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Second Chern class of a smooth Calabi–Yau Weierstrass fourfold

Statement

Let $B$ be a connected smooth compact complex threefold, define its holomorphic canonical line by $K_{B} : = \det Ω_{B / C}^{1}$ , and set $L = K_{B}^{- 1}$ . Let $p : W \to B$ be the analytic weighted projective bundle of type $(2, 3, 1)$ associated with $L$ , with coordinate twists $x \in O_{W} (2) \otimes p^{*} L^{2}$ , $y \in O_{W} (3) \otimes p^{*} L^{3}$ , and $z \in O_{W} (1)$ with the weighted quotient and line-bundle gluing convention specified below. Let $Y \subset W$ be the analytic hypersurface $y^{2} = x^{3} + f x z^{4} + g z^{6}$ , where $f \in H^{0} (B, L^{4})$ and $g \in H^{0} (B, L^{6})$ are holomorphic sections, and write $π = p ∣_{Y} : Y \to B$ . Assume $Y$ is smooth, compact, and connected, and admits a smooth real positive closed $(1, 1)$ -form (the Kähler condition). The canonical-bundle formula then gives $K_{Y} ≅ O_{Y}$ . In this item, a compact Calabi–Yau fourfold means a connected compact Kähler complex fourfold with holomorphically trivial canonical bundle, so $Y$ satisfies that convention. Write $a = c_{1} (B) = c_{1} (L) \in H^{2} (B; Z)$ , and let $σ \in H^{2} (Y; Z)$ be the cohomology class Poincare dual to the section divisor at $z = 0$ . Then $c_{2} (Y) = π^{*} ⁣ (c_{2} (B) + 11 a^{2}) + 12 σ ⌣ π^{*} a in H^{4} (Y; Z) .$

Statement references

Weierstrass model over a line bundle

Canonical bundle of a Weierstrass hypersurface

Facts & Assumptions

Given: A connected smooth compact complex threefold $B$ , its canonical bundle $K_{B}$ , $L = K_{B}^{- 1}$ , and a smooth compact connected Weierstrass hypersurface $Y$ admitting the Kähler form specified in the Statement.

[A1]

Full AC is the choice-function principle of The Axiom of Choice. It is inherited through the Chern-class, CW-type, Thom, and broad Calabi–Yau suppliers used in [F2]–[F5]; the local weighted-Euler and divisor calculations require no further choice.

[F1]

On a holomorphic frame cover of $L$ with $e_{j} = γ_{i j} e_{i}$ , glue $U_{i} \times ((C^{3} ∖ 0) / C^{*})$ using $(x_{j}, y_{j}, z_{j}) = (γ_{i j}^{- 2} x_{i}, γ_{i j}^{- 3} y_{i}, z_{i})$ and the action $λ \cdot (x, y, z) = (λ^{2} x, λ^{3} y, λ z)$ . These transitions commute with the action and satisfy the cocycle identity, so this defines the weighted bundle over the stated complex-manifold base. The Weierstrass equation is $y^{2} = x^{3} + f x z^{4} + g z^{6}$ with $x \in O_{W} (2) \otimes π_{W}^{*} L^{2}$ , $y \in O_{W} (3) \otimes π_{W}^{*} L^{3}$ , $z \in O_{W} (1)$ , and its section at infinity is $z = 0$ (Weierstrass model over a line bundle).

[F2]

Here $L = K_{B}^{- 1}$ and $a = c_{1} (L) = c_{1} (T B)$ . The canonical-line calculation will be made directly from the Euler and normal sequences in step 2.1, so it applies to the complex-manifold base allowed here. The algebraic-base specialization is Canonical bundle of a Weierstrass hypersurface; its algebraicity hypothesis is not assumed for this local calculation.

[F3]

On a path-connected CW base, the total Chern class is natural and multiplicative for Whitney sums; for a line bundle $Q$ , $c (Q) = 1 + c_{1} (Q)$ (Naturality, normalization, and Whitney sum for Chern classes). The smooth manifolds $B$ and $Y$ are paracompact Hausdorff CGWH of CW homotopy type, and their smooth finite-rank bundles are numerable (Smooth manifolds have CW homotopy type). Step 1.2 supplies the transfer and exact-sequence splitting needed here.

[F4]

A closed embedded complex submanifold's integral Poincare dual is represented by its normal Thom class, and the Thom isomorphism, normalized Thom class, and zero-section pullback define the Euler class (The normal Thom class realizes the Poincare dual of a closed submanifold, Thom isomorphism for oriented vector bundles, Thom class by fiberwise normalization, Euler class by zero-section pullback of the Thom class). For a complex line its integral Euler class is its first Chern class, using the complex orientation and the Chern normalization of [F3].

