# Trajectory Control for Differential Drive Mobile Manipulators

Harish Karunakaran, Gopeshh Raaj Subbaraj, Robotics Engineering, WPI

###### Abstract

Mobile manipulator systems are comprised of a mobile platform with one or more
manipulators and are of great interest in a number of applications such as
indoor warehouses, mining, construction, forestry etc. We present an approach
for computing actuator commands for such systems so that they can follow
desired end-effector and platform trajectories without the violation of the
nonholonomic constraints of the system in an indoor warehouse environment. We
work with the Fetch robot which consists of a 7-DOF manipulator with a
differential drive mobile base to validate our method. The major contributions
of our project are, writing the dynamics of the system, Trajectory planning
for the manipulator and the mobile base, state machine for the pick and place
task and the inverse kinematics of the manipulator. Our results indicate that
we are able to successfully implement trajectory control on the mobile base
and the manipulator of the Fetch robot.

###### Index Terms:

Trajectory Control, Mobile Manipulators, Differential Drive

##  I Introduction

A mobile manipulator is defined as a robotic system composed of a mobile
platform and a manipulator mounted on the platform equipped with non-
deformable wheels. This combined system of a manipulator and mobile base is
able to perform manipulation tasks in a much larger workspace than a fixed-
base manipulator such as the Kuka Manipulator. Conventional robot manipulators
with a fixed base have limited reachable workspace. Thus the addition a
wheeled mobile platform to a robot manipulator provides added mobility to the
robotic system. Typically, a mobile manipulator consists of a multi-link
manipulator and a mobile platform. This increases the effective workspace of
the end-effector of the manipulator when compared to fixed base systems. In
recent years, these mobile manipulators have become a subject of interest
because of the mobility and dexterity. Thus, many service robots employ the
similar design which we use in our project.

The integration of a manipulator and a mobile platform, however, gives rise to
difficulties such as decomposing a given task into fine motions to be carried
out by the manipulator and the gross motions to be achieved by the mobile
platform and how to establish the dynamic model of the mobile manipulator
system in a systematic approach. Since, in most cases, the mobile platform is
subjected to nonholonomic constraints and the robot manipulator is a holonomic
mechanical system, how to control the manipulator and the mobile platform to
avoid possible tip over is also important [10]. A nonholonomic system is a
system whose state depends on the path taken in order to achieve it. Also,
unlike conventional industrial manipulators mounted on stationary bases, a
mobile manipulators motions interact dynamically with its base robot, which
degrades system performance, resulting in performance problems like excessive
end-effector errors and poor system stability. These systems operate in highly
unstructured environments, thus limiting the various sensing techniques
available for their control. The above characteristics of mobile manipulators
present challenging control problems.

A Differential Drive robot is a mobile robot whose movement is based on two
separately driven wheels placed on either side of the robot body. It can thus
change its direction by varying the relative rate of rotation of its wheels
and hence does not require an additional steering motion. A differential drive
robot cannot move in the direction along the axis which is a singularity. They
are also very sensitive to slight changes in velocity in each of the wheels.
Even small changes in the relative velocities between the wheels affects the
robot’s trajectory. This makes the control of differential drive system
difficult to implement in practice.

Our project focuses on trajectory following and control for mobile
manipulators and tries to address some of the problems discussed above. We
develop a control method for manipulators mounted on mobile robots which
applies to large manipulator motions when system characteristics are highly
variable and when the accuracy and the availability of sensory information is
limited. The Fetch mobile manipulator platform is used in a simulated
environments which is a differentially driven platform, equipped with a 7-DOF
manipulator is used. The Fetch robot is a combination of two subsystems. The
torso of this robot, called the Fetch, comprises of a camera and other vision
sensors along with the 7-DOF manipulator, while the base, called the Freight,
enables the navigation of the entire robot. The torso is mounted on the
freight. Fetch Robotics’ system uses a relatively large and capable mobile
manipulator to pick items off of warehouse shelves, while Freight, acts as an
autonomous cargo delivery cart. Fetch can pick items continuously, while a
succession of Freights can switch in and out to move different selections of
goods to different parts of the distribution centre. The Fetch robot is shown
in Figure 1.

The Denavit-Hartenburg parameters for the 7-DOF manipulator is found out and
is incorporated as part of the URDF (Unified Robot Description) of the fetch
robot. Lagrange’s method is used to obtain the reduced equations of motion
from the base of the robot to the end-effector. This gives us the dynamics of
the system. Next, we perform trajectory planning for the arm of the fetch
robot to plan from one joint position to another. Trac_ik[11] package
available as an open source implementation is used to find the inverse
kinematics of the arm. Then, for moving the mobile base of the fetch robot, we
used a planning algorithm to find a path from a start position in the
simulated environment to a goal position. In order to make the mobile base and
the trajectory planning work in conjunction, we developed a state-machine for
a pick and place task. The performance of our method has been tested on this
task in a simulated environment in Gazebo and has been discussed in the later
sections.

Figure 1: Fetch Robot

The applications of our system extend into the warehouse domain where
trajectory planning and control is an important aspect of picking and placing
objects in a warehouse or an industrial setting. This mobile manipulator
control method would provide greater autonomy to service robots and extend
their use. Our method tries to solve the problem of control in mobile
manipulators described above using the Fetch robot as an example. The rest of
this paper is divided as follows, the related work is discussed in Section II
followed by the Methodology in Section III and then the results and discussion
are presented in Section IV followed by Conclusion and Future work in Section
V.

##  II Related Work

In the past issues related to Mobile manipulators such as dynamic and static
stability, force development and application control in the presence of base
compliance, dynamic coupling issues, etc. have been studied [2]. Alicja et
al.[4] worked on the problem of trajectory tracking of non-holonomic mobile
manipulators. They presented two versions of the tracking control algorithm,
one for mobile manipulators with a non-holonomic platform only and the other
for a doubly non-holonomic mobile manipulator. They then compared the control
process of these in a simulation environment and made some suggestions about a
choice of drives for the robotic arm. They presented a dynamic control
algorithm for holonomic systems in general which can be used for a mobile
platform with any robot mounted atop of it. This was one of the earlier
approaches to tackle the problem of trajectory tracking as the computing power
was limited back then.

R.Solea et al.[5] in their paper focus on the motion planning problem of
mobile manipulator systems where they propose a methodology for generating
trajectories for both the mobile platform and the manipulator that will take a
system from an initial configuration to a pre-specified final one, without
violating the non-holonomic constraint. In order to ensure the smooth movement
of the considered system they used nonlinear sliding mode control to solve the
motion planning problem. They also present a controller design for the
trajectory-tracking problem using the sliding mode control for a mobile
platform equipped with a manipulator.

Real-time trajectory planning using model predictive control with constraints
was proposed by Ide.S et. al. [6] for mobile manipulators. Their method uses
Quadratic Programming (QP) for optimizing control inputs. The control inputs
and outputs are limited corresponding to the required motion and the hardware
specifications of the mobile manipulator. The required motion is changed
frequently according to the situation and the hardware limitations, concretely
the torque, the angular velocity, the mobile base velocity and the
acceleration, which are subject to the hardware design. Their method was
implemented to a real mobile manipulator model and real-time trajectory
modification was demonstrated on a real mobile robot. But one of the drawbacks
in their approach is that they generalize all manipulators and do not consider
system specific parameters. But, in this project we use system parameters that
are specific to the fetch robot as described in their approach.

Dong, W. et al [7] studied the tracking control problem of mobile manipulators
considering the interaction between the mobile platform and the manipulator.
They proposed a global tracking controller based on the dynamics of the
defined tracking error and the extended Barbalat’s lemma. Their proposed
controller ensures that the full state of the system asymptotically track the
given desired trajectory globally in the presence of the system coupling.

Korayem, M.H. et al [8] in their paper proposed an Open-loop optimal control
method as an approach for trajectory optimization of flexible mobile
manipulator for a given two end-point task in point-to-point motion. Dynamic
equations were derived using combined Euler Lagrange formulation which is
similar to the approach we follow in our project. To solve the optimal control
problem, they adopt an indirect method via establishing the Hamiltonian
function and deriving the optimality condition from Pontryagins minimum
principle. The equations obtained provided a two point boundary value problem
which they then solved by numerical techniques.

Korayem, M.H.,et al [9] in their paper presented a methodology for collision-
free trajectory planning of wheeled mobile manipulators in obstructed
environments by means of potential functions. But in our case we do not
consider obstructed environments making our approach simpler than theirs. In
their method, all mobile manipulator parts and environmental obstacles were
modeled as ellipsoids. Due to collision avoidance, the ellipsoid equations
were expressed in a reference coordinate system and the corresponding
dimensionless potential functions were defined. Then, the trajectory planning
of a spatial mobile robot in cluttered environment was performed, employing
optimal control theory. One of the problems with their approach was that since
they use control theory for trajectory planning, the planning times are higher
than standard methods.

Inverse kinematics is an extensive field of research where numerous
sophisticated algorithms have been presented during the last decades. However,
no universal solution could yet be found. Most prominent approaches either
attempt to directly derive a geometric analytical solution from the kinematic
model or aim to solve the problem iteratively by optimization. Some of the
most widely known and popular methods are based on computing the Jacobian
using the transpose, pseudoinverse or damped least squares method [12]. The
solution is then found by following the gradient, but problems are often
encountered by getting stuck in suboptimal extrema depending on the geometric
complexity of the kinematic model. Constructive approaches successfully
overcome this issue by performing heuristic restarts from random initial
configurations where [11] recently proposed a novel method, TRAC-IK [11] that
is publicly available under the ROS framework[13]. In this, significant
performance increases regarding the success rate could be obtained over the
classical Orocos KDL solver[15]. Their Trac_ik method improves upon several
failure points of KDL’s implementation of joint-limited pseudoinverse Jacobian
inverse kinematics. They are able to outperform the KDL IK implementation by
adding local minima and mitigation by reformulating the Inverse Kinematics
problem as a Sequential Quadratic Programming problem and utilizing Cartesian
tolerances to speed up the IK search. They are able to report a success rate
of around 99.5% on the Fetch robot and therefore is the choice for our IK
solver.

