Let $\pi: Y \to X$ be a smooth projection map between manifolds, and assume the fibres of the map have constant dimension: dim$\left(\pi^{-1}(x)\right)=d ~~ \forall x \in X$. Given a smooth map between manifolds, one can define the pushforward or direct image $\pi_*V$ of a vector bundle $V$. Formally one thinks of the vector bundle as a sheaf and then defines the direct image as for a sheaf.

My question is how to actually compute the result for common, simple vector bundles. Examples are the trivial vector bundle $\mathcal{O}_Y$, the tangent $T_Y$ or cotangent bundle $T_Y^*$, or the canonical $K_Y$ or anti-canonical bundle $K_Y^*$.

I would be interested to know how to work these out using the formal definition of the direct image, or how to cleverly avoid this explicit computation. Even better would be to understand also the answers in the case of the higher direct images $R^p\pi_*V$.

  • $\begingroup$ I know of pulling back a bundle, but how does pushing a bundle forward work in general? $\endgroup$ – Jason DeVito Aug 26 '18 at 19:48
  • $\begingroup$ @JasonDeVito Since bundles can be regarded as sheaves, the pushforward or direct image of a bundle can be defined as for a sheaf. Roughly, the direct image of a sheaf $F$ under a map $f$ is defined by $f_*F(U):=F(f^{-1}(U))$, where $U$ is an open set. $\endgroup$ – diracula Aug 26 '18 at 22:08
  • $\begingroup$ I am only vaguely familiar with the idea of a sheaf. In particular, I can guess how to get a sheaf out a of bundle (using sections), but am at a loss as to how to go backwards. In an event, it is clear I won't be able to help with the question, but I look forward to trying to understand the answers! $\endgroup$ – Jason DeVito Aug 27 '18 at 0:02
  • $\begingroup$ What kind of answer are you looking for(local charts)? what do you mean by projection ($X \subset Y$ such that $X =\pi \circ \pi(Y)$ and $\pi_{\restrection X}=Id$ )? / $\endgroup$ – Elad Jun 19 '19 at 12:22
  • $\begingroup$ @Elad Yes those are properties I expect a projection to have; examples I have in mind include the projection of a product space onto one component, the projection in a fibre bundle, or the projection in a fibration. I am interested in any methods to compute the results, for example explicitly using local charts or using heavier machinery to avoid this. $\endgroup$ – diracula Jun 19 '19 at 21:07

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