-flat.

Proof.

From the previous proposition, the metric $\omega$ is balanced. If $\omega$ happens to be $k$–Gauduchon, for $1\leq k\leq n-1$, then $\omega$ is $k$–Gauduchon and balanced. By [33, Corollary 5.3], a balanced $k$–Gauduchon metric is Kähler. ∎

Proposition 2.4

Let $(M^{n},\omega)$ be a compact Hermitian manifold. If $\text{HSC}_{\omega}\equiv c$, then for any unit vector $v\in\mathbb{R}^{n}\backslash\{0\}$,

[TABLE]

In particular, $\text{HSC}_{\omega}\equiv c$ and $\widetilde{\text{HSC}}_{\omega}\equiv c$ if and only if $c=0$.

Proof.

Suppose the holomorphic sectional curvature of $\omega$ is constant, equal to $c$. The Balas lemma [3] implies that in any unitary frame, we have

[TABLE]

Therefore, for $i=j$, $k=\ell$, and $i\neq k$, we have

[TABLE]

For a unit vector $v=(v_{1},...,v_{n})\in\mathbb{R}^{n}\backslash\{0\}$, we have

[TABLE]

The expressions $\sum_{i\neq k}(R_{i\overline{i}k\overline{k}}+R_{k\overline{k}i\overline{i}})v_{i}v_{k}$ and $\sum_{i\neq k}(R_{k\overline{i}i\overline{k}}+R_{i\overline{k}k\overline{i}})v_{i}v_{k}$ are symmetric, hence,

[TABLE]

Since, by definition, $R_{i\overline{i}i\overline{i}}=c$, we have

[TABLE]

where the last equality uses the fact that $v$ has unit length. ∎

Remark 2.5

Since the $\Gamma_{\circ}$–altered holomorphic sectional curvature $\widetilde{\text{HSC}}_{\omega}^{\circ}$ is comparable to the holomorphic sectional curvature, it is clear that $\widetilde{\text{HSC}}_{\omega}^{\circ}\equiv 0$ if and only if $\text{HSC}_{\omega}\equiv 0$. Moreover, since $\widetilde{\text{HSC}}_{\omega}\succsim\widetilde{\text{HSC}}_{\omega}^{\circ}$, we have that $\widetilde{\text{HSC}}_{\omega}\equiv 0$ implies $\text{HSC}_{\omega}\equiv 0$. In summary:

[TABLE]

Given the success of using the real bisectional curvature to understand the geometry of compact Hermitian manifolds, we introduce the following curvature which measures the difference between the real bisectional curvature and the altered holomorphic sectional curvature:

Definition 2.6

Let $(M^{n},\omega)$ be a Hermitian manifold. The $\Gamma$–altered real bisectional curvature $\widetilde{\text{RBC}}_{\omega}^{\Gamma}$ is the function

[TABLE]

where $v=(v_{1},...,v_{n})\in\Gamma$. We write $\widetilde{\text{RBC}}_{\omega}^{\circ}$ for the $\Gamma_{\circ}$–altered real bisectional curvature. When $\Gamma=\mathbb{R}^{n}\backslash\{0\}$, we write $\widetilde{\text{RBC}}_{\omega}$, and refer to this as the altered real bisectional curvature.

Remark 2.7

Like the real bisectional curvature, the altered real bisectional curvature dominates the holomorphic sectional curvature: Indeed, for any unit $(1,0)$–tangent vector $u\in T^{1,0}M$, we can choose a unitary frame $e=\{e_{1},...,e_{n}\}$ such that $u$ is a scalar multiple of $e_{1}$. Taking $v=(v_{1},...,v_{n})=(1,0,...,0)$ then gives

[TABLE]

In particular,

[TABLE]

Proposition 2.8

Let $(M^{n},\omega)$ be a Hermitian manifold. Then $\widetilde{\text{RBC}}_{\omega}\equiv c$ is equivalent to

[TABLE]

for any Hermitian matrix $\xi=(\xi_{ij})$.