##  III Methodology

The overall work can be divided into the following modules,

  * •

Writing the Dynamics of the system

  * •

Trajectory Planning of the mobile base and the manipulator

  * •

Inverse Kinematics of the manipulator

  * •

Finite State Machine for the pick and place task

  * •

Trajectory Control using PID

###  III-A System Dynamics

Figure 2: The four parameters according to the DH convention are shown in red,
$\theta_{i},d_{i},a_{i},\alpha_{i}$

The first step in the writing the system dynamics involved writing the
Denavit-Hartenberg (DH) parameters for the manipulator arm. DH parameters are
the four parameters associated with a particular convention for attaching
reference frames to the links of a spatial kinematic chain, or robot
manipulator. This convention is a standard method for selecting frames of
reference in robotics applications. This method is very useful to carry out
kinematics operations in a manipulator. The four parameters of DH convention
are used for transformation from one frame to another.

These four parameters are :

  * •

d : offset along previous z to the common normal

  * •

$\theta$ : angle about previous z, from old x to new x

  * •

r : length of the common normal

  * •

$\alpha$ : angle about common normal.

The four parameters that associates two consecutive frames are shown in the
figure 2. In this project we calculated the DH parameters for the fetch robot
as the frames defined in the URDF file of fetch robot was not consistent with
the DH convention. Figure 3 shows the frames for our system defined as per DH
convention.

Figure 3: Robot Frames - DH convention

The next step involves finding the transformation from the base frame to end-
effector from the DH parameters. This gives us the transformation matrix. The
last column from the transformation matrix gives us the forward kinematics of
the system. Once we obtain the transformation matrix we can differentiate it
with respect to each of the each of the joint angles to obtain the Jacobian
matrix which relates the end-effector velocity with the joint velocities. We
can then compute the pseudoinverse of the Jacobian matrix to obtain the joint
velocities if we know the end-effector velocities. The equation relating the
joint velocities and the end-effector velocities is equation 1,

$\begin{bmatrix}\dot{x}\\\ \dot{y}\\\ \dot{z}\\\ \dot{\omega_{x}}\\\
\dot{\omega_{y}}\\\ \dot{\omega_{z}}\\\
\end{bmatrix}=J*\begin{bmatrix}\dot{\theta_{0}}\\\ \dot{\theta_{1}}\\\
\dot{\theta_{2}}\\\ \dot{\theta_{3}}\\\ \dot{\theta_{4}}\\\
\dot{\theta_{5}}\\\ \dot{\theta_{6}}\\\ \end{bmatrix}$ (1)

where, J is the velocity Jacobian and $\dot{\theta}$ represents the joint
velocities and the left hand side is the end-effector velocities.

Now we can compute the Kinetic and Potential energies of the system from the
joint velocities obtained and the parameters of the Fetch robot provided in
their documentation. We can compute the Lagrangian as L = K - P, where K is
the Kinetic Energy and P is the potential energy. The torque can be computed
from the Lagrangian as shown in equation 2,

$\tau=\derivative{t}\partialderivative{L}{\dot{\theta}}-\partialderivative{L}{\theta}$
(2)

From the torque equation, we can compute the Inertia (M), Coriolis (C) and
Gravity (G) matrices and the state space form of the system is written. The
torque represented in the form of Inertia, Coriolis and Gravity matrix is
shown in equation 3,

$M(q)\ddot{q}+C(q,\dot{q})\dot{q}+g(q)=\tau$ (3)

Figure 4 shows the joints on the Fetch robot using which we obtain the
transformation from the robot base to the robot end-effector.

Figure 4: Joint Diagram of the Fetch Robot

###  III-B Trajectory Planning of manipulator

We perform point to point motion for each of the seven joints of the
manipulator. In our case, we have constraints on position, velocity and
acceleration due to the fact that discontinuities in acceleration can lead to
impulsive jerk ( derivative of acceleration), which may excite vibrational
modes in the manipulator and reduce tracking accuracy. Therefore, we add
constraints on the acceleration as well as on the position and velocity thus
giving us a quintic polynomial trajectory planning. So, in our case, we have
six constraints (one each for initial and final configurations, initial and
final velocities, and initial and final accelerations). Therefore we require a
fifth order polynomial as given in the equation below,

$q(t)=a_{0}+a_{1}t+a_{2}t^{2}+a_{3}t^{3}+a_{4}t^{4}+a_{5}t^{5}$ (4)

where, $\tau$ is the time of execution of the trajectory and
$a_{0},a_{1},a_{2},a_{3},a_{4},a_{5}$ are the coefficients of the quintic
polynomial.

The problem here is to find a trajectory that connects an initial to a final
configuration while satisfying other specified constraints at the endpoints
(e.g., velocity and/or acceleration constraints). Without loss of generality,
we will plan the trajectory for a single joint, since the trajectories for the
remaining joints will be created independently and in exactly the same way.
Thus, we will concern ourselves with the problem of determining q(t), where
q(t) is a scalar joint variable. The equations for the initial and final
position, velocities and accelerations are given in equations 5, 6, 7, 8, 9
and 10. The velocities and the accelerations are obtained by differentiating
the joint positions once and twice respectively with respect to time.

$q_{0}=a_{0}+a_{1}t_{0}+a_{2}t_{0}^{2}+a_{3}t_{0}^{3}+a_{4}t_{0}^{4}+a_{5}t_{0}^{5}$
(5) $v_{0}=a_{1}+2a_{2}t_{0}+3a_{3}t_{0}^{2}+4a_{4}t_{0}^{3}+5a_{5}t_{0}^{4}$
(6) $\alpha_{0}=2a_{2}+6a_{3}t_{0}+12a_{4}t_{0}^{2}+20a_{5}t_{0}^{3}$ (7)
$q_{f}=a_{0}+a_{1}t_{f}+a_{2}t_{f}^{2}+a_{3}t_{f}^{3}+a_{4}t_{f}^{4}+a_{5}t_{f}^{5}$
(8) $v_{f}=a_{1}+2a_{2}t_{f}+3a_{3}t_{f}^{2}+4a_{4}t_{f}^{3}+5a_{5}t_{f}^{4}$
(9) $\alpha_{f}=2a_{2}+6a_{3}t_{f}+12a_{4}t_{f}^{2}+20a_{5}t_{f}^{3}$ (10)

This can be represented in matrix form as follows and therefore if we know the
time of planning for each joint, we are able to find the the coefficients of
the polynomial which can then be substituted in the above equations and the
position, velocity and acceleration for each joint can be figured out.

$\begin{bmatrix}q_{0}\\\ v_{0}\\\ \alpha_{0}\\\ q_{f}\\\ v_{f}\\\
\alpha_{f}\\\
\end{bmatrix}=\begin{bmatrix}1&t_{0}&t_{0}^{2}&t_{0}^{3}&t_{0}^{4}&t_{0}^{5}\\\
0&1&t_{0}&3t_{0}^{2}&4t_{0}^{3}&5t_{0}^{4}\\\
0&0&1&6t_{0}&12t_{0}^{2}&20t_{0}^{3}\\\
1&t_{f}&t_{f}^{2}&t_{f}^{3}&t_{f}^{4}&t_{f}^{5}\\\
0&1&t_{f}&3t_{f}^{2}&4t_{f}^{3}&5t_{f}^{4}\\\
0&0&1&6t_{f}&12t_{f}^{2}&20t_{f}^{3}\\\ \end{bmatrix}\begin{bmatrix}a_{0}\\\
a_{1}\\\ a_{2}\\\ a_{3}\\\ a_{4}\\\ a_{5}\\\ \end{bmatrix}$ (11)

Figure 5: Block Diagram of trajectory planner

Figure 5 shows the high level control used for the case of trajectory planning
and control of the manipulator. We get the high level trajectory plan from the
quintic polynomial trajectory for each joint and then a PID controller for
each joint make the joint follow the trajectory generated by the quintic
polynomial. The PID control law is given by equation 12. This torque is given
as feedback to the control law as defined in equation 3.

$\tau=K_{p}\tilde{q}+K_{v}\dot{\tilde{q}}+K_{i}\int_{0}^{t}\tilde{q}(\sigma)d\sigma$
(12)

where, $K_{p},K_{v}$ and $K_{i}$ are the position, velocity and integral gains
respectively and $\tilde{q}$ is the error term.

###  III-C Trajectory Planning of Mobile Base

The trajectory planning for the mobile base is performed with the A*
algorithm. A* is a search algorithm that is widely used in path finding and
graph traversal. Peter Hart, Nils Nilsson and Bertram Raphael first described
the algorithm in 1968. As A* traverses the graph, it follows a path of the
lowest known cost, keeping a sorted priority queue of alternate path segment
along the way. If, at any point, a segment of the path being traversed has a
higher cost than another encountered path segment, it abandons the higher-cost
path segment and traverses the lower-cost path segment instead. This process
continues until the goal is reached [16]. It utilizes a heuristic to focus the
search towards the most promising areas of the search space. A* aims to find
an optimal path which may not be feasible given time constraints and the
dimensionality of the problem. The priority queue is based on f = g + h, where
g is the backward cost and h is the forward cost to goal or called the
heuristic. For the A* algorithm to give an optimal solution in the case of the
tree search, the heuristic should be admissible (i.e. it should never
overestimate the true cost to the goal). It is the user who has to provide the
heuristic which is admissible.

The path returned by the A* planner is then time-parametrized before being
passed onto the controllers which takes the path returned from the planner and
the actual path followed by the robot. The error term for the controller in
this case is the euclidean distance between the time-parametrized path from
the A* planner and the actual path followed by the mobile base.

###  III-D Inverse Kinematics

We make use of the Trac_ik package available in the ROS framework to perform
Inverse Kinematics of the manipulator arm. Trac_ik is an enhancement of the
KDL’s (Kinematics and Dynamics Library) pseudoinverse Jacobian solver which
improves the performance drastically. Some of the issues with the KDL slover
which are addressed, the KDL Inverse Kinematics takes as input a maximum
number of iterations to try which mainly depends on the time to compute the
Jacobian. Thus, there is no need to give the user the criteria to choose the
number of iterations which can be decided by our solver where the user is only
prompted to provide the maximum time allowed for solving. This is functionally
equivalent to running KDL with a number of iterations, but the maximum number
of iterations is dynamically determined based on the user-desired solve time.

