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Results tagged with cotangent-bundles
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user 4428
10
votes
Accepted
Cohomology of the tangent sheaf of $\mathbb{P}(1,2,3)$
If you think about $P(1,2,3)$ as about stack then there is an analogue of the Euler sequence
$$
0 \to O \to O(1) \oplus O(2) \oplus O(3) \to T \to 0.
$$
It allows to compute $h^1 = h^2 = 0$ and $h^0 = …
- 39.3k
4
votes
Accepted
Tangent bundle for orthogonal and isotropic Grassmannians
The tangent bundle to the orthogonal Grassmannian fits into an exact sequence
$$
0 \to T_{\mathrm{OG}(k,V)} \to \mathcal{S}^\vee \otimes \mathcal{Q} \to S^2\mathcal{S}^\vee \to 0.
$$
Taking into accou …
- 39.3k
3
votes
Accepted
Normal bundle of veronese as iteration extension of symmetric powers
Your third exact sequence is incorrect --- the correct form is
$$
0 \to S^{d-1}V \otimes \mathcal{O}(d-1)
\to S^{d}V \otimes \mathcal{O}(d)
\to S^dT
\to 0.
$$
- 39.3k