Proof.

For a fixed local unitary frame $e=\{e_{1},...,e_{n}\}$, any other unitary frame is given by $f_{k}=A_{kj}e_{j}$, where $A=(A_{kj})$ is a unitary matrix. Write

[TABLE]

where we set $\xi_{kt}:=\sum_{i=1}^{n}v_{i}A_{ik}\overline{A_{it}}$. Note that $\xi=(\xi_{kt})$ defines a Hermitian matrix. The condition $\widetilde{\text{RBC}}_{\omega}\equiv c$ is therefore equivalent to

[TABLE]

∎

In [62], it was shown that the if the real bisectional curvature is constant $\text{RBC}_{\omega}\equiv c$, then $c\leq 0$. For the altered real bisectional curvature, we have the following:

Theorem 2.9

Let $(M^{n},\omega)$ be a compact Hermitian manifold with constant altered real bisectional curvature $\widetilde{\text{RBC}}_{\omega}\equiv c$ for some $c\in\mathbb{R}$. Then $c\geq 0$. Further, if $c=0$, then $\omega$ is balanced with vanishing first, second, and third Ricci curvatures. In particular, if $n=3$ and $\widetilde{\text{RBC}}_{\omega}\equiv 0$, then $\omega$ is Chern-flat.

Proof.

Let $\eta=\sum_{j}\eta_{j}\varphi_{j}=\sum_{i,j}T_{ij}^{i}\varphi_{j}$ denote the Gauduchon $1$–form (with respect to a unitary coframe $\{\varphi_{1},...,\varphi_{n}\}$. Let $\tau_{k}=\sum_{i,j}T_{ij}^{k}\varphi_{i}\wedge\varphi_{j}$ denote the torsion $(2,0)$–forms.

From [60, Lemma 7], we have

[TABLE]

Setting $k=i$ and summing over $k$ gives

[TABLE]

Since $\partial(\omega^{n-1})=-2\eta\wedge\omega^{n-1}$ and $M$ is compact, integrating gives

[TABLE]

In a similar manner to [62], if $\widetilde{\text{RBC}}_{\omega}\equiv c$ then

[TABLE]

Hence,

[TABLE]

which implies that

[TABLE]

The remaining claims follow from [62, $\S 3$]. ∎

More generally, if the real bisectional curvature coincides with the altered real bisectional curvature, the metric is balanced:

Proposition 2.10

Let $(M^{n},\omega)$ be a Hermitian manifold. If

[TABLE]

then $\omega$ is balanced.

Proof.

Suppose $\text{RBC}_{\omega}\equiv\widetilde{\text{RBC}}_{\omega}$ at every point on $M$. Then for any local unitary frame, and any vectors $u=(u_{1},...,u_{n})\in\mathbb{R}^{n}\backslash\{0\}$ and $v=(v_{1},...,v_{n})\in\mathbb{R}^{n}\backslash\{0\}$, we have

[TABLE]

Taking $u=v=\frac{1}{\sqrt{n}}(1,...,1)$ gives

[TABLE]

By the well-known balanced criterion of equality of the scalar curvatures, $\omega$ is balanced. ∎

Theorem 2.11

Let $(M^{n},\omega)$ be a Hermitian manifold.

• (i)

If $\widetilde{\text{RBC}}_{\omega}\equiv c$ for some $c\in\mathbb{R}$, then $\text{RBC}_{\omega}\geq 0$ if $c>0$, or $\text{RBC}_{\omega}\leq 0$ if $c<0$.

• (ii)

If $\text{RBC}_{\omega}\equiv c$ for some $c\in\mathbb{R}$, then $\widetilde{\text{RBC}}_{\omega}\geq 0$ if $c>0$, or $\widetilde{\text{RBC}}_{\omega}\leq 0$ if $c<0$.

