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Thanks to @Yosemite Stan, we have a counterexample, and actually many of them: Pick a projective variety $Z$ with some odd-cohomology such that $$c_1(\omega_Z)=-m c_1(H), \text{ for some } m>0,$$ where $H$ is the bundle by which $Z$ embeds to $\mathbb{P}^n,$ and $\omega_Z$ is the canonical bundle.

In other words, we have $c_1(Z)=-c_1(\omega_Z)=m c_1(i^*\mathcal{O}(1)),$ where $i:Z \hookrightarrow \mathbb{P}^n$ is the embedding given by $H.$ In particular, a degree $d$-hypersurface in $\mathbb{P}^n$ having some odd cohomology works, with $m = n+1-d$ (due to adjunction formula).

Denoting by $\widetilde{Z}$ its image via the Veronese map $\mathbb{P}^n\rightarrow\mathbb{P^{\binom{n+m}{m}-1}}$, define $X\subset \mathbb{C}^{\binom{n+m}{m}}$ to be the cone over $\widetilde{Z}$. There is a natural resolution $Y$ of $X$ given by the $Y=\overline{\pi^{-1}(X\setminus 0)},$ where $\pi$ is the blow-up at $\mathcal{0}\subset \mathbb{C}^{\binom{n+m}{m}}.$ As blow-up at $0$ is $\mathcal{O}(-1)\rightarrow \mathbb{P^{\binom{n+m}{m}-1}},$ we have $$Y=\mathcal{O}(-1)|_\widetilde{Z} \cong \mathcal{O}(-m)|_Z,$$ hence it has $$c_1(Y)=c_1(i^*\mathcal{O}(-m))+c_1(Z)=-mc_1(i^*\mathcal{O}(1)+c_1(Z)=0,$$ and cohomology equal to $Z,$ hence not supported in even degrees only.

In particular, choosing the quartic 3-fold $Z_4\subset \mathbb{P}^4$, we have $b_3(Z_4)=60$ and $m=(4+1)-4=1,$ so $X=V(z_0^4+\dots+z_4^4)$ is an isolated singularity and (trivially) a complete intersection as well, and its resolution given by blow up of $\mathbb{C}^5$ at $0$ yields a counterexample.

Thanks to @Yosemite Stan, we have a counterexample, and actually many of them: Pick a projective variety $Z$ with some odd-cohomology such that $$c_1(\omega_Z)=-m c_1(H), \text{ for some } m>0,$$ where $H$ is the bundle by which $Z$ embeds to $\mathbb{P}^n,$ and $\omega_Z$ is the canonical bundle.

In other words, we have $c_1(Z)=-c_1(\omega_Z)=m c_1(i^*\mathcal{O}(1)),$ where $i:Z \hookrightarrow \mathbb{P}^n$ is the embedding given by $H.$ In particular, a degree $d$-hypersurface in $\mathbb{P}^n$ having some odd cohomology works, with $m = n+1-d$ (due to adjunction formula).

Denoting by $\widetilde{Z}$ its image via the Veronese map $\mathbb{P}^n\rightarrow\mathbb{P^{\binom{n+m}{m}-1}}$, define $X\subset \mathbb{C}^{\binom{n+m}{m}}$ to be the cone over $\widetilde{Z}$. There is a natural resolution $Y$ of $X$ given by the $Y=\overline{\pi^{-1}(X\setminus 0)},$ where $\pi$ is the blow-up at ${0}\subset \mathbb{C}^{\binom{n+m}{m}}.$ As the blow-up at $0$ is $\mathcal{O}(-1)\rightarrow \mathbb{P^{\binom{n+m}{m}-1}},$ we have $$Y=\mathcal{O}(-1)|_\widetilde{Z} \cong \mathcal{O}(-m)|_Z,$$ hence it has $$c_1(Y)=c_1(i^*\mathcal{O}(-m))+c_1(Z)=-mc_1(i^*\mathcal{O}(1)+c_1(Z)=0,$$ and cohomology equal to $Z,$ hence not supported in even degrees only.

