# Schubert varieties and zeroes of holomorphic vector fields

by AG14   Last Updated September 11, 2019 09:20 AM - source

I'm trying to make a link between Schubert varieties of a flag manifold G/P (complex, with invariant complex structure fixed), and some zero divisor of some holomorphic vector fields spanning TG/P.

For example, in the case of $$\mathbb{P}^n = G/P$$ (with $$G:=SL_{n+1}$$ and $$P$$ the parabolic subgroup which stabilize $$e_1 \in \mathbb{C}^{n+1}$$), we can consider exponential maps : $$\psi_{i} : \mathfrak{m} \simeq U_i$$ when we choose $$\tau_i \in W(G)$$, corresponding to affine charts on $$\mathbb{P}^n$$. For example, i've taken $$\tau_i = \sigma_{\alpha_i}$$ where $$\alpha_1,\ldots,\alpha_n$$ is a basis of the complementary roots $$\Lambda_P$$. Let's define the vector field on $$U_i$$, $$X_i \in H^0(U_i,TG/P)$$ defined by : $$X_i(\cdot)=\tilde{X}_{\alpha_i}(\psi_{i}^{-1}(\cdot))$$ where $$\tilde{X}_{\alpha_i}$$ stand for the left invariant vector field associated to the element $$X_{\alpha_i}$$ in a Chevalley basis. It then seems to me that it can be extend as a global vector field with zero on $$s_{\tau_i}$$ (with the notations of Bernstein-Gelfand).

1. Does it make sense?
2. Can it be extend to any flag manifold ?
3. Is there a reference which gives a system of global vector fields of $$G/P$$ with a decomposition of its zero divisor in terms of Schubert varieties ?