In particular, if $M$ is compact, then $\text{RBC}_{\omega}\equiv c_{1}$ and $\widetilde{\text{RBC}}_{\omega}\equiv c_{2}$ if and only if $c_{1}=c_{2}=0$.

Proof.

For the case (i), if $\widetilde{\text{RBC}}_{\omega}\equiv c$, we fix a local unitary frame $e=\{e_{1},...,e_{n}\}$, by Proposition 2.11, we have

[TABLE]

For the real bisectional curvature, we have

[TABLE]

This proves case (i); the proof of case (ii) is similar. ∎

$\S 3$. The Holomorphic Bisectional Curvature and its variants

In the hierarchy of curvatures, the holomorphic bisectional curvature sits just under the sectional curvature. In particular, the holomorphic bisectional curvature dominates all curvatures that have been seen thus far. Positive bisectional curvature is extremely restrictive – a compact Kähler manifold with positive bisectional curvature is biholomorphically isometric to $\mathbb{P}^{n}$ with the Fubini–Study metric. This is the famous solution of the Frankel conjecture by Mori [38] and Siu–Yau [50]. Mok’s solution of the generalized Frankel conjecture [37] also classifies those compact Kähler manifolds supporting Kähler metrics of non-negative holomorphic bisectional curvature.

The restriction of the holomorphic bisectional curvature to pairs of orthogonal $(1,0)$–tangent vectors yields the orthogonal bisectional curvature $\text{HBC}_{\omega}^{\perp}$. Algebraically, $\text{HBC}_{\omega}\succsim\text{HBC}_{\omega}^{\perp}$, but it was shown by Gu–Zhang [26] that a compact Kähler manifold with $\text{HBC}_{\omega}^{\perp}\geq 0$ has a metric $\eta$ with $\text{HBC}_{\eta}\geq 0$. Hence, Mok’s classification applies, showing that no new examples are exhibited from the curvature constraint $\text{HBC}_{\omega}^{\perp}\geq 0$ (at least in the Kähler setting).

The holomorphic bisectional curvature is a sum of two sectional curvatures. In this section, introduce a weaker curvature constraint, closer to the real bisectional curvature, and altered real bisectional curvature, given by a sum of two bisectional curvatures:

Definition 3.1

Let $(M^{n},\omega)$ be a compact Hermitian manifold. We define the altered holomorphic bisectional curvature $\widetilde{\text{HBC}}_{\omega}$ to be the function

[TABLE]

for $X,Y\in T^{1,0}M$.

The following theorem shows that if the altered bisectional curvature is constant, then the real bisectional curvature and the altered real bisectional curvature are controlled (in curiously different ways):

Theorem 3.2

Let $(M^{n},\omega)$ be a Hermitian manifold with $\widetilde{\text{HBC}}_{\omega}\equiv c$ for $c\neq 0$. Then for any local unitary frame, we have

[TABLE]

In particular, $\widetilde{\text{RBC}}_{\omega}\equiv c$. Moreover, the sign of the constant $c$ determines the sign of the real bisectional curvature.

Proof.

Suppose $\widetilde{\text{HBC}}_{\omega}\equiv c$. Then for any $X,Y\in T^{1,0}M$,

[TABLE]

For $\lambda,\mu\in\mathbb{C}$, the coefficient of $\overline{\lambda}\mu$ in the expansion of

[TABLE]

yields

[TABLE]

For $X,Y$ and $Z,W$ distinct unitary pairs, we have

[TABLE]

Hence, in any unitary frame, we have

[TABLE]

The remaining statements about the altered real bisectional curvature $\widetilde{\text{RBC}}_{\omega}$ and the real bisectional curvature $\text{RBC}_{\omega}$ are immediate consequences of Proof.. ∎

Proposition 3.3

Let $(M^{n},\omega)$ be a Hermitian manifold with $\widetilde{\text{HBC}}_{\omega}\equiv c$ for some $c\in\mathbb{R}$. Then

[TABLE]

Proof.