The Inverse Jacobian IK implementations can get stuck in local minima during
the iteration process. This is valid in cases where joints have physical
limits on their range. In the KDL implementation, there is nothing to detect
local minima or mitigate this scenario. In this solver, the local minima are
detected and mitigated by changing the next seed. Consequently, the
performance of the KDL IK solver can be significantly improved by simply using
random seeds for q to improve the performance of the iterative algorithm when
local minima are detected. But this method does not handle the constraints of
joint limits very well.

One way to avoid these failures is to solve IK using methods that better
handle constraints like joint limits. IK as a nonlinear optimization problem
can be solved locally using sequential quadratic programming (SQP). SQP is an
iterative algorithm for nonlinear optimization, which can be applied to
problems with objective functions and constraints that are twice continuously
differentiable. SQP minimizes the overall amount of joint movement and only
considers Cartesian error as a constraint. The equation for the SQP problem in
our case is given in equation 13,

$\begin{split}\operatorname*{arg\,min}_{q\in
R^{n}}\quad(q_{seed}-q)^{T}\;(q_{seed}-q)\\\ s.t.\quad f_{i}(q)\leq
b_{i},i=1,...,m\end{split}$ (13)

where $q_{seed}$ is the n-dimensional seed value of the joints, and the
inequality constraints $f_{i}(q)$ are the joint limits, the Euclidean distance
error, and the angular distance error.

SQP method uses dual quaternion error as a combined measure of distance error
and angular error in Cartesian space. Generating the dual quaternion error
takes non-trivial computational time compared to simpler metrics such as the
sum of squares metric. The SQP-SS (Sum of Squares)used the sum of squares
metric when the sum of the squares of the error are considered. The Sum of
Squares equation is given in equation 14,

$\phi_{SS}=p_{err}*p_{err}^{T}$ (14)

Therefore, for a single IK solver request Trac_ik spawns two solvers, one
running SQP-SS and the other running KDL. Once either finishes with a
solution, both threads are immediately stopped, and the resulting solution is
returned. This is the same approach that we are following in our project and
the results for the IK solver are discussed in a later section.

###  III-E State Machine

A finite state machine (FSM) is a mathematical model of computation. It is an
abstract machine that can be in exactly one of a finite number of states at
any given time. The FSM can change from one state to another in response to
some external inputs. the change from one state to another is called a
transition. A FSM is defined by a list of its states, its initial state, and
the conditions for each transition. We have defined a state machine for the
pick and place task using the Fetch robot and it can be defined in Figure 6.
First, we check the robot status which gives us the information of the
location of the base and the location of the end-effector. Various states call
different parts of our system, thus making the execution of the overall task
possible. The transition for every state is provided by a feedback mechanism
as defined in the ROS framework.

Figure 6: State Machine for the pick and place task

###  III-F Implementation Platform

The platform that we use to test our system is the Fetch robot. The Fetch
kinematics are defined by the robot URDF (unified robot description format)
model which specifies the attributes of the joints, links, and frames of the
robot. A link element in the URDF describes a rigid body with inertia, visual
features, and coordinate frames. A joint element in the URDF defines the
kinematics, dynamics, safety limits, and type. We entered these information
based on the DH parameters and the system dynamics derived earlier. Then we
define the direction of the coordinate frames for the Fetch robot.

The coordinate frames for all links in the Fetch and Freight are defined with
positive z-axis up, positive x-axis forward, and positive y-axis to the robot-
left when Fetch is in the home pose. Joint position and velocity limits are
enforced for all the joints in the Fetch robot. This acts the constraints
during trajectory planning and finding inverse kinematic solutions. We make
use of various sensors available on the Fetch robot to perform the trajectory
control. The sensors which are useful in our project are described below.

The Fetch robot has the following sensors which is of use in our system, a
SICK TIM571 scanning range finder, a 6-axis inertial measurement unit (IMU),
Primesense short-range RGBD sensor and a Gripper sensor. The laser has a range
of 25m, 220° field of view, 15Hz update rate and angular resolution of 1/3°.
The laser publishes both distance and RSSI (Received Signal Strength
Indication) to the base_scan topic, The gyroscope within the IMU is capable of
measuring +/-2000 degrees per second, while the accelerometers are capable of
measuring +/-2g. In addition to the position and effort feedback of the
gripper joint, the gripper incorporates a 6-axis inertial measurement unit
(IMU).

We performed the entire task in ROS/Gazebo with the Fetch robot in a
simulation environment. The results obtained after performing our experiments
are explained in Section IV.

##  IV Results

In this project, the first task of deriving the system dynamics began with
formulating the DH parameters for the manipulator of fetch robot as discussed
in the methodology. The DH parameters derived for each of the consecutive
links are shown in figure 7. In figure 7 ’D’ stand for dummy frame. The
Derived DH parameters were validated by performing forward kinematics in
MATLAB.

Figure 7: DH Parameters

The task of creating the simulation world with Fetch robot and the environment
was done in gazebo. The world environment was formed by placing models of the
shelf and object in gazebo world. We created a world with the shelf and the
object and also spawned the fetch robot to this world by importing it’s URDF
file in the simulation environment. Figure 3 & 4 shows screen shot of the
gazebo world with the Fetch robot, Shelf and the object.

Figure 8: Simulation World Figure 9: Visualization of the path found by A* for
an 8-connected Euclidean heuristic is shown in black

After this step, when the base of the robot moves we do inverse kinematics
with the method explained in the previous section. We used the open source
package Trac_ik solver in the ROS framework. Trac_ik performed very well
compared to other classical solvers such as Orocos KDL solver etc in our case.
Trac_ik took an average time of 0.44ms to find an Inverse Kinematic solution
when run for 100 trials against the Orocos KDL solver which took 0.72 ms for
the same 100 trials. The success rate of the Trac_ik solver was also higher
when compared to the Orocos KDL solver.

Next, we did trajectory planning for both the manipulator in the fetch robot
as well as the mobile base. Trajectory planning for the manipulator was done
in joint space considering the joint limit constraints.This trajectory
planning was done in joint space so that there is no need to perform inverse
kinematics operation at each instant of planning which can be costly. This
saves considerable amount of time during planning. Moreover, inverse
kinematics can give multiple solutions and obtaining the correct set of joint
angles will be time consuming. Planning in joint space rather than Cartesian
space saved us time. The position, velocity and acceleration profiles obtained
for one of the joints is shown in figures 10 and 11. Similar profiles were
obtained for all the joints. We performed quintic polynomial planning because
when we don’t consider the acceleration of the joint, it leads to a lot of
jerkiness in the system. The next part was the trajectory planning for the
mobile base of the Fetch robot.

Figure 10: Position for each joint after Trajectory planning

Figure 11: Velocity and Acceleration profile for each joint after Trajectory
Planning

The trajectory planning for the mobile base was done using A* path planning
algorithm as discussed in methodology. In order to visualize the working of
the A* algorithm we tested our algorithm first on a PR2 robot in a simulation
environment known as OpenRAVE. The A* algorithm was written as plugin in the
global planner library available in ROS. The path was generated by the A*
planning algorithm for four connected space as well as eight connected space.
We used the manhattan and euclidean heuristic for both of these scenarios and
tested the performance in OpenRAVE. In case of eight connected we observed
euclidean heuristic to be superior. Number of nodes explored in case of
Manhattan heuristic was 3462. Number of nodes explored in case of Euclidean
heuristic was 4555. Cost in case of Manhattan was 126.70 and the cost in case
of Euclidean was 104.83 for the eight connected space. Therefore, we used
eight connected Euclidean heuristic and performed the planning. The path
generated by A* planning algorithm for an eight connected space with euclidean
distance heuristic is shown in figure 9.

Now, in order to move the base of the fetch robot, we published the
parameterized path found by the A* algorithm on a topic to which the base
subscribed. Before doing this, we found the transformation between the base
frame and the odometry frame. This was found using the tf package available in
ROS where our tf class had a subscriber which waits for the transform between
the odometry and the base frame and records it. Then the geometry_msgs of type
Twist was published on the cmd_vel topic to move the robot base. This method
was used to move the robot base to the desired location according to the

The PID controller is implemeted for both the trajectory planning of the
manipulator and the mobile base. After getting the trajectory for each joint
from the planner, we use a PID controller to ensure that each joint follows
the trajectory generated by the planners. In order to make the system stable
we tune the PID controller by modifying the controller gains. The trajectory
which is generated for the mobile base is time parameterized and given to the
PID controller as input. The PID controller minimizes the euclidean distance
between the time-parameterized path from the A* planner and the actual path
followed by the mobile base. The joint trajectories, Velocity profile and
joint efforts of Shoulder Pan joint, Shoulder Lift joint, Upperarm Roll joint,
Elbow Flex joint and Forearm Roll joint are shown in figures 12, 13 and 14
respectively.

Figure 12: Joint Trajectories of each joint Figure 13: Velocity Profile of
each joint Figure 14: Joint efforts for each joint Figure 15: tf Tree Figure
16: rqt-graph

After the planning step, we implemented a state machine in C++ for the pick
and place task on the fetch robot. Different states are defined in the state
machine and it changes from one state to another when it gets an transition
request in the form of a ROS message after the previous state has been
completed. The State Machine was used to automate the entire process of the
pick and place task. All the tasks which was mentioned above is included in a
ROS node named as fetch_traj_node. When wrote this finite state machine as a
ROS node using the roscpp client library and when we run this node, all the
tasks such as trajectory planning of manipulator, trajectory planning of
mobile base, inverse kinematics etc. are executed sequentially according to
the flowchart explained earlier.

Final task involved controlling the gripping action i.e like controlling the
amount by which the gripper closes to grab an object on the shelf. We used the
gripper_controller or gripper_action interfaces to send gripper commands using
the control_msgs topic. The gripper command takes in position and effort as
parameters. Generally, the gripper is commanded to a fully closed or fully
opened position, so effort is used to limit the maximum effort. As the gripper
never fully reaches the closed position, the grasp strength will be determined
by the maximum effort. So, in order to pick up objects we need to control the
effort generated by the gripper in the fetch robot.

