  The topology of insulators is usually revealed through the presence of
gapless boundary modes: this is the so-called bulk-boundary correspondence.
However, the many-body wavefunction of a crystalline insulator is endowed with
additional topological properties that do not yield surface spectral features,
but manifest themselves as (fractional) quantized electronic charges localized
at the crystal boundaries. Here, we formulate such bulk-corner correspondence
for the physical relevant case of materials with time-reversal symmetry and
spin-orbit coupling. To so do we develop "partial" real-space invariants that
can be neither expressed in terms of Berry phases nor using symmetry-based
indicators. These new crystalline invariants govern the (fractional) quantized
corner charges both of isolated material structures and of heterostructures
without gapless interface modes. We also show that the partial real-space
invariants are able to detect all time-reversal symmetric topological phases of
the recently discovered fragile type.