Restricting $\widetilde{\text{HBC}}_{\omega}$ to the diagonal, proves the asserting about the holomorphic sectional curvature. For the real bisectional curvature,

[TABLE]

∎

Remark 3.4

The above theorems have a few important immediate corollaries:

• (i)

Since $\widetilde{\text{HBC}}_{\omega}\equiv c$ implies $\widetilde{\text{RBC}}_{\omega}\equiv c$, this constant $c$ must be nonnegative if $M$ is compact. It follows that the real bisectional curvature $\text{RBC}_{\omega}$ is not constant if $\widetilde{\text{HBC}}_{\omega}\equiv c$ for some $c\neq 0$.

• (ii)

If $\widetilde{\text{HBC}}_{\omega}\equiv 0$, then $\text{RBC}_{\omega}\equiv 0$ and $\widetilde{\text{RBC}}_{\omega}\equiv 0$. Hence, all the Chern–Ricci curvatures vanish and the manifold is balanced.

• (iii)

From [64], it follows that compact Hermitian threefolds with $\widetilde{\text{HBC}}_{\omega}\equiv 0$ are Chern-flat. Boothby’s theorem [8] then classifies all compact Hermitian threefolds with $\widetilde{\text{HBC}}_{\omega}\equiv 0$: They are compact quotients of complex Lie groups with left-invariant metrics.

• (iv)

Finally, the above theorems do not require compactness of the underlying manifold, indicating the strength of the altered bisectional curvature.

Definition 3.5

The restriction of $\widetilde{\text{HBC}}_{\omega}$ to unitary pairs of $(1,0)$–tangent vectors $X,Y\in T^{1,0}M$ defines the altered orthogonal bisectional curvature $\widetilde{\text{HBC}}^{\perp}_{\omega}$.

Remark 3.6

The altered orthogonal bisectional curvature $\widetilde{\text{HBC}}_{\omega}^{\perp}$ appeared implicitly in [9, Theorem 2.31].

Theorem 3.7

Let $(M^{n},\omega)$ be a compact Hermitian manifold with $\widetilde{\text{HBC}}^{\perp}_{\omega}>0$. Then the Hodge numbers $h^{p,0}=0$ for all $1\leq p\leq n$.

Proof.

Assume there is a non-zero holomorphic $(p,0)$–form $\sigma\in H_{\overline{\partial}}^{p,0}(M)\simeq H^{0}(M,\Omega_{M}^{p})$. Let $x_{0}\in M$ be the point at which the comass $\|\sigma\|_{0}$ attains its maximum. From Ni’s viscosity considerations [40] (c.f., [41, 14]), we have (in a fixed unitary frame $e_{k}$ near $x_{0}$)

[TABLE]

for any $v\in T_{x_{0}}^{1,0}M$. Summing over the choices $v=e_{1},...,e_{k}$ yields

[TABLE]

In particular,

[TABLE]

Since $\widetilde{\text{HBC}}_{\omega}^{\perp}>0$, however, we see that $R_{j\overline{j}k\overline{k}}+R_{k\overline{k}j\overline{j}}>0$ and $R_{i\overline{i}i\overline{i}}>0$, furnishing the desired contradiction. ∎

The altered orthogonal bisectional curvature provides an invariant curvature constraint, weaker than the holomorphic bisectional curvature, but strong enough to establish a theorem of Wu–Yau-type:

Theorem 3.8

Let $(M^{n},\omega)$ be a compact Hermitian manifold with $\widetilde{\text{HBC}}_{\omega}^{\perp}\leq 0$. Then the canonical bundle $K_{M}$ is nef. Moreover, if $\widetilde{\text{HBC}}_{\omega}^{\perp}<0$ at some point, the canonical bundle is ample.

Proof.

