  We present the galaxy shear catalog that will be used for the three-year
cosmological weak gravitational lensing analyses using data from the Wide layer
of the Hyper Suprime-Cam (HSC) Subaru Strategic Program (SSP) Survey. The
galaxy shapes are measured from the $i$-band imaging data acquired from 2014 to
2019 and calibrated with image simulations that resemble the observing
conditions of the survey based on training galaxy images from the Hubble Space
Telescope in the COSMOS region. The catalog covers an area of 433.48 deg$^2$ of
the northern sky, split into six fields. The mean $i$-band seeing is 0.59
arcsec. With conservative galaxy selection criteria (e.g., $i$-band magnitude
brighter than 24.5), the observed raw galaxy number density is 22.9
arcmin$^{-2}$, and the effective galaxy number density is 19.9 arcmin$^{-2}$.
The calibration removes the galaxy property-dependent shear estimation bias to
a level: $|\delta m|<9\times 10^{-3}$. The bias residual $\delta m$ shows no
dependence on redshift in the range $0<z\leq 3$. We define the requirements for
cosmological weak lensing science for this shear catalog, and quantify
potential systematics in the catalog using a series of internal null tests for
systematics related to point-spread function modelling and shear estimation. A
variety of the null tests are statistically consistent with zero or within
requirements, but (i) there is evidence for PSF model shape residual
correlations; and (ii) star-galaxy shape correlations reveal additive
systematics. Both effects become significant on $>1$ degree scales and will
require mitigation during the inference of cosmological parameters using cosmic
shear measurements.
