Let $G := G(k,V)$ denote the Grassmannian of $k$-linear subspaces in a $\mathbb{C}$-vector space $V$ of dimension $n$. Let $S$ denote the tautological bundle over $G$. There is a canonical map on sheaf of sections $$S \otimes S^\vee \to \mathcal{O}_G$$ by evaluative pairing of sections of $S$ and $S^\vee$. What can be said about the kernel of this map? Certainly it is of rank $k^2-1$.
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1$\begingroup$ You mean $k^2-1$, right? $\endgroup$– abxCommented Jan 30 at 17:44
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$\begingroup$ Yes, apologies -- corrected and thank you! $\endgroup$– maxoCommented Jan 30 at 20:39
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3$\begingroup$ What do you want to know? You can compute its Chern classes by the splitting principle. You can apply Borel-Weil-Bott / Bott vanishing to say something about sheaf cohomology of twists. $\endgroup$– Jason StarrCommented Jan 31 at 0:16
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