Note on Small Black Holes in AdS_p x S^q

It is commonly believed that small black holes in AdS_5 x S^5 can be described by the ten dimensional Schwarzschild solution. This requires that the self-dual five-form (which is nonzero in the background) does not fall through the horizon and cause the black hole to grow. We verify that this is indeed the case: There are static solutions to the five-form field equations in a ten dimensional Schwarzschild spacetime. Similar results hold for other backgrounds AdS_p x S^q of interest in supergravity.