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Suppose $\mathcal E$ is a bundle over a rational homogeneous smooth variety $X\subset \mathbb P^n$ such that the zero-locus $Z_s$ of a generic global section $s$ is a reduced zero-dimensional scheme. Then this bundle admits a resolution given by the Koszul complex $$ 0\to \bigwedge^r\mathcal E^\ast\xrightarrow{\phi_r}\bigwedge^{r-1}\mathcal E^\ast\xrightarrow{\phi_{r-1}}\dots\xrightarrow{\phi_1}\mathcal E^\ast\to I_{Z_s}\to 0. $$

I am interested if there are standard techniques to describe the induced map $(\phi_i)_*:H^{m}(\bigwedge^i \mathcal E^\ast)\to H^m(\bigwedge^{i-1} \mathcal E^\ast) $ from the explicit knowledge of the global sections, $H^m(\bigwedge^{i} \mathcal E^\ast), H^m(\bigwedge^{i-1} \mathcal E^\ast)$, and the Koszul maps.

Many thanks in advance.

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  • $\begingroup$ I don't understand the question. The map $\phi_i$ is explicit (interior product with $s$), so don't we know how it acts on global sections? $\endgroup$
    – abx
    Commented Jan 10 at 10:38
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    $\begingroup$ We know how it acts on the global sections, but I am not sure how to translate this to higher cohomologies. I know how to understand what $H^m(\bigwedge^i\mathcal E^\ast)$ is via Bott's theorem, but I cannot see how to translate this map, that I know how it works on the global section level, to higher cohomologies. $\endgroup$
    – ett
    Commented Jan 10 at 10:49
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    $\begingroup$ Sorry, I thought you were asking about $H^0$. I don't think Bott's theorem tells you anything about $H^m$ (or $H^0$, for that matter). $\endgroup$
    – abx
    Commented Jan 10 at 11:18
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    $\begingroup$ If you look at the proof of the Borel--Bott--Weil Theorem (e.g., in the paper of Demazure) you will see that $H^m$ iis identified, eventually, with $H^0$ of a different bundle, and typically $s$ induces a morphism between those. $\endgroup$
    – Sasha
    Commented Jan 10 at 14:35

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