As far as I understand Deligne's far reaching generalisation of Cech cohomology is called cohomological descent and is used to endow any variety with a (mixed) Hodge structure. Again, AFAIU, the idea is to resolve the variety, then take intersections of the exceptional pieces, resolve those and so on.

As you can see I don't understand it well, so can someone please help? Also, what is the most ridiculously easy example one can have? (I guess the thing I'm interested in most is very simple examples illustrating the possible behaviours)

The only places I know that discuss this are an SGA, Brian Conrad's notes and Peters-Steenbrink.

As far as I understand Deligne's far reaching generalisation of Čech cohomology is called cohomological descent and is used to endow any variety with a (mixed) Hodge structure. Again, AFAIU, the idea is to resolve the variety, then take intersections of the exceptional pieces, resolve those and so on.

As you can see I don't understand it well, so can someone please help? Also, what is the most ridiculously easy example one can have? (I guess the thing I'm interested in most is very simple examples illustrating the possible behaviours)

The only places I know that discuss this are an SGA, Brian Conrad's notes and Peters-Steenbrink.