The tf tree and rqt graph showing the transformation from one frame to another
as well as the list of active nodes and their subscribers and publisher topic
are shown in figures 15 and 16 respectively.

##  V Conclusion and Future Work

In this project, we have implemented Trajectory Control for the Fetch robot
which is a differential drive mobile manipulator robot in a simulated
environment. We have shown results and implementation of the dynamics of the
fetch robot, Trajectory planning for the manipulator arm using a quintic
spline and mobile base trajectory planning has been implemented with the A*
planning algorithm and the results have been shown in Section IV. The state
machine for the task of indoor warehouse object manipulator has been carried
out and results of it’s implementation has also been discussed. Also, the
Trac_ik package has been used for performing the inverse kinematic
calculations for the manipulator of the robot. Overall, the project goals
stated in Section I have been implemented and the corresponding methodology
shown in Section III and the results of our experiments carried out discussed
in Section IV.

In the future, we would develop a global planner which would just get the
location of an object in an environment and then autonomously determine the
amount at which the base should be moved and then the end-effector. Right now,
we don’t have this feature as we plan for the mobile base and the manipulator
separately. Also, we would like to make our trajectory control package open
source since there is no robust package that achieves this efficiently. But,
more work is needed to make it generic for all robots. We would implement our
system on an actual hardware platform in an indoor warehouse environment to
test it’s performance. We have preformed the picking of the object but we
would want to incorporate computer vision techniques to make the state machine
more powerful. One more extension of our work would be to make all the
different aspects modular so that it can be ported to any platform easily.

##  VI Acknowledgements

We would like to thank Professor Eugene Eberbach of the department of Robotics
Engineering at Worcester Polytechnic Institute for his patience, guidance,
advice and support over the term of the project, and Junius Santoso for
providing us useful inputs facilitating the successful completion of our
project.

## References

  * [1] Correll, N., Bekris, K.E., Berenson, D., Brock, O., Causo, A., Hauser, K., Okada, K., Rodriguez, A., Romano, J.M. and Wurman, P.R., 2016. Lessons from the amazon picking challenge. arXiv preprint arXiv:1601.05484
  * [2] Papadopoulos, E. and Poulakakis, J., 2000, July. Trajectory planning and control for mobile manipulator systems. In The 8th IEEE Mediterranean Conf. on Control & Automation.
  * [3] Tchon, Krzysztof and Janusz Jakubiak. “An Extended Jacobian Inverse Kinematics Algorithm for Doubly Nonholonomic Mobile Manipulators.” ICRA (2005).
  * [4] Mazur, A. and Arent, K., 2006. Trajectory tracking control for nonholonomic mobile manipulators. In Robot Motion and Control (pp. 55-71). Springer London.
  * [5] Solea, R. and Cernega, D., 2011. Trajectory Tracking Control of Mobile Manipulators based on Kinematics. In ICINCO (2) (pp. 21-27).
  * [6] Ide, S., Takubo, T., Ohara, K., Mae, Y. and Arai, T., 2011, November. Real-time trajectory planning for mobile manipulator using model predictive control with constraints. In Ubiquitous Robots and Ambient Intelligence (URAI), 2011 8th International Conference on (pp. 244-249). IEEE.
  * [7] Dong, W., Xu, Y. and Wang, Q., 2000. On tracking control of mobile manipulators. In Robotics and Automation, 2000. Proceedings. ICRA’00. IEEE International Conference on (Vol. 4, pp. 3455-3460). IEEE.
  * [8] Korayem, M.H. and Nohooji, H.R., 2008, October. Trajectory optimization of flexible mobile manipulators using open-loop optimal control method. In International Conference on Intelligent Robotics and Applications (pp. 54-63). Springer Berlin Heidelberg.
  * [9] Korayem, M.H., Nazemizadeh, M. and Azimirad, V., 2011. Optimal trajectory planning of wheeled mobile manipulators in cluttered environments using potential functions. Scientia Iranica, 18(5), pp.1138-1147.
  * [10] Yu, Q. and Chen, I.M., 2002. A general approach to the dynamics of nonholonomic mobile manipulator systems. TRANSACTIONS-AMERICAN SOCIETY OF MECHANICAL ENGINEERS JOURNAL OF DYNAMIC SYSTEMS MEASUREMENT AND CONTROL, 124(4), pp.512-521.
  * [11] Beeson, P. and Ames, B., 2015, November. TRAC-IK: An open-source library for improved solving of generic inverse kinematics. In Humanoid Robots (Humanoids), 2015 IEEE-RAS 15th International Conference on (pp. 928-935). IEEE.
  * [12] S.R. Buss, Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse and Damped Least Squares methods. Survey, University of California, 2004.
  * [13] P. Beeson, https://bitbucket.org/traclabs/tracik.git, accessed: 2016-07-22.
  * [14] ROS Framework, http://www.ros.org, accessed: 2016-07-22.
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# Betti numbers and compactible almost complex structures on the Hard
Lefschetz condition

Dexie Lin

###### Abstract

A special class of symplectic manifolds is those satisfying the Hard Lefschetz
condition. The algebraic property of $\mathfrak{sl}_{2}(\mathbb{C})$-module
gives a natural constraint on the Betti numbers. However, such a constraint is
only necessary, and it is not so clear what condition on the Betti numbers is
equivalent to the Hard Lefschetz condition. In this paper, we give some
relations between the Betti numbers and the variant Hodge numbers on the Hard
Lefschetz condition. In general, we give a sufficient condition on Betti
numbers and variant Hodge numbers for any almost Kähler manifold satisfying
the Hard Lefschetz condition. In particular, we offer an equivalent condition
on the first Betti number for compact symplectic $4$-manifolds satisfying the
Hard Lefschetz condition.

Keywords: symplectic manifolds, Hard Lefschetz condition, Dolbeault
cohomology, Betti numbers

2010 Math. Subject Class. 32A25, 53C56, 53D35

##  1 Introduction

On a compact Kähler manifold $(M,J,\omega,g)$ with real dimension $2n$, it is
known that the cup products

$[\omega^{k}\wedge]:H^{n-k}(M;\mathbb{R})\to
H^{n+k}(M;\mathbb{R}),~{}k\in\\{1,\ldots,n\\}$ (1.1)

are isomorphic, c.f., [10, Section 6-Chapter 0]. A general compact symplectic
manifold is said to satisfy the Hard Lefschetz condition, provided that the
above cup products are isomorphisms. The Hard Lefschetz condition is a bridge
between the Kähler manifolds and the symplectic manifolds. The isomorphisms in
(1.1) provide a $\mathfrak{sl}_{2}(\mathbb{C})$-module structure on the deRham
cohomology algebra of $(M,\omega)$. By the algebraic property of
$\mathfrak{sl}_{2}(\mathbb{C})$-module, Thurston [16] constructed a compact
symplectic $4$ manifold without Kähler structure by violating the Hard
Lefschetz condition. Thurston’s work showed that the topology of symplectic
manifolds is quite different to Kähler manifold, e.g., $b_{1}$ can be odd.
Such a result disproved Guggenheimer’s conjecture [11]. For more symplectic
manifolds with non-Kähler structure, one can refer [18].

To study the symplectic manifold, the Hard Lefschetz condition also plays an
important role.

  * •

Mathiew [14] and Yan [19] showed that the Hard Lefschetz condition is
equivalent to establish the symplectic Hodge theory, i.e., each deRham
cohomology class contains a symplectic harmonic form. Recall that the notion
of symplectic Hodge decomposition was given by Ehresmann and Libermann [7] and
reintroduced by Brylinski [2]. Yan’s work disproved Brylinski’s conjecture.

  * •

Merkulov [15, Proposition 1.4] showed that the Hard Lefschetz condition is
equivalent to the so called $dd^{\Lambda}$-lemma, see Theorem 2.6 for more
details. Later, Tomassini and Wang [17, Theorem B] generalized this result to
complex manifolds with $J$-symmetric symplectic structures.

  * •

Gompf and Mrowka [8] constructed symplectic $4$ manifolds satisfying the Hard
Lefschetz condition but without a Kähler structure.

A compact symplectic manifolds is called a Lefschetz manifold, if it satisfies
the Hard Lefschetz condition. Summarizing the above results, one has the
following of inclusions:

$\mbox{K\"{a}hler maifolds}\subsetneq\mbox{Lefschetz
manifolds}\subsetneq\mbox{Symplectic manifolds}.$

Hence, it raises a natural question for studying compact symplectic manifolds:

######  Question.

What kind of compact symplectic manifolds satisfies Hard Lefschetz condition?

de Bartolomeis and Tomassini [1] gave a sufficient condition in terms of the
first non-zero egienvalues of Hodge Laplacian operators for compact almost
Kähler manifolds satisfying the Hard Lefschetz condition. Note that an almost
Kähler manifold can be regarded as a symplectic manifold equipped with a
metric.

In this paper, we study the Hard Lefschetz condition in terms of Betti numbers
and almost complex structures. In fact, the algebraic structure of
$\mathfrak{sl}_{2}(\mathbb{C})$-representation gives the natural constraints
of the Betti numbers of Lefschetz manifolds:

$b_{2k+1}\mbox{ are even},b_{k}\leq b_{k+2},~{}b_{2k}>0,\mbox{ for }0\leq
k<n-1,$ (1.2)

However, one can construct a compact symplectic $4$ manifold with an even
first Betti number but violating the Hard Lefschetz condition, see Example
3.1. Moreover, Gompf [9, Theorem 7.1] showed that

######  .

For a given generated group $G$ and a non-negative integer $a$, there is a
compact symplectic $2n(\geq 6)$-dimensional manifolds, whose fundamental group
equals to $G$ and $b_{i}\geq a$ for $2\leq i\leq n-2$ but violating the Hard
Lefschetz condition.

