A product elliptic fourfold over projective three-space

The standard affine charts of ${\mathbb{P}}_{\mathbb{C}}^3$ are copies of ${\mathbb{A}}_{\mathbb{C}}^3$, so $B$ is smooth of dimension three. Projective space is connected and projective. Thus $B$ is a connected nonsingular projective threefold.

The projection $Y\to B$ is smooth of relative dimension one by base change in [F2]; composing with the smooth threefold $B\to \mathrm{Spec}\mathbb{C}$ gives $\dim Y=3+1=4$ by additivity in [F2]. Analytically, products of a holomorphic coordinate chart of $E$ in $\mathbb{C}$ and a standard chart of $B$ in ${\mathbb{C}}^3$ have images open in ${\mathbb{C}}^4$. Their transition maps act separately in the two coordinates, hence are holomorphic with holomorphic inverses. They give a complex four-manifold structure whose underlying real smooth structure is the product structure of [F4]. To prove projectivity, choose a projective embedding $E\hookrightarrow {\mathbb{P}}_{\mathbb{C}}^N$. The Segre map sends $([X_i],[T_j])$ to the matrix $[X_i{T}_j]$. Its image is the closed locus cut out by the $2\times 2$ minors. On a chart where $Z_{i_0{j}_0}\mathrm{\ne }0$, the inverse is $[X_i]=[Z_{i{j}_0}]$ and $[T_j]=[Z_{i_{0}j}]$; the minors imply every entry is proportional to $Z_{i{j}_0}{Z}_{i_{0}j}$, so these regular local inverses give a closed immersion. Base-changing the closed immersion of $E$ shows $E\times B$ is closed in ${\mathbb{P}}^N\times {\mathbb{P}}^3$, hence the Segre embedding makes $Y$ projective. To verify integrality, cover $E$ by nonempty affine opens $\mathrm{Spec}A$ and use standard affine charts ${\mathbb{A}}^3$ of $B$; by [F5] each product chart is $\mathrm{Spec}A[t_1,t_2,t_3]$, whose coordinate ring is a domain. These product opens cover $Y$ and pairwise meet because both factors are integral, so $Y$ is integral. The same argument after every field extension uses geometric integrality of $E$ and ${\mathbb{P}}^3$, so $Y$ is geometrically integral and connected.

The projection $\pi$ is the base change of $E\to \mathrm{Spec}\mathbb{C}$. By [F2], it is smooth and hence flat. The morphism $E\to \mathrm{Spec}\mathbb{C}$ is proper: its projective embedding is a closed immersion, projective space is proper, and proper morphisms compose by [F3]. Properness is preserved by base change by [F3], so $\pi$ is proper.

The map $s:B\to Y$, $s(b)=(O,b)$, is a morphism and satisfies $\pi \circ s={\mathrm{id}}_B$ by the product projections. Thus $\pi$ is surjective and $s$ is a regular section. Each geometric fiber over a geometric point with field $K$ is $E_K$, smooth of pure dimension one by [F2] and geometrically integral by the hypothesis on $E$. The curve $E$ is proper by step 2.1 and ${\mathcal{O}}_E$ is coherent, so [F6] gives $H^1(E_K,{\mathcal{O}}_{E_K})\cong H^1(E,{\mathcal{O}}_E){\otimes }_{\mathbb{C}}K$, of dimension one over $K$. Thus every geometric fiber is genus one, including the geometric generic fiber.

Steps 1.1–3.1 verify that $B$ and $Y$ have the stated dimensions and projectivity, and that $\pi$ is proper, flat, and surjective with the required geometric generic fiber and regular section. Therefore this is an elliptic genus-one fourfold fibration over ${\mathbb{P}}_{\mathbb{C}}^3$ in the mathematical sense of [F1].