It’s probably been awhile since you read the first post in this series, so I’ll quickly remind you of the key point. is a smooth scheme over a field . We fixed connection on . Then given a derivation corresponding to , then for any element of , the sheaf of germs of -derivations we can compose the maps we have and we get .
So every connection gives a map , and we had a relation between and , and any such map satisfying the relation comes from a connection.
Now we’ll go back to the construction of the Gauss-Manin connection from last time. We haven’t actually checked that it is a connection or that it is flat. Recall that it is just one of the maps we get from the spectral sequence associated to the filtration of the complex . Now the filtration is compatible with taking wedge products () and the functors are multiplicative, so we have a product structure on the terms of the spectral sequence as follows.
If we take sections of the sheaves over an open, then by . If you want the actual construction see Godement. The product satisfies a few important properties. We have a type of anti-commutativity . Also we know how it behaves under the differential map: .
In particular, let’s look at what this product rule for the differential is for the Gauss-Manin map. For which is really mapping , the differential is really just . Thus that rule says that . So it is a connection!
The curvature is easily seen to be and since the ‘s are maps of a complex we get that it is , and hence is flat and hence the Gauss-Manin connection is integrable. We’ve now proved the theorem that any smooth -morphism of smooth -schemes gives rise to a canonical integrable connection on the relative de Rham cohomology sheaves that is compatible with the cup product.
If you want a more explicit way to see what the map is see the paper, but it is kind of tedious since writing out how it appears in the spectral sequence you will quickly find that it is the connecting homomorphism when taking the long exact sequence after applying the functor to the exact sequence .
This is the third post on this topic, and I haven’t given you a reason to care yet. Here’s why we should care. One would hope (via a conjecture of Grothendieck) that there is some sort of relative de Rham Leray Spectral Sequence: . For the -term to make any sort of sense we needed to be a complex, and since the Gauss-Manin connection is integrable it is a complex. Also, is defined to be .
It turns out that when is affine such a spectral sequence exists. In case you’re wondering, affineness is needed for a nice proof of this because it makes certain cohomologies vanish. Deligne has proved it in a more complicated way when is not affine (but still with our standing assumptions). This is of great importance in proving every K3 surface lifts from characteristic to characteristic .