Thus, (1.2) is only necessary for a symplectic manifold to become a Lefschetz
one. It is natural to expect a sufficient or an equivalent condition in terms
of Betti numbers for compact symplectic manifolds satisfying the Hard
Lefschetz condition. By observation of (1.2), a rational guess is that such a
condition should be related to Betti numbers and the symplectic form. Since
each symplectic manifold can be equipped with an almost Kähler structure, we
equivalently consider the relation(s) between the Betti numbers and the
compatible almost complex structure on Lefschetz manifolds. The main results
of this paper are as follows.

######  Theorem 1.1.

Let $(M,g,J,\omega)$ be a compact almost Kähler $2n$-dimensional manifold.
Suppose that the equalities

$2\sum_{0\leq p\leq k}\ell^{p,2k+1-p}=b^{2k+1},~{}2\sum_{0\leq
p<k}\ell^{p,2k-p}+\ell^{k,k}=b^{2k}.$ (1.3)

hold for all $k\leq n$. Then, $(M,\omega)$ satisfies the Hard Lefschetz
condition.

Here, $\ell^{p,q}$’s are the generalized Hodge number on compact almost Kähler
manifolds, which are defined by Cirici and Wilson in [4]. One can refer to
Section 3 for their definitions. In particular, we have an equivalent
condition on the first Betti number for a compact symplectic $4$-manifold to
satisfy the Hard Lefschetz condition.

######  Theorem 1.2.

Let $(M,\omega)$ be a compact symplectic $4$ manifold. Then, $(M,\omega)$
satisfies the Hard Lefschetz condition if and only if
$b_{1}=2\tilde{h}^{1,0}_{J}=\ell^{1,0}$ for any compactible almost complex
structure $J$, where $\tilde{h}^{1,0}$ is the refined Hodge number.

Such a result can be used to calculate the Dolbeault cohomology on symplectic
$4$ manifolds modeled on Lie groups, see Example 3.1. Furthermore, for
dimension $\geq 6$ and $b_{1}\neq 0$, the Hard Lefschetz condition implies
either the symplectic manifold is Kählerian or for each compactible almost
complex structure its associated Nijenhuis tensor can not take the maximal
rank at any point, see Proposition 3.6.

The structure of this paper is as follows: In Section 2.1, we review the basic
notions and properties of the Hard Lefschetz condition; In Section 2.2, we
review the Dolbeault cohomology on almost complex manifold and almost Kähler
manifolds and the related properties; In Section 3, we show some technique
lemmas and give the proofs to the above theorems.

######  Acknowledgement.

The author would thank Teng Huang for introducing this topic and valuable
discussion, and Adriano Tomassini for suggesting considering Example 3.1. This
work is partially supported by National Natural Science Foundation of China
No. 12301061 and Fundamental Research Fundsfor the Central Universities No.
2023CDJXY-043.

##  2  Hard Lefschetz condition and Almost complex manifolds

In this section, we review some necessary and related notions and definitions
on the Hard Lefschetz condition.

###  2.1 Hard Lefschetz condition on linear spaces

In this subsection, we give a quick review about the Hard Lefschetz condition
on linear symplectic vector space and some properties on the toy model of
finite dimensional spaces, c.f., [17, Section 2 and 3]. First, we review some
basic notions about the symplectic space.

######  Definition 2.1.

Let $V$ be a $2n$-dimensional real vector space. A bilinear form $\omega$ on
$V$ is called symplectic, if $\omega$ is skew-symmetric and non-degenerate.

Similar to the metric case, the symplectic form $\omega$ defines a star
operator $*_{s}:\wedge^{p}V^{*}\rightarrow\wedge^{2n-p}V^{*}$ by

$\mu\wedge*_{s}\nu=\omega^{-1}(\mu,\nu)\frac{\omega^{n}}{n!},$

where $\omega^{-1}$ is the associated symplectic form on $V^{*}$. Set
$L:\Lambda V^{*}\to\Lambda V^{*}$ as $\alpha\mapsto\omega\wedge\alpha$ and
$\Lambda:=*_{s}^{-1}L*_{s}$.

It is not hard to see,

$\omega^{-1}(Lu,v)=\omega^{-1}(u,\Lambda v),$

i.e., $\Lambda$ is the adjoint of $L$. For the convenience of the later
arguments, we set $L^{r}:=\frac{\omega^{r}}{r!}$.

######  Definition 2.2.

A linear map $J:V\rightarrow V$ is called an almost complex structure on $V$
if $J^{2}=-Id$. An almost complex structure $J$ is said to be tamed by
$\omega$ if

$\omega(u,Ju)>0,$

for each non-zero $u\in V$. We say $J$ is $w$-compatible if it is both taming
and symmetric, i.e., $\omega(u,Jv)=\omega(v,Ju),$ for each $u,v\in V$.

An almost complex structure $J$ induces the decomposition

$V_{\mathbb{C}}:=V\otimes{\mathbb{C}}=V^{1,0}\oplus V^{0,1},$

where $V^{1,0}$ and $V^{0,1}$ are the eigenspaces of $J$ corresponding to
eigenvalues $i$ and $-i$ respectively. This $J$ action can be extended to any
$p$-form $\psi$ on by

$(J\psi)\left(v_{1},\cdots,v_{p}\right)=(-1)^{p}\psi\left(Jv_{1},\cdots,Jv_{p}\right).$

Similarly, we decompose
$V^{*}_{\mathbb{C}}:=V^{*}M\otimes_{\mathbb{R}}\mathbb{C}$ as
$V^{*}_{\mathbb{C}}=V^{*,(1,0)}_{\mathbb{C}}\oplus V^{*,(0,1)}_{\mathbb{C}}$.
Let $\bigwedge^{p,q}V^{*}$ denote the space of forms of
$\bigwedge^{p,q}V^{*}:=\bigwedge^{p}V^{*,(1,0)}_{\mathbb{C}}\otimes\bigwedge^{q}V^{*,(0,1)}_{\mathbb{C}}$.
An almost complex structure $J$ on $V$ compatible with $\omega$ defines an
Hermitian inner product structure on $\bigwedge^{p,q}V^{*},0\leq p,q\leq n$.
The associated Hodge star operator
$*:\bigwedge^{p,q}V^{*}\rightarrow\wedge^{n-q,n-p}V^{*}$ is defined by

$u\wedge*\bar{v}=(u,v)\omega^{n}/n!.$

The following identity gives a relation between $*$ and $*_{s}$,

$*=*_{s}\mathop{\scalebox{0.6}{$\circ$}}J=J\mathop{\scalebox{0.6}{$\circ$}}*_{s}.$

Now we give the definition of Lefschetz space and
$\mathfrak{sl}_{2}(\mathbb{C})$-module, more details can be found in
[[Section6-Chapter0]GH]

######  Definition 2.3.

Let $A=\bigoplus_{k=0}^{2n}A^{k}$ be a direct sum of complex vector spaces
with possibly infinite dimensions. We say that $L\in\operatorname{End}(A)$
satisfies the Hard Lefschetz Condition and $(A,L)$ is a Lefschetz space if

$L\left(A^{l}\right)\subset
A^{l+2},~{}L\left(A^{2n-1}\right)=L\left(A^{2n}\right)=0,$

for $0\leq l\leq 2(n-1)$ and each $L^{k}:A^{n-k}\rightarrow A^{n+k},0\leq
k\leq n$ is an isomorphism.

We review the definitions of Lefschetz space and
$\mathfrak{sl}_{2}(\mathbb{C})$-triple.

######  Definition 2.4.

Set $\Lambda=*_{s}^{-1}L*_{*},B:=[L,\Lambda]$. $(L,\Lambda,B)$ is called an
$\mathfrak{sl}_{2}(\mathbb{C})$-triple on $(A,L)$.

The general Kähler identity implies the following result.

######  Theorem 2.5 (c.f. [12]).

Let $(A,L)$ be a Lefschetz space. Let $d$ be a $\mathbb{C}$-linear
endomorphism of $A$ such that $d\left(A^{l}\right)\subset A^{l+1}$ and
$d^{\Lambda}:=(-1)^{l+1}*_{s}d*_{s}:A^{l}\to A^{l-1}$. If $[d,L]=0$, then

  * (1)

$(\ker\left.d\cap\operatorname{ker}d^{\Lambda},L\right)$ and
$(\ker\left.d\cap\operatorname{ker}d^{\Lambda},\Lambda\right)$ are Lefschetz.

  * (2)

$\left(\operatorname{im}d+\operatorname{im}d^{\Lambda},L\right)$ and
$\left(\operatorname{im}d+\operatorname{im}d^{\Lambda},\Lambda\right)$ are
Lefschetz.

  * (3)

Assume further that $d^{2}=0$. Then,
$\left(\operatorname{im}dd^{\Lambda},L\right)$ and
$\left(\operatorname{im}dd^{\Lambda},\Lambda\right)$ are Lefschetz.

Let $(A,L)$ be a Lefschetz space. Let $d$ be a C-linear endomorphism of $A$
such that $d\left(A^{l}\right)\subset A^{l+1}$. We call $(A,L,d)$ a Lefschetz
complex if $d^{2}=0$. Set

$H_{d}=\bigoplus_{k=0}^{2n}H_{d}^{k},\quad
H_{d}^{k}:=\frac{\operatorname{ker}d\cap A^{k}}{\operatorname{Im}d\cap
A^{k}},\quad H_{d^{\Lambda}}=\bigoplus_{k=0}^{2n}H_{d^{\Lambda}}^{k},\quad
H_{d^{\Lambda}}^{k}:=\frac{\operatorname{ker}d^{\Lambda}\cap
A^{k}}{\operatorname{Im}d^{\Lambda}\cap A^{k}}.$

One has the following theorem.

######  Theorem 2.6 (c.f. [17, Theorem 3.5]).

Let $(A,L,d)$ be a Lefschetz complex. Assume that $[d,L]=0$. Then the
followings are equivalent:

  * (1)

$(A,L,d)$ satisfies the $dd^{\Lambda}$-Lemma.

  * (2)

The natural map $\ker d\cap\ker d^{\Lambda}\rightarrow H_{d}$ is surjective
and the $\mathfrak{sl}_{2}(\mathbb{C})$-triple $(L,\Lambda,B)$ on $\ker
d\cap\ker d^{\Lambda}$ induces the $\mathfrak{sl}_{2}(\mathbb{C})$-triple on
$\mathrm{H}_{d}$.

