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 ?

Thanks for reading,

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