Question

In the paper hep-th/9712042v2, p. 20, the following setup is given:

A complex manifold M and an n+1-dimensional vector bundle V on it. V has an underlying real bundle $V_{\mathbb{R}}$ with a flat connection $\nabla$. Also, there is a holomorphic inclusion of a line bundle $L$ in $V$.

Finally, we have the section $M \to \mathbb{P}[(V_{\mathbb{R}})_{\mathbb{C}}],\ m \mapsto L_m$

My question is what is meant by the statement that this section is assumed to be "an immersion with respect to $\nabla$"? I just can't make sense of how the existence of a connection has something to do with an immersion condition for the section.

Answer 1

Just read further down the same page: "The immersion condition in (b) states that $m \to L_m$ is an immersion into the 2n + 1 dimensional projective space of a local trivialization of $P(V_R)_C$." You can also say it without reference to the flatness of the connection: the section is transverse to the horizontal distribuition of the connection. Ie, L, thought of as a section of $P(V_R)_C$, is "maximally non parallel". Etc etc.