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Using $2.93\,\rm fb^{-1}$ of $e^+e^-$ collision data taken at acenter-of-mass energy of 3.773\,GeV with the BESIII detector, we report thefirst measurements of the absolute branching fractions of fourteen hadronic$D^{0(+)}$ decays to exclusive final states with an $\eta$, e.g., $D^0\toK^-\pi^+\eta$, $K^0_S\pi^0\eta$, $K^+K^-\eta$, $K^0_SK^0_S\eta$,$K^-\pi^+\pi^0\eta$, $K^0_S\pi^+\pi^-\eta$, $K^0_S\pi^0\pi^0\eta$, and$\pi^+\pi^-\pi^0\eta$; $D^+\to K^0_S\pi^+\eta$, $K^0_SK^+\eta$,$K^-\pi^+\pi^+\eta$, $K^0_S\pi^+\pi^0\eta$, $\pi^+\pi^+\pi^-\eta$, and$\pi^+\pi^0\pi^0\eta$. Among these decays, the $D^0\to K^-\pi^+\eta$ and$D^+\to K^0_S\pi^+\eta$ decays have the largest branching fractions, which are$\mathcal{B} (D^0\to K^-\pi^+\eta )=(1.853\pm0.025_{\rm stat}\pm0.031_{\rmsyst})\%$ and $\mathcal{B}(D^+\to K^0_S\pi^+\eta)=(1.309\pm0.037_{\rmstat}\pm0.031_{\rm syst})\%$, respectively. We also determine the $CP$asymmetries for the six decays with highest event yields. No evidence of $CP$violation is found.

Measurements of Absolute Branching Fractions of Fourteen Exclusive Hadronic $D$ Decays to $\eta$