A nice way of constructing a non-projective Calabi-Yau threefold is to take the total space $$Y:= \mathrm{Tot}(\omega_S) = \mathbf{Spec}(\mathrm{Sym}^\bullet(\omega_S^\vee))$$ of the canonical bundle of a surface $S$. Here $\mathbf{Spec}$ is relative Spec. This is Calabi-Yau because it can be shown that $\omega_Y \cong \mathcal{O}_Y$.

The projective version of the total space construction is $$ \mathbb{P}(\mathcal{F}) = \mathbf{Proj}(\mathrm{Sym}^\bullet(\mathcal{F})) $$ where $\mathcal{F}$ is a (finite rank) locally free coherent sheaf. My question is: can such a sheaf $\mathcal{F}$ (on some variety $X$) be picked such that $\omega_{\mathbb{P}(\mathcal{F})} \cong \mathcal{O}_{\mathbb{P}(\mathcal{F})}$, i.e. such that $\mathbb{P}(\mathcal{F})$ is Calabi-Yau?

Thank you for any answers.


1 Answer 1


No, this construction will never give a Calabi--Yau variety.

For simplicity assume $X$ is smooth. By construction, the variety $V=\mathbb P_X(F)$ is a projective bundle, so it is uniruled (except possibly in the trivial case when $F$ has rank 1). It is a basic property of smooth projective uniruled varieties $V$ that

$$H^0(V,K_V^m)=0 \quad \text{ for all } m>0$$

On the other hand, if $Y$ is Calabi--Yau, then $K_Y$ is the trivial bundle, so $H^0(Y,K_Y)=1$.

A reference for the above is Kollár, Rational Curves on Algebraic Varieties, Corollary IV.1.11.

  • 1
    $\begingroup$ It is easier to just say that $K_Y$ restricts to a fiber $\mathbb{P}^m$ of the projective bundle as $\mathcal{O}(-m-1)$, hence is non-trivial. $\endgroup$
    – Sasha
    Commented Oct 3, 2019 at 5:27

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