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The degree-four Chern parity of a compact complex threefold

Statement

Assume full AC. Let $B$ be a connected compact smooth complex threefold. For the underlying real tangent bundle, write $w_{i} (B)$ for its Stiefel–Whitney classes with coefficients in $F_{2}$ , and write $c_{i} (B)$ for the integral Chern classes of its holomorphic tangent bundle. Then $c_{2} (B) + c_{1} (B) ⌣ c_{1} (B) \in 2 H^{4} (B; Z) .$

Statement references

Mod-two reduction of Chern classes

Wu formula for a closed smooth manifold

Facts & Assumptions

Given: Full AC and a connected compact smooth complex threefold $B$ , with its complex orientation and mod-two fundamental class $[B]_{2}$ .

[A1]

Full AC is the choice-function principle of The Axiom of Choice. It is inherited from the Wu-class, Poincare-duality, and characteristic-class suppliers.

[F1]

The complex tangent bundle has rank three. The underlying smooth manifold is paracompact Hausdorff CGWH of CW homotopy type, and its tangent bundles are numerable (Complex manifolds, holomorphic vector bundles and Hermitian metrics, Smooth manifolds have CW homotopy type). For a numerable complex bundle on a path-connected paracompact Hausdorff CW complex, the comparison theorem gives $w_{2 i + 1} = 0$ and $w_{2 i} = ρ_{2} c_{i}$ (Mod-two reduction of Chern classes); step 1.2 transfers this assertion to $B$ . Stiefel–Whitney classes on $B$ use Stiefel–Whitney classes from the projective-bundle relation.

[F2]

The Wu classes satisfy $⟨ v_{k} x, [B]_{2} ⟩ = ⟨ S q^{k} (x), [B]_{2} ⟩$ , and the mod-two cup pairing is nonsingular (Wu classes of a closed manifold, Poincaré duality gives a nonsingular cup pairing).

[F3]

The Wu formula is $w_{i} (B) = \sum_{j = 0}^{i} S q^{j} (v_{i - j} (B))$ in each degree (Wu formula for a closed smooth manifold).

[F4]

Steenrod squares are unstable, satisfy $S q^{d} (x) = x^{2}$ for degree- $d$ classes, and $S q^{1}$ is the connecting operation for $0 \to F_{2} \overset{2}{\to} Z / 4 \to F_{2} \to 0$ (Steenrod normalization, instability, suspension, and top square, $S q^{1}$ is the mod-two Bockstein, Bockstein connecting operation).

[F5]

For a short exact coefficient sequence, singular cohomology has its long exact sequence, whose exactness identifies the kernel of coefficient reduction with the image of the preceding coefficient map (The long exact sequence in cohomology, Singular cochain complex with coefficients).

[F6]

On paracompact Hausdorff CGWH bases of CW type, Chern classes are the unique coefficients of the monic relation for the Euler class of the tautological complex line; the projective-bundle theorem applies to these bases (Chern classes from the projective-bundle relation, Integral complex projective bundle theorem). The complex-oriented Euler class is natural for pullbacks in this Thom scope (Naturality, orientation sign, and Whitney product for Euler classes).

[F7]

Stiefel–Whitney classes are natural on paracompact Hausdorff CGWH bases of CW type (Naturality of Stiefel–Whitney classes). A homotopy equivalence induces cohomology isomorphisms, and coefficient reduction commutes with pullback (Homotopic maps induce equal maps in singular cohomology, Singular cohomology is contravariantly functorial).

Proof

technique · expand the Wu formula in degrees one through four, then use exactness of integral coefficient reduction

1.1A1F2F4algebra

Since $B$ has real dimension six, $v_{k} (B) = 0$ for $k > 3$ : for such $k$ , every class $x \in H^{6 - k} (B; F_{2})$ has degree $6 - k < k$ , so instability gives $S q^{k} x = 0$ ; the defining pairing in [F2] and its nonsingularity force $v_{k} = 0$ . Also $v_{0} = 1$ by the Wu-class definition. The full AC hypothesis is inherited from these suppliers.