[F5]

For a smooth divisor inclusion $i : Z ↪ X$ , define the cohomological Thom-Gysin map $i_{!} : H^{k} (Z; Z) \to H^{k + 2} (X; Z)$ by the Thom isomorphism of its oriented real rank-two normal bundle, followed by relative-to-absolute and tubular-neighborhood maps. Its value $i_{!} (1)$ is the Poincare-dual class of $Z$ . If $p : D (ν_{Z}) \to Z$ is a tubular disk bundle with zero section $s$ , then $s$ is a homotopy inverse to $p$ . Thus for $α \in H^{*} (X; Z)$ , restriction to the disk bundle is $p^{*} i^{*} α$ . The Thom isomorphism sends $i^{*} α$ to $p^{*} i^{*} α ⌣ u_{ν_{Z}}$ ; the relative cup product and its map to absolute cohomology therefore give $i_{!} (i^{*} α) = α ⌣ i_{!} (1)$ . This is the divisor self-intersection identity used below, with the orientation sign fixed by the tangent-first Thom convention (Thom isomorphism for oriented vector bundles, Thom class by fiberwise normalization, Relative cup product for an excisive triad, Relative cup products are natural and connector-compatible, The normal Thom class realizes the Poincare dual of a closed submanifold).

[F6]

Chern classes on paracompact Hausdorff CGWH bases of CW type are the unique coefficients of the monic projective-bundle relation; their line normalization is the complex-oriented Euler class (Chern classes from the projective-bundle relation, Integral complex projective bundle theorem). Euler classes are natural for oriented pullbacks in this scope, and a homotopy equivalence induces cohomology isomorphisms (Naturality, orientation sign, and Whitney product for Euler classes, Homotopic maps induce equal maps in singular cohomology).

[F7]

On these same CW-type bases, $c_{1} (Q \otimes R) = c_{1} (Q) + c_{1} (R)$ and $c_{1} (Q^{*}) = - c_{1} (Q)$ for complex line bundles; $c_{1} (E) = c_{1} (\det E)$ for a positive-rank complex bundle (First Chern class of tensor, dual, and conjugate lines, First Chern class of a complex bundle equals that of its determinant). In particular $L = \det T B$ gives $a = c_{1} (L) = c_{1} (T B)$ .

Proof

technique · compute the total Chern class by the weighted Euler and hypersurface sequences, then use the section's normal bundle to reduce its square

1.1F1givenalgebra

The singular locus of the weighted projective fiber lies among points with at least one vanishing coordinate and nontrivial stabilizer. On $Y$ , if $z \neq 0$ the weight-one coordinate makes the stabilizer trivial. If $z = 0$ , the equation is $y^{2} = x^{3}$ , so both $x$ and $y$ are nonzero; a stabilizer must satisfy $λ^{2} = λ^{3} = 1$ , hence $λ = 1$ . Thus $Y$ lies in the smooth quotient locus, where the three coordinate twists restrict to line bundles and the vertical Euler sequence is $0 ⟶ O_{Y} ⟶ Q_{2} \oplus Q_{3} \oplus Q_{1} ⟶ T_{W / B} ∣_{Y} ⟶ 0,$ with $Q_{2} = O_{Y} (2) \otimes π^{*} L^{2}$ , $Q_{3} = O_{Y} (3) \otimes π^{*} L^{3}$ , and $Q_{1} = O_{Y} (1)$ . The first map is the infinitesimal weighted scaling $(2 x, 3 y, z)$ ; its quotient is the tangent space to each weighted-projective fiber. Let $h = c_{1} (O_{Y} (1))$ and identify $a$ with its pullback when it occurs on $Y$ .

1.2A1F3F6F7chooseconstructalgebra

We justify the Chern identities on the actual manifold bases. Let $E, F \to X$ be numerable finite-rank complex bundles and $b : X^{'} \to X$ a continuous map between path-connected paracompact Hausdorff CGWH bases of CW type. The projective charts identify $P (b^{*} E) = X^{'} \times_{X} P (E)$ and pull the tautological line back. Euler naturality in [F6] pulls its class back; pulling back the monic relation and using uniqueness on $X^{'}$ gives $c_{i} (b^{*} E) = b^{*} c_{i} (E)$ . The rank-zero conventions give the same identity. Choose a path-connected CW model $q : K \to X$ ; CW complexes are paracompact Hausdorff and CGWH, and a pulled-back numeration numerates $q^{*} E$ and $q^{*} F$ . Applying [F3] on $K$ and the naturality just proved yields $q^{*} (c (E \oplus F) - c (E) c (F)) = 0$ . Since $q^{*}$ is an integral cohomology isomorphism, Whitney multiplicativity holds on $X$ . For any exact sequence $0 \to E^{'} \to E \to E^{''} \to 0$ here, sum the local standard Hermitian metrics using a subordinate partition from a numeration of $E$ . This defines a positive continuous Hermitian metric; the orthogonal complement of $E^{'}$ is a subbundle mapped isomorphically to $E^{''}$ , so $E ≅ E^{'} \oplus E^{''}$ as topological complex bundles. Thus $c (E) = c (E^{'}) c (E^{''})$ on these bases. Line normalization follows directly from [F6], and [F7] supplies the tensor powers used below.

