  Recent high-resolution and sensitivity ALMA observations have unveiled the
carbon isotope ratios ($^{12}$C/$^{13}$C) of Complex Organic Molecules (COMs)
in a low-mass protostellar source. To understand the $^{12}$C/$^{13}$C ratios
of COMs, we investigated the carbon isotope fractionation of COMs from
prestellar cores to protostellar cores with a gas-grain chemical network model.
We confirmed that the $^{12}$C/$^{13}$C ratios of small molecules are bimodal
in the prestellar phase: CO and species formed from CO (e.g., CH$_{3}$OH) are
slightly enriched in $^{13}$C compared to the local ISM (by $\sim$ 10 $\%$),
while those from C and C$^{+}$ are depleted in $^{13}$C owing to isotope
exchange reactions. COMs are mainly formed on the grain surface and in the hot
gas ($>$ 100 K) in the protostellar phase. The $^{12}$C/$^{13}$C ratios of COMs
depend on which molecules the COMs are formed from. In our base model, some
COMs in the hot gas are depleted in $^{13}$C compared to the observations.
Thus, We additionally incorporate reactions between gaseous atomic C and
H$_{2}$O ice or CO ice on the grain surface to form H$_2$CO ice or \ce{C2O}
ice, as suggested by recent laboratory studies. The direct C-atom addition
reactions open pathways to form \ce{^13C}-enriched COMs from atomic C and CO
ice. We find that these direct C-atom addition reactions mitigate
$^{13}$C-depletion of COMs, and the model with the direct C-atom addition
reactions better reproduces the observations than our base model. We also
discuss the impact of the cosmic ray ionization rate on the $^{12}$C/$^{13}$C
ratio of COMs.
