Gravity is the result of the curvature of space, so could the curvature actually send objects away from the source?
Apart from the field-theoretical standpoint presented by Stan, one can repel objects in a sense, when taking orbital mechanics into account.
The slingshot maneuver extracts angular momentum and energy of an orbiting mass by the use of gravity. The trick here is that the probe's velocity just gets redirected in the planet's reference frame. But this results in an acceleration in the solar rest-frame that the probe essentially tries to leave.
As you were asking about curvature, I don't know how and if centrifugal potentials that play a role here are described by general relativity.
I'm not sure how accepted this is in the physics community, but a book by Andrew Thomas called Hidden in Plain Sight posits a theory that gravity may be repulsive within black holes (i.e. on scales smaller than the Schwartzchild radius for a given mass).
It's an interesting theory, and it avoids the need for a singularity, but to my knowledge it's completely untestable and therefore not really science.
As far as our current understanding of gravity, no. The common analogy is a rubber sheet with marbles. The sheet can only be pulled downwards, so you cannot have a repelling force.
To pull the sheet upwards would require something like inverse gravity, which is not yet known to current physics.
Of course dark energy seems to look like this, but as the origin of dark energy is currently completely unknown, it is not useful to speculate on this at this point in time.