2.1A1F1F2F3F7step 1.1step 1.2algebra

The tangent sequence of the smooth bundle projection and step 1.1 give $c (T W ∣_{Y}) = π^{*} c (T B) (1 + 2 h + 2 a) (1 + 3 h + 3 a) (1 + h) .$ The hypersurface equation is a section of $O_{Y} (6) \otimes π^{*} L^{6}$ , so its smooth zero locus has normal line with first Chern class $6 h + 6 a$ and exact tangent-normal sequence. Taking determinants of these same sequences gives $K_{Y} ≅ π^{*} (K_{B} \otimes L) ≅ O_{Y}$ : the Euler determinant is $O_{Y} (6) \otimes π^{*} L^{5}$ , and the normal line is $O_{Y} (6) \otimes π^{*} L^{6}$ , so their tautological factors cancel in adjunction. This is a holomorphic line-bundle identity for the stated analytic base. Since $a = c_{1} (T B)$ , the degree-one part of $c (T W ∣_{Y})$ is $a + (6 h + 5 a) = 6 h + 6 a$ . The degree-two part is $c_{2} (B) + a (6 h + 5 a) + ((2 h + 2 a) (3 h + 3 a) + (2 h + 2 a) h + (3 h + 3 a) h) = π^{*} c_{2} (B) + 11 h^{2} + 23 h a + 11 a^{2} .$ The normal first Chern class equals the ambient first Chern class, so dividing the total Chern class by $1 + (6 h + 6 a)$ leaves its degree-two part unchanged. Consequently $c_{1} (T Y) = 0, c_{2} (T Y) = π^{*} c_{2} (B) + 11 h^{2} + 23 h π^{*} a + 11 π^{*} a^{2} .$

3.1A1F2F3F4F5F7step 1.2step 2.1algebra

The divisor $σ$ is the smooth section cut out by $z = 0$ , and $h$ is its Poincare-dual divisor class. The normal sequence $0 \to T σ \to T Y ∣_{σ} \to N_{σ / Y} \to 0$ and determinants give $K_{σ} ≅ K_{Y} ∣_{σ} \otimes N_{σ / Y}$ . Because $σ ≅ B$ , $K_{Y}$ is trivial by step 2.1, and $K_{σ} ≅ K_{B}$ , this identifies $N_{σ / Y} ≅ K_{B} = L^{- 1}$ . The divisor normal line is $O_{Y} (σ) ∣_{σ}$ , so $i^{*} h = c_{1} (N_{σ / Y}) = - a$ . With the tangent-first convention in [F4], the Thom sign is $(- 1)^{2 \cdot 6} = 1$ , so $i_{!} (1) = h$ ; [F5] gives $i_{!} (i^{*} h) = h ⌣ h$ . Since $i^{*} h = - a = i^{*} (- π^{*} a)$ and $π \circ i = {id}_{B}$ , the Thom-Gysin identity in [F5] gives $h^{2} = i_{!} (i^{*} h) = i_{!} (- a) = - π^{*} a ⌣ i_{!} (1) = - h ⌣ π^{*} a .$ Here $i : σ ↪ Y$ is the divisor inclusion.

4.1step 2.1step 3.1algebra∎

Substitute $h^{2} = - h π^{*} a$ from step 3.1 into step 2.1: $c_{2} (Y) = π^{*} c_{2} (B) + 11 (- h π^{*} a) + 23 h π^{*} a + 11 π^{*} a^{2} = π^{*} (c_{2} (B) + 11 a^{2}) + 12 h π^{*} a .$ Since $h = σ$ by definition, this is the claimed integral class identity. This is a cohomology calculation for the smooth global Weierstrass specialization only; no formula for singular or resolved fourfolds follows.

The line bundle twist $L = K_{B}^{- 1}$ and smoothness of the total hypersurface are essential inputs. The identity is in integral cohomology and does not assume torsion-free cohomology.

Remarks