  In this paper, we study the qualitative behavior at a vortex blow-up point
for Chern-Simon-Higgs equation. Roughly speaking, we will establish an energy
identity at a each such point, i.e. the local mass is the sum of the bubbles.
Moreover, we prove that either there is only one bubble which is a singular
bubble or there are more than two bubbles which contains no singular bubble.
Meanwhile, we prove that the energies of these bubbles must satisfy a quadratic
polynomial which can be used to prove the simple blow-up property when the
multiplicity is small. As is well known, for many Liouville type system,
Pohozaev type identity is a quadratic polynomial corresponding to energies
which can be used directly to compute the local mass at a blow-up point. The
difficulty here is that, besides the energy's integration, there is a
additional term in the Pohozaev type identity of Chern-Simon-Higgs equation. We
need some more detailed and delicated analysis to deal with it.
