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maxo
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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-1$$k^2-1$.

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 $k-1$.

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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LSpice
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trace Trace map for universal bundle of Grassmannian

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maxo
maxo
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trace map for universal bundle of Grassmannian

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 $k-1$.