Let $E\to X$ be a rank $r$ holomorphic vector bundle on a $n$-dimensional compact complex manifold. Then, it is well known that one can recover the Segre classes of $E$ as follows. Let $\pi\colon P(E)\to X$ the projectivized bundle of lines of $E$, and $\mathcal O_E(1)\to P(E)$ the corresponding (anti)tautological line bundle.

The cohomology ring $H^\bullet(P(E),\mathbb Z)$ is then given by $$ H^\bullet(P(E),\mathbb Z)\simeq H^\bullet(X,\mathbb Z)[\xi]/(\xi^r+\pi^*c_1(E)\cdot\xi^{r-1}+\cdots+\pi^*c_r(E)), $$ where $\xi=c_1(\mathcal O_E(1))$. Therefore, since $\pi_*\xi^{r-1}=1$, and the total Chern class $c_\bullet(E)$ is the formal inverse of the total Segre class $s_\bullet(E)$, one can show that $$ \pi_*\xi^{r-1+k}=s_k(E). $$ Now, Segre classes are just particular Schur polynomials in the Chern classes of $E$, namely those corresponding to the partitions of the form $1+\cdots+1$.

**Question.** Is there an analogous geometric construction to obtain all the Schur polynomials in the Chern classes of $E$ as a direct image of (the appropriate self-intersection of) the first Chern class of a "tautological" line bundle?

For instance, what happens if we consider the (complete or an incomplete) flag manifold of $E$ with its tautological line bundles?

Thanks in advance.