Castelnuovo–de Franchis theorem
- ω1 and ω2
be two differentials of the first kind on X which are linearly independent but with wedge product 0. Then this data can be represented as a pullback of an algebraic curve: there is a non-singular algebraic curve C, a morphism
- φ: X → C,
and differentials of the first kind ω′1 and ω′2 on C such that
- φ*(ω′1) = ω1 and φ*(ω′2) = ω2.
The converse, that two such pullbacks would have wedge 0, is immediate.