It suffices to show that one can apply the Hermitian Chern–Lu inequality in [9, 10, 62]. To this end, assume $\widetilde{\text{HBC}}_{\omega}^{\perp}\leq-\delta$ for some $\delta\geq 0$. Then, in any unitary frame, we have

[TABLE]

We therefore have control of the $\Gamma_{\circ}$–real bisectional curvature:

[TABLE]

Applying the argument in [56, 57, 54, 21, 62, 9] proves the theorem. ∎

Remark 3.9

The proof of the above theorem amounts to the observation that

[TABLE]

$\S 4$. The Quadratic Orthogonal Bisectional Curvature and Its variants

Let us start with a reminder:

Definition 4.1

Let $(M^{n},\omega)$ be a Hermitian manifold. The quadratic orthogonal bisectional curvature is the function

[TABLE]

This curvature constraint first appeared implicitly in [7]. As discussed in [15, 16], the QOBC is the Weitzenböck curvature operator (c.f., [44, 45, 46]) acting on real $(1,1)$–forms. The first formulation of the QOBC as in Definition 4.1 was given by Wu–Yau–Zheng [58], where they showed that every nef class on a compact Kähler manifold with $\text{QOBC}_{\omega}\geq 0$ has a smooth semi-positive representative. In contrast with the orthogonal bisectional curvature, compact Kähler manifolds with $\text{QOBC}_{\omega}\geq 0$ are a more general class of manifolds than those supporting metrics with $\text{HBC}_{\omega}\geq 0$ [31]. Further studies of the QOBC were carried out in [15, 16, 13]

Remark 4.2

For the QOBC, we again have a quadratic form, but the QOBC is not a Rayleigh quotient. Hence, the extrema of the QOBC are not determined by the eigenvalues of $\mathcal{R}$. The most appropriate language for understanding this quadratic form comes from convex optimization and combinatorics. To explain this, let us recall:

Definition 4.3

A real symmetric matrix $\Sigma$ is said to be a Euclidean distance matrix (EDM) (of embedding dimension one) if there is a vector $v=(v_{1},...,v_{n})\in\mathbb{R}^{n}$ such that the components of $\Sigma$ are $\Sigma_{\alpha\gamma}=(v_{\alpha}-v_{\gamma})^{2}$. The set of Euclidean distance matrices in a given dimension form a convex cone which we call the EDM cone, denoted $\mathcal{EDM}$. If we wish to emphasize dimension, we will write $\mathcal{EDM}_{n}$.

Let $\Sigma_{v}$ denote the EDM corresponding to a vector $v\in\mathbb{R}^{n}$. The quadratic orthogonal bisectional curvature is therefore non-negative if $\text{tr}(\mathcal{R}\Sigma_{v})\geq 0$ for all $v\in\mathbb{R}^{n}$. Since the trace defines the Frobenius duality pairing, this is precisely the statement that $\mathcal{R}$ lies in the dual EDM cone:

Proposition 4.4

The quadratic orthogonal bisectional curvature of a Kähler-like metric is non-negative if and only if $\mathcal{R}$ lies in the dual EDM cone.

Remark 4.5

To understand the relation with the real bisectional curvature, note that the real bisectional curvature is non-negative if and only if (in the frame which minimizes the real bisectional curvature) the smallest eigenvalue of $\widehat{\mathcal{R}}$ is non-negative. In particular, $\widehat{\mathcal{R}}$ must be an element of the cone $\mathcal{PSD}$ of positive semi-definite matrices. The cone $\mathcal{PSD}$ is self-dual, while the EDM cone intersects the PSD cone only at the zero matrix.

