Castelnuovo–de Franchis theorem

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: XC,

and differentials of the first kind ω1 and ω2 on C such that

φ*(ω1) = ω1 and φ*(ω2) = 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

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References

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  • Coen, S. (1991), Geometry and Complex Variables, Lecture Notes in Pure and Applied Mathematics, vol. 132, CRC Press, p. 68, ISBN 9780824784454.
  • 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. S2CID 122748633.