Castelnuovo–de Franchis theorem explained

In mathematics, the Castelnuovo–de Franchis theorem is a classical result on complex algebraic surfaces. Let X be such a surface, projective and non-singular, and let

ω₁ and ω₂be 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 ω₁ and ω₂ on C such that

φ*(₁) = ω₁ and φ*(₂) = ω₂.

This result is due to Guido Castelnuovo and Michele de Franchis (1875–1946).

The converse, that two such pullbacks would have wedge 0, is immediate.

See also

• de Franchis theorem

References

• .
• Moduli and classification of irregular Kaehler manifolds (And algebraic varieties) with Albanese general type fibrations . Inventiones Mathematicae . 1991 . 104 . 2 . 263–290 . Catanese . Fabrizio . 10.1007/BF01245076 . 122748633 .