Hence, the notions of $\text{QOBC}_{\omega}\geq 0$ and $\text{RBC}_{\omega}\geq 0$ are opposite in the sense that $\text{QOBC}_{\omega}\geq 0\iff\widehat{\mathcal{R}}\in\mathcal{EDM}^{\ast}$ and $\text{RBC}_{\omega}\geq 0\iff\widehat{\mathcal{R}}\in\mathcal{PSD}^{\ast}$, where $\mathcal{EDM}\cap\mathcal{PSD}=\{0\}$. That is, the curvature constraints are equivalent to $\widehat{\mathcal{R}}$ being in two cones whose dual have no non-trivial intersection. Of course, this does not mean that there are no Hermitian metrics with $\text{RBC}_{\omega}\geq 0$ and $\text{QOBC}_{\omega}\geq 0$. The cones $\mathcal{PSD}^{\ast}=\mathcal{PSD}$ and $\mathcal{EDM}^{\ast}$ have non-trivial intersection.

We also have the following eigenvalue characterization of $\text{QOBC}_{\omega}\geq 0$:

Theorem 4.6

([11, 12]). Let $(M^{n},\omega)$ be a Kähler-like Hermitian manifold. Let $\lambda_{1}\geq\cdots\geq\lambda_{n}$ denote the eigenvalues of $\widehat{\mathcal{R}}$. Then $\text{QOBC}_{\omega}\geq 0$ if and only if, for every Euclidean distance matrix $\Sigma_{v}$, the Perron weights $r_{2},...,r_{n}$ of $\Sigma$ satisfy

[TABLE]

in all frames.

Remark 4.7

Here, the Perron weights $r_{2},...,r_{n}$ of an $n\times n$ Euclidean distance matrix $\Sigma$, with eigenvalues $\delta_{1}\geq\cdots\geq\cdots\geq\delta_{n}$, are defined to be the ratios $r_{k}:=-\delta_{k}/\delta_{1}$. Note that the Perron weights of a Euclidean distance matrix always satisfy $0\leq r_{2}\leq r_{3}\leq\cdots\leq r_{n}\leq 1$.

In [15, 42], it is shown that a compact Kähler manifold with $\text{QOBC}_{\omega}\geq 0$ has non-negative scalar curvature. This extends to compact Hermitian manifolds with Kähler-like metrics without change. In the general Hermitian category, we have:

Theorem 4.8

Let $(M^{n},\omega)$ be a complete Hermitian manifold with $\text{QOBC}_{\omega}\geq 0$. Then for any point $p\in M$ and any unitary pair $v,w\in T_{p}^{1,0}M$ we have

[TABLE]

Moreover, we have the scalar curvature satisfies

[TABLE]

Proof.

Suppose $\text{QOBC}_{\omega}\geq 0$. Then for any $\xi=(\xi_{1},...,\xi_{n})\in\mathbb{R}^{n}$, and any unitary frame, we have

[TABLE]

For distinct indices $j,k,\ell$, set $\xi_{k}=0$, $\xi_{\ell}=2$, and $\xi_{j}=1$. This gives

[TABLE]

Let $f_{k}=\frac{1}{\sqrt{2}}(e_{k}-e_{\ell})$, $f_{\ell}=\frac{1}{\sqrt{2}}(e_{k}+e_{\ell})$ and $f_{j}=e_{j}$. Then (Proof.) in this frame gives

[TABLE]

Similarly, setting $f_{k}=\frac{1}{\sqrt{2}}(e_{k}-\sqrt{-1}e_{\ell})$, $f_{\ell}=\frac{1}{\sqrt{2}}(e_{k}+\sqrt{-1}e_{\ell})$ and $f_{j}=e_{j}$ gives

[TABLE]

Adding these equations together, we get

[TABLE]

Observe that

[TABLE]

Hence, for $k\neq\ell$, we have

[TABLE]

For the statement concerning the scalar curvature, we observe that

[TABLE]

∎

From the perspective of quadratic forms, it is natural to introduce the following variation of the quadratic orthogonal bisectional curvature:

Definition 4.9

Let $(M^{n},\omega)$ be a Hermitian manifold. The altered quadratic orthogonal bisectional curvature is the function

[TABLE]

The non-negativity of the altered QOBC, together with non-negativity of the QOBC is enough to control the scalar curvature:

Proposition 4.10

Let $(M^{n},\omega)$ be a complete Hermitian manifold. If $\text{QOBC}_{\omega}\geq 0$ and $\widetilde{\text{QOBC}}_{\omega}\geq 0$, then $\text{Scal}_{\omega}\geq 0$. If $\text{QOBC}_{\omega}\geq 0$ and $\widetilde{\text{QOBC}}_{\omega}>0$, then $\kappa(M)=-\infty$.

