Measurements of $C P$ observables in $B^0 \to DK^{*0}$ decays are presented, where $D$ represents a superposition of $D^0$ and $\bar{D}^0$ states. The $D$ meson is reconstructed in the twobody final states $K^+\pi^$, $\pi^+ K^$, $K^+K^$ and $\pi^+\pi^$, and, for the first time, in the fourbody final states $K^+\pi^\pi^+\pi^$, $\pi^+ K^\pi^+\pi^$ and $\pi^+\pi^\pi^+\pi^$. The analysis uses a sample of neutral $B$ mesons produced in protonproton collisions, corresponding to an integrated luminosity of 1.0, 2.0 and 1.8 $\rm fb^{1}$ collected with the LHCb detector at centreofmass energies of $\sqrt{s} = $ 7, 8 and 13 TeV, respectively. First observations of the decays $B^0 \to D(\pi^+ K^)K^{*0}$ and $B^0 \to D(\pi^+\pi^\pi^+\pi^)K^{*0}$ are obtained. The measured observables are interpreted in terms of the $C P$violating weak phase $\gamma$.
Feynman diagrams of (left) $ B ^0 \rightarrow D ^0 K ^{*0} $ and (right) $ B ^0 \rightarrow \overline{ D } {}^0 K ^{*0} $. 
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Invariantmass distributions (data points with error bars) and results of the fit (lines and coloured areas) for the twobody GLW modes (top left) $\overline{ B } {}^0 \rightarrow D( K ^+ K ^ )\overline{ K } {}^{*0} $, (top right) $ B ^0 \rightarrow D( K ^+ K ^ ) K ^{*0} $, (bottom left) $\overline{ B } {}^0 \rightarrow D(\pi ^+ \pi ^ )\overline{ K } {}^{*0} $ and (bottom right) $ B ^0 \rightarrow D(\pi ^+ \pi ^ ) K ^{*0} $. 
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Invariantmass distributions (data points with error bars) and results of the fit (lines and coloured areas) for the twobody ADS modes (top left) $\overline{ B } {}^0 \rightarrow D( K ^ \pi ^+ )\overline{ K } {}^{*0} $, (top right) $ B ^0 \rightarrow D( K ^+ \pi ^ ) K ^{*0} $, (bottom left) $\overline{ B } {}^0 \rightarrow D(\pi ^ K ^+ )\overline{ K } {}^{*0} $ and (bottom right) $ B ^0 \rightarrow D(\pi ^+ K ^ ) K ^{*0} $. The bottom distributions are shown on a logarithmic scale. 
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Invariantmass distributions (data points with error bars) and results of the fit (lines and coloured areas) for the fourbody GLW mode (left) $\overline{ B } {}^0 \rightarrow D(\pi ^+ \pi ^ \pi ^+ \pi ^ )\overline{ K } {}^{*0} $, (right) $ B ^0 \rightarrow D(\pi ^+ \pi ^ \pi ^+ \pi ^ ) K ^{*0} $. 
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Invariantmass distributions (data points with error bars) and results of the fit (lines and coloured areas) for the fourbody ADS modes (top left) $\overline{ B } {}^0 \rightarrow D( K ^ \pi ^+ \pi ^ \pi ^+ )\overline{ K } {}^{*0} $, (top right) $ B ^0 \rightarrow D( K ^+ \pi ^ \pi ^+ \pi ^ ) K ^{*0} $, (bottom left) $\overline{ B } {}^0 \rightarrow D(\pi ^ K ^+ \pi ^+ \pi ^ )\overline{ K } {}^{*0} $ and (bottom right) $ B ^0 \rightarrow D(\pi ^+ K ^ \pi ^ \pi ^+ ) K ^{*0} $. The bottom distributions are shown on a logarithmic scale. 
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Contour plots showing 2D scans of (left) $\delta_B^{D K ^{*0} }$ versus $\gamma$ and (right) $\delta_B^{D K ^{*0} }$ versus $r_B^{D K ^{*0} }$. The lines represent the $\Delta\chi^2 =$ 2.30, 6.18 and 11.8 contours, corresponding to 68.6%, 95.5% and 99.7% confidence levels (C.L.), respectively. 
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Animated gif made out of all figures. 
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Summary of signal yields. The uncertainties are statistical. 
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Systematic uncertainties for the observables. Uncertainties are shown if they are larger than 1% of the statistical uncertainty. The total systematic uncertainty is calculated by summing all sources in quadrature. Statistical uncertainties are given for reference. 
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Measured observables split by LHC running period. Observables relating to $ B ^0 \rightarrow D(\pi ^+ \pi ^ \pi ^+ \pi ^ ) K ^{*0} $ decays are not presented for Run 1, as this decay channel was not selected in the Run 1 data. 
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Combined statistical and systematic correlation matrix for the principal observables. 
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Combined statistical and systematic correlation matrix for the principal observables in Run 1 data only. 
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Combined statistical and systematic correlation matrix for the principal observables in Run 2 data only. 
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Correlation matrix for the principal observables between Run 1 and Run 2 data. 
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Supplementary material full pdf 
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Supplementary material full pdf 
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This ZIP file contains supplementary material for the publication LHCbPAPER2019021. The files are: Supplementary.pdf : An overview of the extra figures *.pdf, *.png, *.eps : The figures in various formats 
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Created on 14 January 2021.