nLab Picard-Fuchs equation

Redirected from "Picard-Fuchs equations".
Contents

Contents

Idea

Picard-Fuchs equation is a linear ordinary differential equation of degree 2g2g for 2g2g periods of a nonsingular projective curve of genus gg over a field KK of characteristic 00. 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

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

Last revised on October 15, 2023 at 16:20:19. See the history of this page for a list of all contributions to it.