Let $E$ be a vector bundle (i.e. locally free $\mathcal{O}_X$-module) on some smooth algebraic variety $X$ and let $\nabla: E \to E \otimes \Omega^1_X$ be an integrable connection.
I have seen that $\nabla$ induces a connection $\nabla^\vee$ on the dual vector bundle $E^\vee=\mathcal{Hom}_{\mathcal{O}_X}(E, \mathcal{O}_X)$.
1) How is $\nabla^\vee$ defined?
2) If $X$ is not complete and $\nabla$ has regular singularities along the boundary, is the same true for $\nabla^\vee$?
3) Are the local systems of horizontal solutions of $\nabla$ and $\nabla^\vee$ dual?
Best Answer
The dual connection $\nabla^\vee$ on $E^\vee$ is defined by $$\langle \nabla^\vee \phi, s\rangle = d\langle \phi, s\rangle - \langle\phi, \nabla s\rangle,\quad \forall \phi\in E^\vee, s\in E,$$ where $\langle-,-\rangle:E^\vee\otimes E\rightarrow \mathcal{O}$ is the pairing. One can check that the formula is $\mathcal{O}$-linear in terms of $s$ and that $\nabla^\vee$ satisfies the Leibniz rule.
A connection on $\overline{X}$ with regular singularities along $D=\overline{X}-X$ is the same as the datum of $\nabla_v: E\rightarrow E$ for all vector fields $v\in T_{\overline{X}}$ tangent to $D$. If the original connection $\nabla$ is defined for such vector fields, so is the dual connection. It does not extend to an ordinary connection: on $\mathbf{A}^1$ the dual of the connection $d+\frac{dz}{z}$ on the trivial line bundle is $d-\frac{dz}{z}$.
Yes, this is the case when Riemann-Hilbert is an equivalence, i.e. when $E \cong dR(E)\otimes \mathcal{O}_X$, where $dR(E)$ is the associated local system. Indeed, in this case there are maps between $dR(E^\vee)$ and $dR(E)^\vee$ in both directions (one map is given by the obvious pairing, the other map is given by the $\mathcal{O}$-linear extension of functionals).