nLab Picard-Fuchs equation

Idea

Picard-Fuchs equation is a linear ordinary differential equation of degree $2g$ for $2g$ periods of a nonsingular projective curve of genus $g$ over a field $K$ of characteristic $0$. For curves over complex numbers it has been first derived by Lazarus Fuchs. It may be more abstractly defined using a Gauss-Manin connection which makes sense in a more general context (beyond curves and algebraic/analytic geometry).

Literature

A description in the language of algebraic geometry is in the introduction to

• Nicholas M. Katz, On the differential equations satisfied by period matrices, Publications Mathématiques de l’IHÉS 35 (1968) 71–106 numdam
• Pierre Deligne, Equations différentielles à points singuliers réguliers, Lecture Notes in Math. 163, Springer-Verlag 1970

Mirror maps for families of Calabi-Yau are also obtained from corresponding Picard-Fuchs equations.

See also

• wikipedia Picard-Fuchs equation

Doran has clarified in which cases the corresponding mirror map is the Hauptmoduln (j-function) for the elliptic curve:

• Charles F. Doran, Picard-Fuchs uniformization: modularity of the mirror map and mirror-moonshine, arxiv:math.AG/9812162

category: algebraic geometry