In particular, choosing the quartic 3-fold $Z_4\subset \mathbb{P}^4$, we have $b_3(Z_4)=60$ and $m=(4+1)-4=1,$ so $X=V(z_0^4+\dots+z_4^4)$ is an isolated singularity and (trivially) a complete intersection as well, and its resolution given by the blow up of $\mathbb{C}^5$ at the origin yields a counterexample.

Thanks to @Yosemite Stan, we have a counterexample, and actually many of them: Pick a projective variety $Z$ with some odd-cohomology such that $$c_1(\omega_Z)=-m c_1(H), \text{ for some } m>0,$$ where $H$ is the bundle by which $Z$ embeds to $\mathbb{P}^n,$ and $\omega_Z$ is the canonical bundle.

In other words, we have $c_1(Z)=-c_1(\omega_Z)=m c_1(i^*\mathcal{O}(1)),$ where $i:Z \hookrightarrow \mathbb{P}^n$ is the embedding given by $H.$ In particular, a degree $(n+1-m)$-hypersurface having some odd cohomology works (due to adjunction formula).

Denoting by $\widetilde{Z}$ its image via the Veronese map $\mathbb{P}^n\rightarrow\mathbb{P^{\binom{n+m}{m}-1}}$, define $X\subset \mathbb{C}^{\binom{n+m}{m}}$ to be the cone over $\widetilde{Z}$. There is a natural resolution $Y$ of $X$ given by the $Y=\overline{\pi^{-1}(X\setminus 0)},$ where $\pi$ is the blow-up at $\mathcal{0}\subset \mathbb{C}^{\binom{n+m}{m}}.$ As blow-up at $0$ is $\mathcal{O}(-1)\rightarrow \mathbb{P^{\binom{n+m}{m}-1}},$ we have $$Y=\mathcal{O}(-1)|_\widetilde{Z} \cong \mathcal{O}(-m)|_Z,$$ hence it has $$c_1(Y)=c_1(i^*\mathcal{O}(-m))+c_1(Z)=-mc_1(i^*\mathcal{O}(1)+c_1(Z)=0,$$ and cohomology equal to $Z,$ hence not supported in even degrees only.

In particular, choosing the quartic 3-fold $Z_4\subset \mathbb{P}^4$, we have $b_3(Z_4)=60$ and $m=(4+1)-4=1,$ so $X=V(z_0^4+\dots+z_4^4)$ is an isolated singularity and (trivially) a complete intersection as well, and its resolution given by blow up of $\mathbb{C}^5$ at $0$ yields a counterexample.

Thanks to @Yosemite Stan, we have a counterexample, and actually many of them: Pick a projective variety $Z$ with some odd-cohomology such that $$c_1(\omega_Z)=-m c_1(H), \text{ for some } m>0,$$ where $H$ is the bundle by which $Z$ embeds to $\mathbb{P}^n,$ and $\omega_Z$ is the canonical bundle.

In other words, we have $c_1(Z)=-c_1(\omega_Z)=m c_1(i^*\mathcal{O}(1)),$ where $i:Z \hookrightarrow \mathbb{P}^n$ is the embedding given by $H.$ In particular, a degree $d$-hypersurface in $\mathbb{P}^n$ having some odd cohomology works, with $m = n+1-d$ (due to adjunction formula).

Denoting by $\widetilde{Z}$ its image via the Veronese map $\mathbb{P}^n\rightarrow\mathbb{P^{\binom{n+m}{m}-1}}$, define $X\subset \mathbb{C}^{\binom{n+m}{m}}$ to be the cone over $\widetilde{Z}$. There is a natural resolution $Y$ of $X$ given by the $Y=\overline{\pi^{-1}(X\setminus 0)},$ where $\pi$ is the blow-up at $\mathcal{0}\subset \mathbb{C}^{\binom{n+m}{m}}.$ As blow-up at $0$ is $\mathcal{O}(-1)\rightarrow \mathbb{P^{\binom{n+m}{m}-1}},$ we have $$Y=\mathcal{O}(-1)|_\widetilde{Z} \cong \mathcal{O}(-m)|_Z,$$ hence it has $$c_1(Y)=c_1(i^*\mathcal{O}(-m))+c_1(Z)=-mc_1(i^*\mathcal{O}(1)+c_1(Z)=0,$$ and cohomology equal to $Z,$ hence not supported in even degrees only.

