Let $E \to X$ be a vector bundle over a decent space $X$. Then there is a space $Z$ together with a map $p: Z \to X$ which induces a split injection on cohomology and such that $p^* E$ splits as a direct sum of line bundles (take e.g. the flag bundle of $E$). Is the analog true for holomorphic vector bundles (if we stay purely in the category of complex manifolds)? That is, if $X$ is a complex manifold and $E$ a holomorphic vector bundle, can we get a holomorphic map $p: Z \to X$ (with $Z$ a complex manifold) with the same properties: the map on cohomology is a split injection, and $p^*E$ splits in the holomorphic category as a sum of line bundles?
(As a side question, I'm curious what additional invariants one can construct for holomorphic vector bundles, which don't make sense for an ordinary complex vector bundle. I'm vaguely aware of the Atiyah class, but are there other examples?)
Best Answer
The answer is positive. Let $P$ be the principal $\mathrm{GL}_n$-bundle associated with $E$; then the space of flags is the quotient $P/B$, where $B$ is the Borel subgroup of $\mathrm{GL}_n$ consisting of upper triangular matrices. Set $Z = P/T$, where $T$ is the maximal torus consisting of diagonal matrices. A point of $Z$ is a point of $X$, plus $n$ independent 1-dimensional linear subspaces of the fiber of $E$. The projection $Z \to P/B$ is a fibration with contractible fibers, hence the pullback from the cohomology of $P/B$ to that of $Z$ is an isomorphism. Since the cohomology of $X$ injects into the cohomology of $P/B$, it also injects into the cohomology of $Z$.