Castelnuovo–de Franchis theorem

Short description: When differentials on an algebraic surface represent as a pullback of an algebraic curve

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 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.

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

• Coen, S. (1991), Geometry and Complex Variables, Lecture Notes in Pure and Applied Mathematics, 132, CRC Press, p. 68, ISBN 9780824784454, https://books.google.com/books?id=tIDzJMEQOQUC&pg=PA68 .
• Catanese, Fabrizio (1991). "Moduli and classification of irregular Kaehler manifolds (And algebraic varieties) with Albanese general type fibrations". Inventiones Mathematicae 104 (2): 263–290. doi:10.1007/BF01245076. http://eudml.org/doc/143883.