Tangent Chern classes of a smooth quintic threefold

Let $Q=({\mathbb{P}}^4\times {\mathbb{C}}^5)\mathrm{/}\gamma$ be the holomorphic quotient bundle. In the standard projective charts, the tangent space at a line $\mathrm{\ell }$ is identified holomorphically with ${\mathrm{Hom}}_{\mathbb{C}}(\mathrm{\ell },{\mathbb{C}}^5\mathrm{/}\mathrm{\ell })$. Tensoring the tautological sequence $0\to \gamma \to {\mathbb{P}}^4\times {\mathbb{C}}^5\to Q\to 0$ by ${\gamma }^{\ast }=\mathcal{O}(1)$ gives the holomorphic Euler sequence $0\to \mathcal{O}\to \mathcal{O}(1{)}^{\oplus 5}\to T{\mathbb{P}}^4\to 0$. For a holomorphic short exact sequence $0\to E^{\mathrm{\prime }}\to E\to E^{\mathrm{\prime }\mathrm{\prime }}\to 0$, local holomorphic frames of $E^{\mathrm{\prime }}$ and $E^{\mathrm{\prime }\mathrm{\prime }}$, with local holomorphic lifts of the latter, give the determinant-line isomorphism $\det E^{\mathrm{\prime }}\otimes \det E^{\mathrm{\prime }\mathrm{\prime }}\cong \det E$: wedging the subbundle frame with the lifted quotient frame is unchanged when a lift is altered by a subbundle section. Applying this to the Euler sequence gives $\det T{\mathbb{P}}^4\cong \mathcal{O}(1{)}^{\otimes 5}=\mathcal{O}(5)$ and hence $K_{{\mathbb{P}}^4}\cong \mathcal{O}(-5)$ as holomorphic line bundles. By [F3], the base is also paracompact Hausdorff, so [F5] supplies a continuous complex-bundle splitting for the later Chern-class calculation; that splitting is not used to infer holomorphicity.

Let $M$ be a connected component of either base ${\mathbb{P}}^4$ or $X$. By [F3], $M$ is paracompact Hausdorff, has CW homotopy type, and admits a homotopy equivalence $q:K\to M$ from a path-connected CW complex $K$. The Chern classes of the smooth bundles over $M$ are defined by [F7]; each bundle is numerable by [F3], and its pullback to $K$ is numerable by [F4]. For any continuous map $f:N\to M$ between the CW-type bases used here, [F8] identifies the pulled-back projective bundle and tautological line. Euler naturality [F9] pulls back the tautological Euler class. Pulling back the unique defining relation [F7] therefore gives a monic relation for $f^{\ast }E$ with coefficients $f^{\ast }{c}_i(E)$; uniqueness in [F7] proves $c_i(f^{\ast }E)=f^{\ast }{c}_i(E)$. Thus Chern classes are natural for both $q$ and $X\hookrightarrow {\mathbb{P}}^4$ in this proof. By [F6], $q^{\ast }:H^{\ast }(M;\mathbb{Z})\to H^{\ast }(K;\mathbb{Z})$ is an isomorphism. Applying the Whitney formula [F1] on $K$ to $q^{\ast }E,q^{\ast }G$ and using this proved pullback naturality shows $q^{\ast }c(E\oplus G)=q^{\ast }(c(E)c(G))$; injectivity of $q^{\ast }$ gives $c(E\oplus G)=c(E)c(G)$ on $M$. This supplies the exact bridge from the CW-base Whitney hypothesis of [F1] to the smooth bases in steps 1.1, 2.1.

The degree-five homogeneous polynomial $f$ is a holomorphic section of $\mathcal{O}(5)$. In a local trivialization, changing its representative to $uf$ for a nowhere-zero holomorphic function $u$ changes $df{\mathrm{\mid }}_X$ to $udf{\mathrm{\mid }}_X$, since $f=0$ on $X$; thus these differentials define a holomorphic bundle map $T{\mathbb{P}}^4{\mathrm{\mid }}_X\to {\mathcal{O}}_X(5)$. Smoothness of the hypersurface makes the map surjective, with kernel $TX$, so $0\to TX\to T{\mathbb{P}}^4{\mathrm{\mid }}_X\to {\mathcal{O}}_X(5)\to 0$ is a holomorphic exact sequence. Its determinant-line isomorphism from step 1.1 gives $\det TX\otimes {\mathcal{O}}_X(5)\cong \det T{\mathbb{P}}^4{\mathrm{\mid }}_X$ and therefore $K_X\cong K_{{\mathbb{P}}^4}{\mathrm{\mid }}_X\otimes {\mathcal{O}}_X(5)\cong {\mathcal{O}}_X$ holomorphically. Each component of $X$ is paracompact Hausdorff by [F3], so [F5] also supplies the continuous splitting needed for the later Chern-class computation.

Write $H_{{\mathbb{P}}^4}=c_1(\mathcal{O}(1))$. The rank-one convention in [F7] gives $c(\mathcal{O}(1))=1+H_{{\mathbb{P}}^4}$. The continuous splittings from [F5] and the Whitney identity from [F1] give $c(T{\mathbb{P}}^4)=(1+H_{{\mathbb{P}}^4}{)}^5$ and $c(TX)c({\mathcal{O}}_X(5))=c(T{\mathbb{P}}^4{\mathrm{\mid }}_X)$; the pullback naturality proved in step 1.2 gives $c(T{\mathbb{P}}^4{\mathrm{\mid }}_X)=(1+H{)}^5$ on each component. Since ${\mathcal{O}}_X(5)={\mathcal{O}}_X(1{)}^{\otimes 5}$, [F2] gives $c_1({\mathcal{O}}_X(5))=5H$, and [F7] gives $c({\mathcal{O}}_X(5))=1+5H$. Hence $c(TX)(1+5H)=(1+H{)}^5$. The complex rank of $TX$ is three, so [F7] gives $c(TX)=1+c_1(TX)+c_2(TX)+c_3(TX)$ with no higher Chern classes. Comparing cohomological degrees two, four, and six yields successively $c_1(TX)=0$, $c_2(TX)=10H^2$, and $c_3(TX)+50H^3=10H^3$, hence $c_3(TX)=-40H^3$. The asserted total class follows without an unproved cohomological-dimension truncation. The numerical normalization ${\int }_X{H}^3=5$ remains the explicit hypothesis and is not inferred from the normal sequence.