  * (3)

$\left(H_{d},L\right)$ satisfies the Hand Lefschetz condition.

  * (4)

$\left(H_{d^{\Lambda}},\Lambda\right)$ satisfies the Hard Lefschetz condition.

Here a Lefschetz complex $(A,L,d)$ satisfies the $dd^{\Lambda}$-Lemma if

$\ker d\cap\ker
d^{\Lambda}\cap\left(\operatorname{im}d+\operatorname{im}d^{\Lambda}\right)=\operatorname{im}dd^{\Lambda},$

on each $A^{k},0\leq k\leq 2n$.

For a compact symplectic manifold $(M,\omega)$, setting $A=\mathcal{A}^{*}$,
$L=\omega\wedge$ and $d$=deRahm differential, we say $(M,\omega)$ satisfies
the Hard Lefschetz condition or $(M,\omega)$ is a Lefschetz manifold, if it
satisfies one of the equivalent statements in Theorem 2.6.

###  2.2 Cohomology groups on almost complex manifolds

In this subsection, we review the Dolbeault cohomology on almost complex
manifolds and generalized Hodge number on almost Kähler manifolds. We first
review some basic definitions and notions.

######  Definition 2.7.

For a smooth manifold $M$ of real dimension $2n$. If there exists a smooth
section $J\in\Gamma(End(TM))$ on $TM$ satisfying $J^{2}=-1$, then we call
$(M,J)$ an almost complex manifold with almost complex structure $J$.

Similar to the decompositions on linear toy model $V$ of the previous
subsection, the parallel decompositions on
$T_{\mathbb{C}}M=TM\otimes_{\mathbb{R}}\mathbb{C}$ and
$T^{*}_{\mathbb{C}}M:=T^{*}M\otimes_{\mathbb{R}}\mathbb{C}$ also hold.
Denoting $\mathcal{A}^{*}(M)$ by the space of sections of
$\bigwedge^{*}T^{*}_{\mathbb{C}}M$, it gives that:

$\mathcal{A}^{r}(M)=\bigoplus_{p+q=r}\mathcal{A}^{p,q}(M),$

where $\mathcal{A}^{p,q}(M)$ denotes the space of forms of
$\bigwedge^{p,q}T^{*}_{\mathbb{C}}M:=\bigwedge^{p}T^{*,(1,0)}_{\mathbb{C}}M\otimes\bigwedge^{q}T^{*,(0,1)}_{\mathbb{C}}M$.
By the formula

$d\psi\in\mathcal{A}^{p+2,q-1}(M)+\mathcal{A}^{p+1,q}(M)+\mathcal{A}^{p,q+1}(M)+\mathcal{A}^{p-1,q+2}(M),$

for any $\psi\in\mathcal{A}^{p,q}(M)$, the deRham differential can be
decomposed as follows:

$d=\mu+\partial+\bar{\mu}+\bar{\partial}\mbox{ on }\mathcal{A}^{p,q},$

where each component is a derivation. The bi-degrees of the four components
are given by

$|\mu|=(2,-1),~{}|\partial|=(1,0),~{}|\bar{\partial}|=(0,1),~{}|\bar{\mu}|=(-1,2).$

Observe that the operators $\partial,~{}\bar{\partial}$ are of the first order
and the operators $\mu,~{}\bar{\mu}$ are of the zero order. Similar to the
case of Riemannian manifolds, we introduce the definitions of almost Hermitian
manifold and almost Kähler manifold.

######  Definition 2.8.

A metric $g$ on an almost complex manifold $(M,J)$ is called Hermitian, if $g$
is $J$-invariant, i.e., $g(J-,J-)=g(-,-)$. The imaginary component $\omega$ of
$g$ defined by $\omega=g(J-,-)$, is a real non-degenerate $(1,1)$-form. The
quadruple $(M,J,g,\omega)$ is called an almost Hermitian manifold. Moreover,
when $\omega$ is $d$-closed, the quadruple $(M,J,g,\omega)$ is called an
almost Kähler manifold.

Recall that by doing polar decomposition, one can equip an almost Kähler
structure on any symplectic manifold. The Hermitian metric $g$ induces a
unique $\mathbb{C}$-linear Hodge-$*$ operator

$*:\Lambda^{p,q}\rightarrow\Lambda^{n-q,n-p},$

$\mbox{defined by
}g\left(\phi_{1},\phi_{2}\right)dV=\phi_{1}\wedge*\overline{\phi_{2}}$, where
$dV$ is the volume form $\frac{\omega^{n}}{n!}$ and
$\phi_{1},\phi_{2}\in\Lambda^{p,q}$. For any two forms $\varphi_{1}\mbox{ and
}\varphi_{2}$, we define their product by,

$(\varphi_{1},\varphi_{2})=\int_{M}\varphi_{1}\wedge*\overline{\varphi_{2}}.$

Let $\delta^{*}$ denote the formal-$L^{2}$-adjoint of $\delta$ for
$\delta\in\\{\mu,\partial,\bar{\partial},\bar{\mu},d\\}$ with respect to the
metric $g$. By the integration by part, one has the identities

$\delta^{*}=-*\bar{\delta}*\mbox{ for
}\delta\in\\{\mu,\partial,\bar{\partial},\bar{\mu},d\\}.$

By the identity
$\Delta_{\bar{\partial}}+\Delta_{\mu}=\Delta_{\partial}+\Delta_{\bar{\mu}}$ on
almost Kähler manifold, c.f. [1, Lemma 3.6], one has the following theorem.

######  Theorem 2.9 (Cirici and Wilson [4, Theorem 4.1]).

On any compact almost Kähler manifold with dimension $2n$, it holds

$\mathcal{H}_{d}^{p,q}=\mathcal{H}_{\bar{\partial}}^{p,q}\cap\mathcal{H}_{\mu}^{p,q}=\mathcal{H}_{\partial}^{p,q}\cap\mathcal{H}_{\bar{\mu}}^{p,q}=\ker(\Delta_{\bar{\partial}}+\Delta_{\mu}),$

for all $(p,q)$. Moreover, we also have the following identities:

  * $(1)$

(Complex conjugation)

$\mathcal{H}_{\bar{\partial}}^{p,q}\cap\mathcal{H}_{\mu}^{p,q}=\mathcal{H}_{\bar{\partial}}^{q,p}\cap\mathcal{H}_{{\mu}}^{q,p}.$

  * $(2)$

(Hodge duality)

$*:\mathcal{H}_{\bar{\partial}}^{p,q}\cap\mathcal{H}_{\mu}^{p,q}\rightarrow\mathcal{H}_{\bar{\partial}}^{n-q,n-p}\cap\mathcal{H}_{\mu}^{n-q,n-p}.$

  * $(3)$

(Serre duality)

$\mathcal{H}_{\bar{\partial}}^{p,q}\cap\mathcal{H}_{\mu}^{p,q}\cong\mathcal{H}_{\bar{\partial}}^{n-p,n-q}\cap\mathcal{H}_{\mu}^{n-p,n-q}.$

For the convenience of later arguments in the next section, we set
$\ell^{p,q}=\dim_{\mathbb{C}}\mathcal{H}^{p,q}_{\bar{\partial}}\cap\mathcal{H}^{p,q}_{\mu}$.
One has the following constrains on Betti numbers for compact almost Kähler
manifolds.

######  Theorem 2.10 (Cirici and Wilson [4, Theorem 4.3]).

For any compact almost Kähler manifold of dimension $2n$, the following are
satisfied:

  * (1)

In odd degrees, it establishes

$\sum_{p+q=2k+1}\ell^{p,q}=2\sum_{0\leq p\leq k}\ell^{p,2k+1-p}\leq b^{2k+1}.$

  * (2)

In even degrees, it holds that

$\sum_{p+q=2k}\ell^{p,q}=2\sum_{0\leq p<k}\ell^{p,2k-p}+\ell^{k,k}\leq
b^{2k},$

with $\ell^{k,k}\geq 1$ for all $k\leq n$.

On the other hand, expanding the relation $d^{2}=0$ implies the following
identities:

$\bar{\mu}\bar{\partial}+\bar{\partial}\bar{\mu}=0,~{}\bar{\mu}^{2}=0,\mbox{
and }\bar{\partial}^{2}|_{\ker(\bar{\mu})}\equiv 0(\bmod~{}im(\bar{\mu})).$

These identities induce a Dolbeault-type cohomology.

######  Definition 2.11 (Cirici and Wilson [5, Definition 3.1]).

The Dolbeault cohomology of an almost complex $2n$-dimensional manifold
$(M,J)$ is given by

$H^{p,q}_{Dol}=H^{q}(H^{p,*}_{\bar{\mu}},\bar{\partial})=\frac{\ker(\bar{\partial}:H^{p,q}_{\bar{\mu}}\to
H^{p,q+1}_{\bar{\mu}})}{im(\bar{\partial}:H^{p,q-1}_{\bar{\mu}}\to
H^{p,q}_{\bar{\mu}})},$

where
$H^{p,q}_{\bar{\mu}}=\frac{\ker(\bar{\mu}:\mathcal{A}^{p,q}\to\mathcal{A}^{p-1,q+2})}{im(\bar{\mu}:\mathcal{A}^{p+1,q-2}\to\mathcal{A}^{p,q})}$.

Cirici and Wilson gave the definition of the Hodge-type filtration.

On the other hand, the author gave another Dolbeault cohomology on almost
complex manifolds.

