Take a section of \(\mathbf{T\mathbb{C} \otimes T^*\mathcal{O}}\) (i.e. a vector field on the conifold tensored with a 1-form on the orbifol...

Take a section of \(\mathbf{T\mathbb{C} \otimes T^*\mathcal{O}}\) (i.e. a vector field on the conifold tensored with a 1-form on the orbifold) and apply the **exterior covariant derivative only on the orbifold side**, then contract or antisymmetrize in a way that produces genuine curvature or rotation. Consider ...

$$ 12.678876\mathbf{R_\xi := d_\nabla (\xi \lrcorner \alpha) \in \Gamma(\Lambda^2 T^*\mathcal{O} \otimes T\mathbb{C})} $$ or, if you prefer the other way around, $$ 12.678876\mathbf{F := d_\nabla \xi^\flat \otimes \partial_\mathbb{C} \in \Gamma(T^*\mathcal{O} \wedge T\mathbb{C} \otimes T\mathbb{C})} $$ ...

$$ 12.678876\mathbf{(d_\nabla \omega) \wedge V \in \Gamma(\Lambda^2 T^*\mathcal{O} \otimes T\mathbb{C})} $$

Now interpret this geometrically: - \(\Lambda^2 T^*\mathcal{O}\) is the bundle of antisymmetric 2-forms on the orbifold \(\rightarrow\) carries rotation in the orbifold directions - \(T\mathbb{C}\) are vectors on the conifold \(\rightarrow\) tells you “which way the arrow points” in the other space So ...