The positive energy of Hawking radiation, detected by the observer at infinity, implies that the the black hole loses the corresponding energy, that's all. This loss can be described as the contribution of particles of negative energy that are absorbed by the black hole. 

Since the event horizon is a light-like surface, ``near the event horizon'' (or ``far from the horizon'') doesn't make sense. What's meant by the ``near horizon geometry'' is a specific spacetime geometry. 

Gravity isn't repulsive or attractive depending on the sign of the energy, since energy isn't a good observable in curved spacetime. Gravity is always attractive, because the graviton has even spin. 

In the Hawking radiation model described by you, in order to preserve the total energy, it is assumed that the particle that falls into the black hole must have had a negative energy. This causes the black hole to lose the mass. Gravity is always attractive.

In another model, negative particle energy is not assumed but virtual particle pairs are created near event horizon which are boosted by black hole's gravitational energy into becoming real particles. Therefore escape of one of the particles lowers the mass of the black hole.

In a third model involving quantum tunnelling, one of the particle tunnel outside the event horizon.

 I’m sorry I can’t speak English well.

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Now, I would expect that gravity is repulsive for particles of positive and negative energy, therefore the anti-particle with negative energy should be repelled by the black hole instead of falling in.

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Your deduction is wrong because of that inertial mass of negative mass is negative. 

  O ----------------- O

-m1                  +m2

Negative mass & Positive mass : Negative mass is accelerated in the direction of positive mass, and positive mass is accelerated in the direction to be far away from negative mass.

The direction of acceleration a1 worked on negative mass –m1 is –r^, so -m1 moves in the direction of reducing distance r, and the direction of acceleration a2 worked on positive mass +m2 is +r^, so positive mass +m2 is accelerated in the direction that distance r increases, namely the direction of being far away from negative mass.

If the absolute value of positive mass is bigger than that of negative mass, they will meet within finite time(attractive effect), and if the absolute value of positive mass is smaller than that of negative mass, the distance between them will be bigger, and they cannot meet(repulsive effect). On earth, a ball with negative mass fall to the earth.

The type of force(-G(-m1)m2/r^2=+Gm1m2/r^2) is repulsion, so the potential energy has + value.

However, I do not agree with hawking radiation mechanism.^^

1.Computer simulation video for negative mass’s movement: https://youtu.be/MZtS7cBMIc4

2.Paper

https://www.researchgate.net/publication/275056453_Pair_Creation_Model_of_the_Universe_From_Positive_and_Negative_Energy

Research Pair Creation Model of the Universe From Positive and Negative Energy

Negative or positive particles can fall in.  Either way we have negative energy

By convention when a particle pair is produced out of vacuum due to quantum fluctuations, one of them has a negative energy and the counterpart has positive energy, such that if you sum energies up, you get zero. Note that it is a virtual pair (not a pair production process for which one needs to cross an energy threshold of 2m)  of quantum fluctuation which exists only for a very very tiny amount of time.

It is not the negatively charged  particle which falls inside the event horizon; but  it is the negative energy particle which falls into the event horizon. The positive energy counter part may or may not escape, it depends whether on it has a sufficient amount of positive energy which will let it escape. If it does escape then Hawking radiation is formed.

When the particle anti-particle annihilate, the sum of their energy is not zero but twice the value of one of the particle. In relativity theories we have rest mass and the relativistic mass. My conclusion is that the rest mass is always positive but the relativistic mass can have positive or negative energy. This is because when the particle anti-particle pair is created, they fly away from each other in opposite direction so they have opposite momentum and opposite direction for their velocity vectors. This is consistent with the fact that the relativistic mass of the particle is due to the relative motion between two particles. This is discussed in detail at eqns. (1) and (2) in following quantum gravity theory and at eqn.(2.35) in the quarkonium article.

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