Proof.

The above theorem implies that if $\text{QOBC}_{\omega}\geq 0$, then

[TABLE]

If $\widetilde{\text{QOBC}}_{\omega}\geq 0$, then $\sum_{i,j=1}^{n}R_{i\overline{j}j\overline{i}}(\xi_{i}-\xi_{j})^{2}\geq 0$ for all $\xi=(\xi_{1},...,\xi_{n})\in\mathbb{R}^{n}$. Setting $\xi_{k}=1$, $\xi_{j}=0$ for any $j\neq k$, we see that

[TABLE]

In particular, the right-hand side of (0.10) is non-negative. ∎

Theorem 4.11

Let $(M^{n},\omega)$ be a complete Hermitian manifold with $\widetilde{\text{QOBC}}_{\omega}\geq 0$. Then for any point $p\in M$ and any unitary pair $v,w\in T_{p}^{1,0}M$ we have

[TABLE]

Moreover, the altered scalar curvature satisfies

[TABLE]

Proof.

Suppose $\widetilde{\text{QOBC}}_{\omega}\geq 0$. Then, in each unitary frame,

[TABLE]

for all $\xi=(\xi_{1},...,\xi_{n})\in\mathbb{R}^{n}$. Let $\xi_{k}=0$, $\xi_{\ell}=2$, and $\xi_{j}=1$ for $k\neq\ell$, $j\neq k$. Then

[TABLE]

Let $f_{k}=\frac{1}{\sqrt{2}}(e_{k}-e_{\ell})$, $f_{\ell}=\frac{1}{\sqrt{2}}(e_{k}+e_{\ell})$, and $f_{j}=e_{j}$. Then

[TABLE]

Similarly, setting $f_{k}=\frac{1}{\sqrt{2}}(e_{k}-\sqrt{-1}e_{\ell})$, $f_{\ell}=\frac{1}{\sqrt{2}}(e_{k}+\sqrt{-1}e_{\ell})$, $f_{j}=e_{j}$, we have

[TABLE]

Hence, we see that

[TABLE]

Since

[TABLE]

it follows that

[TABLE]

For the statement concerning the altered scalar curvature, simply observe that

[TABLE]

∎

Corollary 4.12

Let $(M^{n},\omega)$ be a complete Hermitian manifold. If $\text{QOBC}_{\omega}\geq 0$ and $\widetilde{\text{QOBC}}_{\omega}\geq 0$, then $\widetilde{\text{Scal}}_{\omega}\geq 0$.

Given the (mostly conjectural) rigidity of compact Hermitian manifolds with constant holomorphic sectional curvature, it is natural to ask whether similar rigidity theorems may hold for the QOBC and the altered QOBC. The standard metric on the Hopf surface shows that $\widetilde{\text{QOBC}}_{\omega}\equiv 0$ does not impose a significant constraint on the other curvatures:

Proposition 4.13

The standard metric on the Hopf surface $\mathbb{S}^{3}\times\mathbb{S}^{1}$ has constant $\widetilde{\text{QOBC}}_{\omega}$ and non-constant $\text{QOBC}_{\omega}\geq 0$ with

[TABLE]

Moreover, these bounds are sharp, and independent of the choice of unitary frame.

Proof.

