Section IV Summary

Get out the microscope, because we’re going through this poem line-by-line.

The eyes are not here
There are no eyes here
In this valley of dying stars
In this hollow valley
This broken jaw of our lost kingdoms

The Hollow Men are still worried about those eyes. The eyes from heaven are not present, but the lines also suggest that the Hollow Men have no vision.
There is another way to interpret this line. "Eyes" sounds like "Is", as in, "The Is are not here." There are no independent personalities or selves among the group.
Hope continues to fade, as the stars fade or "die" away.
The "valley" leads us to think of one a famous Psalm from the Bible, that goes, "Even though I walk through the valley of the shadow of death, I will fear no evil, for you are with me; your rod and your staff, they comfort me" (Psalm 23).
They are in a valley of death, but there is no one there to comfort them because they never joined with God.
The Hollow Men each used to have their own kingdoms – literally or metaphorically – but these kingdoms have been lost or broken like a jaw. Why a jaw?
We're not sure…maybe you can tell us!
At any rate, here the only true kingdom is the Kingdom of God, and they had their chance to join it but did not.

In this last of meeting places
We grope together
And avoid speech
Gathered on this beach of the tumid river

We finally learn where the Hollow Men are gathered: on the banks of a swollen or "tumid" river.
They are huddled together as if they were going to be washed away. The river is practically overflowing with water, in contrast to the dryness of the men and the desert around them.
This is the last place that they will meet before they face some more terrible fate.
The river most likely represents Acheron, branch of the mythical River Styx in Greece that souls must cross into death.
To make the trip, you would have to pay Charon, the ferryman, a coin to take you on his boat.
Unfortunately, no one has arrived to take these souls across. They are stranded.
There's nothing left to say about their dire situation, so they "avoid speech."
In Canto 3 of Dante's Inferno, Dante asks his guide Virgil why souls are so eager to get across Acheron, and Virgil responds that God's justice "spurs them on" so that they actually want to get to Hell sooner.
But the Hollow Men can't even get to Hell.

Sightless, unless
The eyes reappear
As the perpetual star
Multifoliate rose
Of death's twilight kingdom
The hope only
Of empty men.

The Hollow Men are "sightless," like a bunch of underground worms, but if the "eyes" return their vision could be restored.
Their only hope is if the heavenly eyes come back as a star.
This star would be "perpetual" or eternal, unlike the "fading" or "dying" stars in the desert. By now you've probably noticed that Eliot is throwing around symbols like candy at a Fourth of July parade.
A "multifoliate" rose has many petals. Here again Eliot is referring to – guess who? – Dante Alighieri.
In Dante's Paradiso, the final vision of paradise is of a flower made up of saints, angels, and other examples of goodness and virtue. The community of Heaven is like a rose with petals made of people. Dante also compares Mary, the mother of Jesus, to a rose.
The point of these lines is that the Hollow Men cannot save themselves. They have no hope except for the Heavenly souls to come down and restore their vision of truth and goodness.