In particular, choosing the quartic 3-fold $Z_4\subset \mathbb{P}^4$, we have $b_3(Z_4)=60$ and $m=(4+1)-4=1,$ so $X=V(z_0^4+\dots+z_4^4)$ is an isolated singularity and (trivially) a complete intersection as well, and its resolution given by blow up of $\mathbb{C}^5$ at $0$ yields a counterexample.

Thanks to @Yosemite Stan, we have a counterexample, and actually many of them: Pick a projective variety $Z\subset \mathbb{P}^n$ with some odd-cohomology and non-vanishing $c_1(Z)=m>0.$

Denoting by $\widetilde{Z}$ its image via the Veronese map $\mathbb{P}^n\rightarrow\mathbb{P^{\binom{n+m}{m}-1}}$, define $X\subset \mathbb{C}^{\binom{n+m}{m}}$ to be the cone over $\widetilde{Z}$. There is a natural resolution $Y$ of $X$ given by the closure of its preimage under the blowup at $0\subset \mathbb{C}^{\binom{n+m}{m}}.$ As blowup at $0$ is $O(-1)\rightarrow \mathbb{P^{\binom{n+m}{m}-1}},$ $Y$ is precisely equal to $O(-1)|_\widetilde{Z} \cong O(-m)|_Z,$ hence has $c_1=-m+c_1(Z)=0,$ and cohomology equal to $Z,$ hence not supported in even degrees only.

In particular, choosing the quartic 3-fold $Z$, we have $b_3(Z)=60$ and $c_1(Z)=1,$ and $X=V(z_0^4+\dots+z_4^4)$ is an isolated singularity and (trivially) a complete intersection.

Thanks to @Yosemite Stan, we have a counterexample, and actually many of them: Pick a projective variety $Z$ with some odd-cohomology such that $$c_1(\omega_Z)=-m c_1(H), \text{ for some } m>0,$$ where $H$ is the bundle by which $Z$ embeds to $\mathbb{P}^n,$ and $\omega_Z$ is the canonical bundle.

In other words, we have $c_1(Z)=-c_1(\omega_Z)=m c_1(i^*\mathcal{O}(1)),$ where $i:Z \hookrightarrow \mathbb{P}^n$ is the embedding given by $H.$ In particular, a degree $(n+1-m)$-hypersurface having some odd cohomology works (due to adjunction formula).

Denoting by $\widetilde{Z}$ its image via the Veronese map $\mathbb{P}^n\rightarrow\mathbb{P^{\binom{n+m}{m}-1}}$, define $X\subset \mathbb{C}^{\binom{n+m}{m}}$ to be the cone over $\widetilde{Z}$. There is a natural resolution $Y$ of $X$ given by the $Y=\overline{\pi^{-1}(X\setminus 0)},$ where $\pi$ is the blow-up at $\mathcal{0}\subset \mathbb{C}^{\binom{n+m}{m}}.$ As blow-up at $0$ is $\mathcal{O}(-1)\rightarrow \mathbb{P^{\binom{n+m}{m}-1}},$ we have $$Y=\mathcal{O}(-1)|_\widetilde{Z} \cong \mathcal{O}(-m)|_Z,$$ hence it has $$c_1(Y)=c_1(i^*\mathcal{O}(-m))+c_1(Z)=-mc_1(i^*\mathcal{O}(1)+c_1(Z)=0,$$ and cohomology equal to $Z,$ hence not supported in even degrees only.

In particular, choosing the quartic 3-fold $Z_4\subset \mathbb{P}^4$, we have $b_3(Z_4)=60$ and $m=(4+1)-4=1,$ so $X=V(z_0^4+\dots+z_4^4)$ is an isolated singularity and (trivially) a complete intersection as well, and its resolution given by blow up of $\mathbb{C}^5$ at $0$ yields a counterexample.