The quadratic orthogonal bisectional curvature of the standard metric on the Hopf surface is

[TABLE]

From the scale invariance, assume $v=(v_{1},v_{2})$ is a unit vector, then

[TABLE]

this is maximized when $v_{1}=-\frac{1}{\sqrt{2}}$ with value $\text{QOBC}_{\omega}(v)=\frac{8}{|z|^{4}}$, and is minimized at $v_{1}=\frac{1}{\sqrt{2}}$ with value $\text{QOBC}_{\omega}(v)$. The altered quadratic orthogonal bisectional curvature of the standard metric on the Hopf surface is

[TABLE]

∎

Remark 4.14

In particular, the altered quadratic orthogonal bisectional curvature is much too weak to control the Ricci curvatures.

$\S 5$. An example of frame-dependence

Let $\mathcal{H}\subset\mathbb{C}$ denote the upper half-plane. We consider the Tricerri metric on the Inoue surface $\mathcal{S}_{M}:=(\mathbb{C}\times\mathcal{H})/\Sigma$. Here, $\Sigma$ is a group of automorphisms of $\mathbb{C}\times\mathcal{H}$ which we now describe:

The Inoue Surface

Following [23], let $M\in\text{SL}_{3}(\mathbb{Z})$ be an integral matrix with one real eigenvalue $\lambda>1$ and two distinct complex conjugate eigenvalues $\eta,\overline{\eta}$. Write $v=(v_{1},v_{2},v_{3})$ and $u=(u_{1},u_{2},u_{3})$ for a vector in the eigenspace of $\lambda$ and $\eta$, respectively. Let $z$ denote the coordinate on $\mathbb{C}$ and $w$ denote the coordinate on $\mathcal{H}$. The automorphism group $\Sigma$ is generated by the automorphisms

[TABLE]

The group $\Sigma$ acts on $\mathbb{C}\times\mathcal{H}$ properly discontinuously with compact quotient. For the calculations here, we work in a single compact fundamental domain for $\mathcal{S}_{M}$ in $\mathbb{C}\times\mathcal{H}$, using $(z,w)$ as local coordinates. We assume $(z,w)$ are uniformly bounded and that $\text{Im}(w)$ is uniformly bounded away from zero.

The Tricerri metric

On $\mathbb{C}\times\mathcal{H}$, define the non-negative $(1,1)$–forms

[TABLE]

These forms are invariant under the action of $\Sigma$, and thus descend to $(1,1)$–forms on the Inoue surface. The Tricerri metric is defined $\omega_{\text{T}}:=4\alpha+\beta$.

Curvature of the Tricerri metric

The non-zero component of the Chern curvature tensor is

[TABLE]

Changing the unitary frame, specified by the unitary matrix $U=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$, we see that

[TABLE]

and

[TABLE]

Form the matrices

[TABLE]

The Real Bisectional Curvature

The eigenvalues of the symmetric part of $\widetilde{\mathcal{R}}$ are

[TABLE]

Since the matrix $U$ is unitary, we have $|b|^{2}\leq 1$ and $|d|^{2}\leq 1$. It is easily checked that

[TABLE]

Similarly,

[TABLE]

Hence,

[TABLE]

and

[TABLE]

In particular, we have the following sharp pinching of the real bisectional curvature of the Tricerri metric:

[TABLE]

The $\Gamma_{\circ}$–real bisectional curvature $\text{RBC}_{\omega_{\text{T}}}^{\circ}$ is non-positive, with

[TABLE]

The Altered Real Bisectional Curvature

The eigenvalues of $\widetilde{\mathcal{P}}$ are

[TABLE]

Hence, we have the following sharp pinching of the altered real bisectional curvature

[TABLE]

The $\Gamma_{\circ}$–altered real bisectional curvature is non-positive with

[TABLE]

The Altered Holomorphic Sectional Curvatures

To compute the altered holomorphic sectional curvature, we compute the eigenvalues of

[TABLE]

To this end, we have

[TABLE]

Moreover,

[TABLE]

and

[TABLE]

The $\Gamma_{\circ}$–altered holomorphic sectional curvature is

[TABLE]