Weierstrass model over a line bundle

Let $B$ be a smooth complex algebraic variety, meaning an integral separated scheme of finite type over $\mathbb{C}$ that is smooth over $\mathbb{C}$ as in Flat genus-one fibration, and write $B^{\mathrm{a}\mathrm{n}}$ for its associated complex manifold. Let $L\to B^{\mathrm{a}\mathrm{n}}$ be a holomorphic line bundle, and put $E=L^{\otimes 2}\oplus L^{\otimes 3}\oplus {\mathcal{O}}_{B^{\mathrm{a}\mathrm{n}}}$. The analytic weighted projective bundle of type $(2,3,1)$ is the coarse complex-analytic quotient $W=(E\setminus 0)\mathrm{/}{\mathbb{C}}^{\ast },\lambda \cdot (v_2,v_3,v_1)=({\lambda }^2{v}_2,{\lambda }^3{v}_3,\lambda v_1),$ with projection $p:W\to B^{\mathrm{a}\mathrm{n}}$. Equivalently, choose an analytic-open cover $(U_i)$ trivializing $L$ and holomorphic frames $e_i$ with $e_j={\gamma }_{ij}{e}_i$. The bundle is glued from $U_i\times {\mathbb{P}}^{\mathrm{a}\mathrm{n}}(2,3,1)$, where ${\mathbb{P}}^{\mathrm{a}\mathrm{n}}(2,3,1)=({\mathbb{C}}^3\setminus \{0\})\mathrm{/}{\mathbb{C}}^{\ast }$ for the displayed weighted action, by $x_j={\gamma }_{ij}^{-2}{x}_i,y_j={\gamma }_{ij}^{-3}{y}_i,z_j=z_i\mathrm{.}$ These holomorphic maps commute with the weighted action and satisfy the cocycle identity. Let $W^{\circ }$ be the open locus where the weighted action has trivial stabilizer. Since $B^{\mathrm{a}\mathrm{n}}$ is smooth, $W^{\circ }$ is a complex manifold. On $W^{\circ }$, let ${\mathcal{O}}_W(n)$ denote the tautological line bundle whose local coordinate sections transform with weight $n$ under the displayed ${\mathbb{C}}^{\ast }$ action. The stabilizer of any point divides one of $1,2,3$, hence has order dividing $6$; the weight-$6$ character is trivial on every stabilizer, so ${\mathcal{O}}_W(6)$ extends as an invertible sheaf to all of $W$. On $W^{\circ }$ the tautological coordinate sections have twists $x\in {\mathcal{O}}_W(2)\otimes p^{\ast }{L}^{\otimes 2}$, $y\in {\mathcal{O}}_W(3)\otimes p^{\ast }{L}^{\otimes 3}$, and $z\in {\mathcal{O}}_W(1)$.

A Weierstrass model over $L$ is the analytic hypersurface $Y\subset W$ with local equation $y^2=x^3+fx{z}^4+g{z}^6,$ where $f\in H^0(B^{\mathrm{a}\mathrm{n}},L^{\otimes 4})$ and $g\in H^0(B^{\mathrm{a}\mathrm{n}},L^{\otimes 6})$. If their local coefficients in the frame $e_i$ are $f_i,g_i$, then $f_j={\gamma }_{ij}^{-4}{f}_i$ and $g_j={\gamma }_{ij}^{-6}{g}_i$. Thus each term of the equation transforms by ${\gamma }_{ij}^{-6}$ and the equation is a section of ${\mathcal{O}}_W(6)\otimes p^{\ast }{L}^{\otimes 6}$. Its discriminant section is $\mathrm{\Delta }=4f^3+27g^2\in H^0(B^{\mathrm{a}\mathrm{n}},L^{\otimes 12})$; its analytic zero locus is an effective divisor when $\mathrm{\Delta }$ is not identically zero. The point at infinity $[1:1:0]$ is a distinguished holomorphic section: on an overlap its coordinates become $[{\gamma }_{ij}^{-2}:{\gamma }_{ij}^{-3}:0]$, which is the same weighted-projective point by scaling with $\lambda ={\gamma }_{ij}^{-1}$. Every point of $Y$ lies in $W^{\circ }$: this holds when $z\mathrm{\ne }0$, and if $z=0$ the equation forces $x,y\mathrm{\ne }0$, so a stabilizer satisfies ${\lambda }^2={\lambda }^3=1$ and is trivial.

If an algebraic line bundle $\mathcal{L}\to B$ satisfies $L\cong {\mathcal{L}}^{\mathrm{a}\mathrm{n}}$ and $f,g$ are analytifications of algebraic sections of ${\mathcal{L}}^{\otimes 4}$ and ${\mathcal{L}}^{\otimes 6}$, choose algebraic frames on a Zariski-open cover trivializing $\mathcal{L}$ and glue $U_i\times \mathrm{Proj}\mathbb{C}[x,y,z]$, graded by $\mathrm{deg}x=2$, $\mathrm{deg}y=3$, $\mathrm{deg}z=1$, by the same coordinate transition maps, now with regular invertible ${\gamma }_{ij}$. This is the algebraic weighted projectivization of ${\mathcal{L}}^{\otimes 2}\oplus {\mathcal{L}}^{\otimes 3}\oplus {\mathcal{O}}_B$ with weights $(2,3,1)$; its analytification is the analytic bundle above, and the algebraic equation analytifies to $Y$. For holomorphic data that do not arise from algebraic line-bundle and section data, the construction is analytic and no algebraic scheme structure is asserted.

If ${\dim }_{\mathbb{C}}B=3$ and $L\cong K_{B^{\mathrm{a}\mathrm{n}}}^{-1}$, call this the Calabi–Yau Weierstrass specialization; the theorem below proves that its invertible analytic dualizing sheaf is trivial. This name does not assert smoothness of $Y$.

Flat genus-one fibration

Compact Calabi–Yau manifold conventions

Complex manifolds, holomorphic vector bundles and Hermitian metrics

Canonical bundle of a Weierstrass hypersurface

T. Weigand, TASI Lectures, §2.2, equations (2.50)–(2.60), and §3.1

V. Braun and D. R. Morrison, §2 and §5.1

The discriminant formula fixes its vanishing locus only up to the conventional nonzero scalar factor. A nonzero discriminant section gives a divisor; if $\mathrm{\Delta }\equiv 0$, the analytic zero locus is the whole base and is not called a divisor here. The general hypersurface is not asserted to be flat or smooth over every point of $B^{\mathrm{a}\mathrm{n}}$. The broad Calabi–Yau convention requires a connected compact Kähler manifold with holomorphically trivial canonical bundle; triviality of the analytic dualizing sheaf of a possibly singular hypersurface alone does not meet that definition.

The extension of ${\mathcal{O}}_W(6)$ can be seen on each weighted-projective fiber. The degree-six homogeneous sheaf on $\mathbb{P}(2,3,1)$ is trivialized on the standard opens $x\mathrm{\ne }0$, $y\mathrm{\ne }0$, and $z\mathrm{\ne }0$ by the nonvanishing monomials $x^3$, $y^2$, and $z^6$, respectively. On overlaps their ratios are invertible degree-zero functions. These opens cover the fiber, so the degree-six homogeneous sheaf is a line bundle there. The base transition maps $(x_i,y_i,z_i)\mapsto ({\gamma }_{ij}^{-2}{x}_i,{\gamma }_{ij}^{-3}{y}_i,z_i)$ are graded automorphisms and therefore glue these local line bundles to ${\mathcal{O}}_W(6)$ on all of $W$.