1.2F1F4F5F6F7givenchoosealgebra

Choose a homotopy equivalence $q : K \to B$ from a path-connected CW complex; CW complexes are paracompact Hausdorff and CGWH. The numeration of $T B$ pulls back along $q$ , so [F1] applies to $q^{*} T B$ on the actual CW base $K$ . To compare its Chern classes with those on $B$ , the bundle charts give $P (q^{*} T B) = K \times_{B} P (T B)$ , and the induced map of projective bundles pulls back the tautological line. Euler naturality in [F6] therefore pulls its tautological class back. Pulling back the monic relation for $T B$ and using coefficient uniqueness on $K$ gives $c_{i} (q^{*} T B) = q^{*} c_{i} (T B)$ . By [F7], realification commutes with pullback and $w_{j} (q^{*} T B_{R}) = q^{*} w_{j} (T B_{R})$ , while $q^{*} ρ_{2} = ρ_{2} q^{*}$ . The comparison on $K$ thus says $q^{*} (w_{2 i} - ρ_{2} c_{i}) = 0$ and $q^{*} w_{2 i + 1} = 0$ . Since $q^{*}$ is an isomorphism over $F_{2}$ , the same identities hold on $B$ . In particular $w_{1} = 0$ , $w_{2} = ρ_{2} c_{1}$ , $w_{3} = 0$ , and $w_{4} = ρ_{2} c_{2}$ . The class $w_{2}$ has an integral lift, so its Bockstein for $0 \to F_{2} \to Z / 4 \to F_{2} \to 0$ vanishes: reduce $c_{1}$ modulo four to obtain a lift in $H^{2} (B; Z / 4)$ , then apply exactness. By [F4], $S q^{1} w_{2} = 0$ .

2.1F3F4step 1.2algebra

In degree one the Wu formula [F3] gives $w_{1} = v_{1} + S q^{1} (v_{0}) = v_{1}$ , since $S q^{1} (1) = 0$ . Hence $v_{1} = 0$ . In degree two it gives $w_{2} = v_{2} + S q^{1} (v_{1}) + S q^{2} (v_{0}) = v_{2}$ , since $v_{1} = 0$ and positive squares of the unit vanish. Thus $v_{2} = w_{2}$ .

3.1F3F4step 1.1step 1.2step 2.1algebra

In degree three, [F3] gives $w_{3} = v_{3} + S q^{1} (v_{2}) + S q^{2} (v_{1}) + S q^{3} (v_{0})$ . By steps 1.1, 1.2, and 2.1, $v_{3} = 0$ because $w_{3} = 0$ , $S q^{1} (v_{2}) = S q^{1} (w_{2}) = 0$ , $v_{1} = 0$ , and $S q^{3} (1) = 0$ .

4.1F3F4step 1.1step 1.2step 2.1step 3.1algebra

In degree four, [F3] gives $w_{4} = v_{4} + S q^{1} (v_{3}) + S q^{2} (v_{2}) + S q^{3} (v_{1}) + S q^{4} (v_{0}) = S q^{2} (w_{2}) = w_{2}^{2} .$ Here $v_{4} = 0$ by step 1.1, $v_{3} = 0$ by step 3.1 and $v_{1} = 0$ by step 2.1, $S q^{4} (1) = 0$ , and the top-square identity applies to the degree-two class $w_{2}$ .

5.1F5step 1.2step 4.1algebra∎

Substituting the comparison proved in step 1.2 into step 4.1 yields $ρ_{2} (c_{2} + c_{1}^{2}) = w_{4} + w_{2}^{2} = 0$ (addition and subtraction agree over $F_{2}$ ). For the coefficient sequence $0 \to Z \overset{2}{\to} Z \overset{ρ_{2}}{\to} F_{2} \to 0$ , exactness in [F5] says $\ker (ρ_{2} : H^{4} (B; Z) \to H^{4} (B; F_{2})) = im (2 : H^{4} (B; Z) \to H^{4} (B; Z))$ . Hence there is an integral class $η$ with $c_{2} + c_{1}^{2} = 2 η$ . This makes no torsion-free assumption.

The degree-four Chern parity of a compact complex threefold

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources