Hi,


just a short question on the theory of D-Modules:


if one has $p:X\times Y \rightarrow Y$ the projection of two smooth projective complex varieties to the second factor, then what is a $p^{\*}D\_Y$-Module?


Is this the same as a vectorbundle on $X\times Y$ with an integrable connection relative (!) to $X$?



Pullback of D-Modules
This was basically answered in the comments. A $p^\*D\_Y$-module is a vector bundle with an integrable relative connection on $X \times Y$ if and only if the underlying $\mathcal{O}\_{X \times Y}$-module is locally free, i.e., a vector bundle. In other words, if you are given a vector bundle on $X \times Y$, an action of $p^\*D\_Y$ is essentially the same data as an integrable relative connection. 


You can find a discussion of some of these ideas in [these notes on crystals](http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19%28Crystals%29.pdf). The action of $p^\*D\_Y$ is equivalent to an identification of infinitesimally nearby fibers in the $Y$ direction.


