  We study quantum Hall (QH) states on a punctured Riemann sphere. We compute
the Berry curvature under adiabatic motion in the moduli space in the large N
limit. The Berry curvature is shown to be finite in the large N limit and
controlled by the conformal dimension of the cusp singularity, a local property
of the mean density. Utilizing exact sum rules obtained from a Ward identity,
we show that for the Laughlin wave function, the dimension of a cusp
singularity is given by the central charge, a robust geometric response
coefficient in the QHE. Thus, adiabatic transport of curvature singularities
can be used to determine the central charge of QH states. We also consider the
effects of threaded fluxes and spin-deformed wave functions. Finally, we give a
closed expression for all moments of the mean density in the integer QH state
on a punctured disk.
