Leray-Serre spectral sequence for projective bundles

Question

Let $\mathcal{E} \rightarrow X$ be a complex vector bundle of rank $r+1$ and let $F=\mathbb{P}^r \rightarrow E = \mathbb{P}\mathcal{E}\rightarrow X$ be the associated projective bundle. We know that under the assumption that $H^{\bullet}(E) \rightarrow H^{\bullet}(F)$ is surjective, by the Leray-Serre spectral sequence we have for any field $k$ $$H^{\bullet}(E, k) \simeq H^{\bullet}(X,k) \otimes H^{\bullet}(F,k)$$ Where the tensor product on the right is graduated. Can we say anything more about the algebra structure of left hand side when we take $\mathbb{Z}$ coefficients, maybe in terms of the Chern classes of the bundle, unravelling the definitions of the Leray-Serre spectral sequence?

Answer 1

As you say, if the map $H(E)\to H(F)$ is surjective, the Leray-Serre spectral sequence degenerates at the $E^2$-page which is therefore isomorphic to the $E^\infty$-page. Unwrapping the definitions, this means that there is a filtration on $H^*(E)$ such that the associated graded is $H^*(X)\otimes H^*(F)\cong H^*(X)[x]/x^{r+1}$. Choosing a representative of the class $x$, we obtain that $H^*(E)$ is generated as a $H^*(X)$-algebra (with $H^*(X)$ in filtration degree $0$) by a single element $x$ in degree and filtration degree $2$ such that $x^{r+1}$ sits in degree less than $2r+2$. This can only happen if there is a relationship of the form $x^{r+1} = \sum_{k=0}^r p^*\gamma_{r+1-k}x^k$ with $\gamma_k\in H^{2k}(X)$. Determining the concrete values of the $c_k$ amounts to solving the extension problem of the Leray-Serre spectral sequence, which is in a sense precisely the part of the problem the spectral sequence doesn't capture. However, in this case we can solve this problem geometrically: By construction, the pullback of $\mathcal E$ to $E$ splits off a line bundle $L$, and we may take $x = -c_1(L)$. Since $p^*\mathcal E/L$ is $r$-dimensional, its $(r+1)$-th Chern class vanishes, and the sum formula for the total Chern class yields $$ c(p^*\mathcal E/L) = c(p^*\mathcal E)(1-x)^{-1}. $$ Expanding the second factor as $\sum_{k=0}^r x^k$ and taking the resulting expression for the $(r+1)$-th Chern class yields $\gamma_k = c_k(\mathcal E)$. In fact, this can be used to define Chern classes for general complex oriented cohomology theories.