# Tangent bundle of product of projective spaces

A product of projective spaces $\mathbb{P}^r \times \mathbb{P}^s$ has Chow ring $A(\mathbb{P}^r \times \mathbb{P}^s) \cong \mathbb{Z}[\alpha, \beta]/(\alpha^{r+1}, \beta^{s+1})$, where $\alpha$ and $\beta$ are the pullbacks of the hyperplane classes, respectively (see e.g. 3264 & All That, Products of Projective Spaces).

How can we identify the tangent bundle of $\mathbb{P}^r \times \mathbb{P}^s$? Can we express it in terms of other bundles? Is there an analogue here of the Euler sequence?

How can we describe the positions in the Chow ring of the Chern classes $c_i(\mathcal{T}_{\mathbb{P}^r \times \mathbb{P}^s})$, in particular $c_2(\mathcal{T}_{\mathbb{P}^r \times \mathbb{P}^s})$?

Any help is appreciated! Thanks.

• The tangent bundle should be the sum of the pullbacks of the tangent bundles of the factors. This is true in general.
– Hoot
Aug 8, 2016 at 19:35
• More generalities: you can get the action of the Chern classes of bundles as above on stuff that looks like $\alpha \times \beta$ inside $A_*(X\times Y)$. In general this doesn't seem like enough since there's no Kunneth formula, but here everything is so simple that it seems like this should work. Anyway, Sasha's answer seems like the best thing.
– Hoot
Aug 8, 2016 at 19:49

First, $$T_{P^r \times P^s} \cong p^*T_{P^r} \oplus q^*T_{P^s},$$ where $p$ and $q$ are projections. Pulling back Euler sequences from the factors, one gets $$0 \to O \oplus O \to O(\alpha)^{r+1} \oplus O(\beta)^{s+1} \to T_{P^r \times P^s} \to 0.$$ One can compute Chern classes from any of the above formulas.