  We discovered a fractional Chern structure in chiral superconducting
Sr$_2$RuO$_4$ nanofilms by employing electric transport. By using Sr$_2$RuO$_4$
single crystals with nanoscale thickness, a fractional Hall conductance was
observed without an external magnetic field. The Sr$_2$RuO$_4$ nanofilms
enhanced the superconducting transition temperature to about 3 K. We found an
anomalous induced voltage with temperature and thickness dependence, and the
switching behavior of the induced voltage appeared when we applied a magnetic
field. We suggest that there was fractional magnetic-field-induced electric
polarization in the interlayer. These anomalous results are related to
topological invariance. The fractional axion angle $\theta=\pi/6$ is determined
by observing the topological magneto-electric effect in Sr$_2$RuO$_4$
nanofilms.
