Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces

Si Li Harvard University Bong H. Lian Brandeis University Shing-Tung Yau Harvard University


Distinguished Paper Award in 2017

American Journal of Mathematics, 134, (5), 1345-1384, 2012
We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a toric Calabi-Yau hypersurface, and deriving a general formula for the d -exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for Abel-Jacobi map, which has played an important role in recent study of D-branes [25]. Using the variation formalism, we prove that the relative periods of toric B-branes on a toric Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature [6], and discuss the relations between the works [25] [21] [6] in recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.
Picard-Fuchs Equations, Relative Periods, Abel-Jacobi Map, Calabi-Yau Hypersurfaces
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  title={Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces},
  author={Si Li, Bong H. Lian, and Shing-Tung Yau},
  booktitle={American Journal of Mathematics},
Si Li, Bong H. Lian, and Shing-Tung Yau. Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces. 2012. Vol. 134. In American Journal of Mathematics. pp.1345-1384. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160828145039225696480.
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