Evenness of c2 for smooth global Weierstrass Calabi–Yau fourfolds

Let $B_3$ be a smooth connected integral separated scheme of finite type over $\mathbb{C}$ and of dimension three, let $\mathcal{L}\to B_3$ be an algebraic line bundle, write $B_3^{\mathrm{a}\mathrm{n}}$ for its analytic manifold and $L={\mathcal{L}}^{\mathrm{a}\mathrm{n}}$, and let $p:W\to B_3$ be the algebraic weighted projective bundle of type $(2,3,1)$ associated with $\mathcal{L}$. The coordinate twists are $x\in {\mathcal{O}}_W(2)\otimes p^{\ast }{\mathcal{L}}^2$, $y\in {\mathcal{O}}_W(3)\otimes p^{\ast }{\mathcal{L}}^3$, and $z\in {\mathcal{O}}_W(1)$. Let $Y_4\subset W$ be the hypersurface $y^2=x^3+fx{z}^4+g{z}^6$ for $f\in H^0(B_3,{\mathcal{L}}^4)$ and $g\in H^0(B_3,{\mathcal{L}}^6)$, and put $\pi =p{\mathrm{\mid }}_{Y_4}$. Assume $\pi$ is proper (separated, of finite type, and universally closed), flat (each local ring ${\mathcal{O}}_{Y_4,y}$ is flat over ${\mathcal{O}}_{B_3,\pi (y)}$), and surjective, with the regular section $s$ given by the distinguished point $[1:1:0]$. For the generic point $\eta$ of $B_3$, set $Y_{4,\bar{\eta }}:=Y_4{\times }_{B_3}\mathrm{Spec}\overline{\kappa (\eta )}$; assume this geometric generic fiber is a smooth proper geometrically integral curve with ${\dim }_{\overline{\kappa (\eta )}}{H}^1(Y_{4,\bar{\eta }},{\mathcal{O}}_{Y_{4,\bar{\eta }}})=1$. Write $Y_4^{\mathrm{a}\mathrm{n}}$ and ${\pi }^{\mathrm{a}\mathrm{n}}:Y_4^{\mathrm{a}\mathrm{n}}\to B_3^{\mathrm{a}\mathrm{n}}$ for the analytifications. Assume $Y_4^{\mathrm{a}\mathrm{n}}$ is smooth, connected, and compact, admits a smooth real positive closed $(1,1)$-form, and has a holomorphic line-bundle isomorphism $K_{Y_4^{\mathrm{a}\mathrm{n}}}:=\det {\mathrm{\Omega }}_{Y_4^{\mathrm{a}\mathrm{n}}\mathrm{/}\mathbb{C}}^1\cong {\mathcal{O}}_{Y_4^{\mathrm{a}\mathrm{n}}}$. These are the broad Calabi–Yau conditions used here. Then the second Chern class of the holomorphic tangent bundle $T^{1,0}{Y}_4^{\mathrm{a}\mathrm{n}}$ satisfies $c_2(Y_4^{\mathrm{a}\mathrm{n}})\in 2H^4(Y_4^{\mathrm{a}\mathrm{n}};\mathbb{Z})$. The conclusion is restricted to this smooth global-Weierstrass setting and is not asserted for a singular or resolved fourfold.

Weierstrass model over a line bundle

Second Chern class of a smooth Calabi–Yau Weierstrass fourfold

The degree-four Chern parity of a compact complex threefold

The exactness of the coefficient sequence $0\to \mathbb{Z}\stackrel{2}{\to }\mathbb{Z}\to {\mathbb{F}}_2\to 0$ makes the parity statement valid even when the integral cohomology has torsion. The result does not extend by itself to a singular Weierstrass hypersurface or to an arbitrary crepant resolution.