######  Definition 2.12 (c.f. [13, Definition 3.8]).

On an almost complex manifold $(M,J)$, we define the subspace
$\mathcal{A}^{*,*}_{Dol}$ of $\mathcal{A}^{*,*}$ by

$\mathcal{A}^{*,*}_{Dol}=\ker(\mu)\cap\ker(\bar{\mu})\cap\ker((\bar{\partial}+\mu)\bar{\partial})\cap\mathcal{A}^{*,*}.$

The refined Dolbeault cohomology $\tilde{H}^{*,*}_{Dol}$ is defined by

$\tilde{H}^{*,*}_{Dol}=\frac{\ker(\bar{\partial})\cap\mathcal{A}^{*,*}_{Dol}}{\bar{\partial}(\mathcal{A}^{*,*-1}_{Dol})}.$

By observation, the author showed $\ell^{1,0}=h^{1,0}=\tilde{h}^{1,0}$ on
compact almost Kähler $4$ manifold and
$\tilde{h}^{p,0}=h^{p,0}\leq\ell^{p,0}$, for $0\leq p\leq n$ on general
compact almost Hermitian $2n$-manifolds, c.f., [13, Proposition 3.5 and Lemma
4.11]. Moreover, on compact almost complex $4$ manifolds with non-trivial
almost complex structure, it holds that $\tilde{h}^{0,1}<h^{0,1}=\infty$, c.f.
[13, Proposition 3.9] and [3]. We end this section by showing that the Hodge
number $h^{1,0}$ is equal to $\ell^{1,0}$ on any compact almost Kähler
manifold.

######  Lemma 2.13.

Let $(M,g,J,\omega)$ be a compact almost Kähler manifold. Then, it holds that

$h^{1,0}=\tilde{h}^{1,0}=\ell^{1,0}.$

Proof By the identities
$\Delta_{\bar{\partial}}+\Delta_{\mu}=\Delta_{\partial}+\Delta_{\bar{\mu}}$
and $\Delta_{\mu}\mid_{\mathcal{A}^{1,0}}=0$, any $u\in\tilde{H}^{1,0}$
implies
$u\in\ker(\Delta_{\partial}+\Delta_{\bar{\mu}})=\ker(\Delta_{\partial})\cap\ker(\Delta_{\bar{\mu}})$.

Conversely, any
$u\in\mathcal{H}^{1,0}_{\bar{\partial},\mu}=\mathcal{H}^{1,0}_{\partial,\bar{\mu}}$
induces $u\in\ker(\bar{\partial})\cap\ker(\bar{\mu})$ by the same arguments.

##  3 Proofs

In this section, we show the proofs to the main theorems. First, we review
some basic properties of the Laplacian operator. Since $\Delta_{d}$ is an
elliptic operator, it is well-known that $\Delta_{d}$ is Fredholm. Hence,
$\Delta_{d}$ admits a Green-function $G$ on a compact manifold. Before moving
on, we give two elementary lemmas.

######  Lemma 3.1.

Let $(M,g)$ be a compact Riemannian manifold. Under the above notations, it
holds that

$[G,*]=0,~{}[G,d]=0,~{}[G,d^{*}].$

Proof By the decomposition $\mathcal{A}^{k}=\mathcal{H}^{k}_{d}\oplus
im(\Delta_{d})$, it suffices to check the three formulas for the forms in
$\mathcal{H}^{k}_{d}$ and $im(\Delta_{d})$ separately. Here, we just show the
first one, since the other two are similar. Recall that for any
$\phi\in\mathcal{H}^{k}_{d}$, its Hodge dual $*\phi$ is a harmonic
$(\dim_{\mathbb{R}}M-k)$-form, thus

$G*\phi=*G\phi=0.$

For a $k$-form $\psi=\Delta_{d}\alpha$ with $\alpha\perp\mathcal{H}^{k}_{d}$,
it holds that

$\displaystyle G*\psi$ $\displaystyle=$ $\displaystyle G*\Delta_{d}\alpha$
$\displaystyle=$ $\displaystyle G\Delta_{d}*\alpha$ $\displaystyle=$
$\displaystyle*\alpha=*G\Delta_{d}\alpha=*G\psi,$

where we use the identity $[\Delta_{d},*]=0$ on the second equality. We find a
form $\alpha\in(\mathcal{H}^{p,q}_{\bar{\partial}})^{\perp}$ such that
$\Delta_{\bar{\partial}}\alpha=\phi$. One has that

$G\bar{\partial}\phi=G\bar{\partial}(\bar{\partial}\bar{\partial}^{*}+\bar{\partial}^{*}\bar{\partial})\alpha=G\bar{\partial}^{2}\bar{\partial}^{*}\alpha+G\bar{\partial}\bar{\partial}^{*}\bar{\partial}\alpha=G(\bar{\partial}^{2}\bar{\partial}^{*}-\bar{\partial}^{*}\bar{\partial}^{2})\alpha+\bar{\partial}\alpha=\bar{\partial}G\phi-G(\bar{\partial}^{*}\bar{\partial}^{2}f)$
$\displaystyle\Delta_{d}|_{\mathcal{A}^{0,1}}=(\partial+\bar{\partial})(\bar{\partial}^{*})+(\bar{\partial}^{*}+\partial^{*}+\mu^{*}+\bar{\mu}^{*})(\mu+\bar{\partial}+\partial)$
$\displaystyle=\bar{\partial}\bar{\partial}^{*}+\partial\bar{\partial}^{*}+\bar{\partial}^{*}\bar{\partial}+\bar{\partial}^{*}\partial+\partial^{*}\mu+\partial^{*}\partial+\mu^{*}\mu+\bar{\mu}^{*}\bar{\partial}$
$\displaystyle=\Delta_{\bar{\partial}}+\Delta_{\mu}+\Delta_{\partial}+(\partial\bar{\partial}^{*}+\bar{\partial}^{*}\partial+\bar{\mu}^{*}\bar{\partial})$

######  Lemma 3.2.

Let $(V,\omega)$ be a $2n$-dimensional symplectic vector space, and $J$ be a
compactible (almost)complex structure on $(V,\omega)$. Then, for any
$\alpha\in V^{*}$, the identity

$*(J\alpha)=\alpha\wedge\frac{\omega^{n-1}}{(n-1)!},$

where $*$ is the Hodge operator with respect to the metric $\omega(-,J-)$.

Proof By the linear model of the previous section, we can find a unitary basis
$\\{e_{1},e_{2},\cdots e_{2n}\\}$ of $V$ such that

$\omega=e^{1}\wedge e^{2}+e^{3}\wedge e^{4}+\cdots+e^{2n-1}\wedge e^{2n},$

and $J(e_{2i-1})=e_{2i},~{}J(e_{2i})=-e_{2i-1}$, where $e^{i}$ is dual to
$e_{i}$ for each $1\leq i\leq n$. It suffices to check the identity of the
cases $\alpha=e^{2j-1}$ and $\alpha=e^{2j}$. The formulas
$*J(e^{2j-1})=*e^{2j}=-e^{1}\wedge\cdots\widehat{e^{2j}}\wedge
e^{2n}=e^{2j-1}\wedge\frac{\omega^{n-1}}{(n-1)!}$ and
$*J(e^{2j})=-*e^{2j-1}=e^{1}\wedge\cdots\widehat{e^{2j-1}}\wedge
e^{2n}=e^{2j}\wedge\frac{\omega^{n-1}}{(n-1)!}$ yield the identity of the
lemma.

Recall that $L^{p}=\frac{\omega^{p}}{p!}:H^{n-p}(M;\mathbb{R})\to
H^{n+p}(M;\mathbb{R})$. Before showing Theorem 1.1 and 1.2, we also need the
following proposition.

######  Proposition 3.3.

Let $(M,g,J,\omega)$ be a $2n$-dimensional compact almost Kähler manifold.
Then the equality $b_{1}=2\tilde{h}^{1,0}=2\ell^{1,0}$ is equivalent to the
isomorphism of the map

$L^{n-1}:H^{1}(M;\mathbb{R})\to H^{2n-1}(M;\mathbb{R}).$ (3.1)

Proof Recall that in general, only the inequality $b_{1}\geq 2h^{1,0}$ holds.
The equality$b_{1}=2h^{1,0}$ holds if and only for each $\Delta_{d}$-harmonic
$1$-form $\alpha$ can be written as $\alpha=u+\bar{u}$ for a unique
$u\in\ker(\Delta_{d})\cap\mathcal{A}^{1,0}$. It suffices to verity that for
each $\Delta_{d}$-harmonic $1$-form $\alpha$, the $2n-1$-form
$L^{n-1}\alpha=\alpha\wedge\frac{\omega^{n-1}}{(n-1)!}$ is also
$\Delta_{d}$-harmonic. Lemma 3.2 yields

$d^{*}(L^{n-1}\alpha)=*d(J\alpha)=*id(u-\bar{u})=0.$

Conversely, assume that the isomorphism (3.1) holds. Recall that any real-
valued harmonic $1$-form $\alpha$ can be uniquely written as
$\alpha=u+\bar{u}$ with $u\in\mathcal{A}^{1,0}$. Moreover, such $u$ belongs to
$\mathcal{H}^{1,0}_{\bar{\partial},\mu}=\mathcal{H}^{1,0}_{d}$ if and only if
$J\alpha$ is also $\Delta_{d}$-harmonic. We prove the equality by
contradiction. If the equality $b_{1}=2h^{1,0}$ does not hold, then by
applying Lemma 3.4 below to
$\mathcal{A}^{1}_{\mathbb{R}}=\mathcal{H}^{1}_{d}\oplus(\mathcal{H}^{1}_{d})^{\perp}$,
there is a $\Delta_{d}$-harmonic $1$-form still denoted by $\alpha$, such that
$\alpha=u+\bar{u}$, with

$J\alpha=iu-i\bar{u}\perp\ker(\Delta_{d})\cap\mathcal{A}^{1},$

and the harmonic part of $L^{n-1}\alpha$ is nonzero. The Hodge-decomposition
shows

$L^{n-1}\alpha=(L^{n-1}\alpha)_{H}+G\Delta_{d}(L^{n-1}\alpha).$

Here, $(L^{n-1}\alpha)_{H}$ denotes the harmonic part of $L^{n-1}\alpha$,
which is nonzero by the isomorphism of (3.1). On the other hand, we calculate

$\displaystyle G\Delta_{d}(L^{n-1}\alpha)$ $\displaystyle=$ $\displaystyle
G\Delta_{d}(*J\alpha)$ $\displaystyle=$ $\displaystyle*G\Delta_{d}(J\alpha)$
$\displaystyle=$ $\displaystyle*G\Delta_{d}(iu-i\bar{u})=*(iu-i\bar{u}),$

where we use Lemma 3.1 for the second equality and
$(u-\bar{u})\perp\mathcal{H}^{1}_{d}$ for the last one. Hence, the above
formula gives

$\displaystyle\|\alpha\|^{2}_{L^{2}}=\|J\alpha\|^{2}_{L^{2}}=\|*J\alpha\|^{2}_{L^{2}}=\|L^{n-1}\alpha\|^{2}_{L^{2}}=\|G\Delta_{d}(L^{n-1}\alpha)\|^{2}_{L^{2}}.$

It implies $(L^{n-1}\alpha)_{H}=0$, contradicting to the assumption on $u$.
Therefore, we proved the proposition.

######  Lemma 3.4.

Let $V$ be a possibly infinite dimensional real vector space with a complex
structure $J$ and a $J$-compactible metric $g$. Suppose that there is a
decomposition $V=V_{1}\oplus V_{2}$ with $V_{1}\perp V_{2}$ with respect to
$g$ and $\dim V_{1}<\infty$. Then, $J$ preserves $V_{1}$, if and only if there
is no non-trivial subspace $V^{\prime}_{1}\subset V_{1}$ such that
$J(V^{\prime}_{1})\subset V_{2}$.

Proof When $J$ preserves $V_{1}$, it is clear that there is no non-zero
element in $V_{1}$ which maps onto $V_{2}$. Here, we only show the “if” part,
by using the proof of contradiction.

Assume that $J$ does not preserve $V_{1}$. Then, the intersection
$W:=J(V_{1})\cap V_{1}$ is a proper subspace of $V_{1}$. Since $V_{1}$ is of
finite dimension, one can find an orthonormal basis $\\{e_{j}\\}_{1\leq
j\leq\dim V_{1}}$ of $V_{1}$ and a maximal integer $n$ such that

$J(e_{2i-1})=e_{2i},~{}J(e_{2i})=-e_{2i-1},\mbox{ for }1\leq i\leq n,\mbox{
and }W=\mbox{span}\\{e_{j}\\}_{1\leq j\leq 2n}$

and

$J(e_{s})\not\in V_{1},\mbox{for }2n<s\leq\dim V_{1}.$

If $J(e_{s})$ has a non-zero $V_{1}$ component, then $J(e_{s})\cap V_{1}(\neq
0)\subset W$. This violates the choice of $n$. Thus,
$V^{\prime}_{1}:=\mbox{span}\\{e_{s}\\}_{2n<s\leq N}$ is a proper subspace of
$V_{1}$ such that $J(V^{\prime}_{1})\subset V_{2}$, which is a contradiction.

Now, we give a sufficient condition for compact symplectic manifolds
satisfying the Hard Lefschetz condition in terms of Betti numbers and
$\ell^{p,q}$s.

Proof of Theorem 1.1 Since the two equalities hold, each $2k+1$-degree real
harmonic form $\alpha_{2k+1}$ with $0\leq k<n-1$ can be expressed as

$\alpha_{2k+1}=\sum_{p=0}^{k}u_{p,2k+1-p}+\bar{u}_{p,2k+1-p},$

where
$u_{p,2k+1-p}\in\mathcal{H}^{p,2k+1}_{\bar{\partial},\mu}=\mathcal{H}^{p,2k+1-p}_{d}$,
and each $2k$-degree real harmonic form $\alpha_{2k}$ can be expressed as

$\alpha_{2k}=\sum_{0\leq p\leq k}u_{p,2k-p}+\bar{u}_{p,2k-p}+u_{2k}$

By the formula $[d^{*},L]=d^{c}$ in [1, Formula (15)], where $d^{c}=J^{-1}dJ$,
it yields that

$\displaystyle dd^{*}(L^{n-k-1}\alpha_{2k+1})$ $\displaystyle=$
$\displaystyle(n-k-1)(L^{n-k-2}dd^{c}\alpha_{2k+1})$ $\displaystyle=$
$\displaystyle(n-k-1)\frac{\omega^{n-k-2}}{(n-k-2)!}dJ^{-1}d\sum_{0\leq p\leq
k}(i^{2p-2k-1}u_{p,2k+1-p}-i^{2p-2k-1}\bar{u}_{p,2k+1-p})$ $\displaystyle=$
$\displaystyle 0.$

Thus, the term $L^{n-2k-1}\alpha$ is a real harmonic form, and the map

$L^{n-k-1}:H^{2k+1}_{dR}(M;\mathbb{R})\to H^{2n-2k-1}_{dR}(M;\mathbb{R}),$

is an isomorphism. By the similar arguments, the map

$L^{n-k}:H^{2k}_{dR}(M;\mathbb{R})\to H^{2n-2k}_{dR}(M;\mathbb{R}),$

is also an isomorphism for each $k$.

Proof of Theorem 1.2 For any compactible almost complex structure $J$,
Proposition 3.3 and Lemma 2.13 show that the Hard Lefschetz condition is
equivalent to the relation $b_{1}=2\tilde{h}^{1,0}=2\ell^{1,0}$, which
finishes the proof.

The following example shows that the even condition on $b_{1}$ is not
sufficient to make compact symplectic manifolds satisfy the Hard Lefschetz
condition.

######  Example 3.1.

Let $N$ be a nil-potent simple connected $4$ dimensional Lie group, whose non-
trivial structure equations on its associated Lie algebra
$\mathfrak{g}=\mbox{span}(e_{1},e_{2},e_{3},e_{4})$ are given by

$[e_{1},e_{2}]=e_{3},~{}[e_{1},e_{3}]=e_{4}.$

Then, on the dual algebra $\mathfrak{g}^{*}$, it holds that

$de^{1}=0,de^{2}=0,de^{3}=-e^{1}\wedge e^{2},de^{4}=-e^{1}\wedge e^{3}.$

Recall that the Lie group $N$ admits a lattice $\Gamma$. Let
$M=\Gamma\backslash N$ and the form $\omega=e^{1}\wedge e^{4}+e^{2}\wedge
e^{3}$ be a symplectic structure. It is clear that $b_{1}(M)=4$. But
$(M,\omega)$ does not satisfy the Hard Lefschetz condition, since

$[\omega\wedge e^{1}]=[e^{1}\wedge e^{2}\wedge e^{3}]=[d(e^{4}\wedge
e^{2})]=0.$

Moreover, we also have $\tilde{h}^{1,0}=1$.

We end this paper by showing the following proposition for higher dimensional
symplectic satisfying the Hard Lefschetz condition. First, we give a technique
lemma.

######  Lemma 3.5.

Let $(M,J)$ be a compact almost complex $2n$-manifold $n\geq 3$. If the
Nijenhuis tensor takes the maximal rank at some point of $M$, then
$H^{1,0}_{Dol}=\tilde{H}^{1,0}_{Dol}=\\{0\\}$. Moreover, if the Nijenhuis
tensor is nowhere vanishing, then
$H^{1,0}_{Dol}=\tilde{H}^{1,0}_{Dol}=\tilde{H}^{0,1}_{Dol}=\\{0\\}$.

Proof An easy calculus yields

$rank_{\mathbb{C}}\mathcal{A}^{1,0}=rank_{\mathbb{C}}\mathcal{A}^{0,1}=n,~{}rank_{\mathbb{C}}\mathcal{A}^{2,0}=rank_{\mathbb{C}}\mathcal{A}^{0,2}=\frac{n(n-1)}{2}.$

For an Hermitian metric $g$, it is clear that $\bar{\partial}^{*}u=0$ and
$u\in\ker(\bar{\partial})$ is equivalent to
$u\in\ker(\Delta_{\bar{\partial}})$ for any $u\in\mathcal{A}^{1,0}$. Hence,
all elements in $\tilde{H}^{1,0}$ are $\Delta_{\bar{\partial}}$-harmonic.
Since the maximal rank condition is open, there exists a small open subset $U$
such that the restriction $N|_{U}$ is also of the maximal rank. When $n\geq
3$, it is clear that
$\bar{\mu}|_{U}:\mathcal{A}^{1,0}(U)\to\mathcal{A}^{0,2}(U)$ is either
isomorphic($n=3$) or injective($n>3$) by the algebraic calculus. The equation
$\bar{\mu}(u)=0$ gives that $u|_{U}=0$ for any $u\in\mathcal{A}^{1,0}$. Hence,
the unique-continuity-property for harmonic forms shows that $u=0$ for any
$u\in\tilde{H}^{1,0}_{Dol}$.

Similarly, when $N$ is nowhere vanishing,
$\mu:\mathcal{A}^{0,1}\to\mathcal{A}^{2,0}$ is either isomorphic($n=3$) or
injective($n>3$). Therefore, $\mu(v)=0$ implies $v=0$ for any
$v\in\mathcal{A}^{0,1}$, which proves the last statement.

######  Proposition 3.6.

Let $(M,\omega)$ be a compact symplectic manifold with dimension $\geq 6$. If
$b_{1}\neq 0$ and satisfies the Hard Lefschetz condition, then either
$(M,\omega)$ is Kählerian or for each compactible almost complex structure its
associated Nijenhuis tensor can not take the maximal rank at any point.

Proof We argue by contradiction. Assume that $(M,\omega)$ is non-Kählerian.
For any $\omega$-compactible almost complex structure $J$,
$b_{1}=2\tilde{h}^{1,0}$ by Proposition 3.3.

If the associated Nijenhuis tensor $N$ takes maximal rank at a point $p$, then
there exists a small neighborhood $U_{p}$ such that $N\mid_{U_{p}}$ is of
maximal rank by the open condition of the maximal rank. Again, by the relation
$\bar{\mu}|_{U_{p}}:\mathcal{A}^{1,0}(U)\to\mathcal{A}^{0,2}(U)$ is either
isomorphic($n=3$) or injective($n>3$), any $u\in\tilde{H}^{1,0}_{Dol}$ is zero
inside $U_{p}$. Thus, $u\equiv 0$ by the unique continuity property, which
contradicts to the assumption $b_{1}\neq 0$.

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College of Mathematics and Statistics, Chongqing University, Huxi Campus,
Chongqing, 401331, P. R. China

Chongqing Key Laboratory of Analytic Mathematics and Applications, Chongqing
University, Huxi Campus, Chongqing, 401331, P. R. China

E-mail<EMAIL_ADDRESS>