  We derive some exact results concerning the anomalous U(1)$_A$ symmetry in
the chirally symmetric phase of QCD at high temperature. We discuss the
importance of topology and finite-volume effects on the U(1)$_A$ symmetry
violation characterized by the difference of chiral susceptibilities. In
particular, we present a reliable method to measure the anomaly strength in
lattice simulations with fixed topology. We also derive new spectral sum rules
and a novel Banks-Casher-type relation. Through our spectral analysis we arrive
at a simple alternative proof of the Aoki-Fukaya-Taniguchi "theorem" on the
effective restoration of the U(1)$_A$ symmetry at high temperature.
