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$I$-Maximal Variation of Hodge Structure and Jacobian Rings

Mounir Nisse

Abstract

We investigate higher--order variation of Hodge structure for families of smooth hypersurfaces and complete intersections through the notion of $I$--maximal variation. Using Griffiths' description of primitive cohomology, we interpret the infinitesimal variation of Hodge structure and the $n$--fold Yukawa coupling as graded multiplication maps in the Jacobian ring. Our main result shows that the Strong Lefschetz property of the Jacobian ring provides the algebraic mechanism ensuring $I$--maximal variation. In particular, we prove that smooth hypersurfaces of degree $d\ge n+2$ and smooth complete intersections with $κ>0$ exhibit $I$--maximal variation. We further establish that for complete intersections of general type the infinitesimal Torelli property is equivalent to the nondegeneracy of the Yukawa coupling. Finally, we analyze degenerations and show that the failure of the Strong Lefschetz property leads to degeneration of the Yukawa coupling and the loss of $I$--maximal variation. These results identify the Lefschetz property of the Jacobian ring as the fundamental algebraic structure governing maximal variation of Hodge structure.

$I$-Maximal Variation of Hodge Structure and Jacobian Rings

Abstract

We investigate higher--order variation of Hodge structure for families of smooth hypersurfaces and complete intersections through the notion of --maximal variation. Using Griffiths' description of primitive cohomology, we interpret the infinitesimal variation of Hodge structure and the --fold Yukawa coupling as graded multiplication maps in the Jacobian ring. Our main result shows that the Strong Lefschetz property of the Jacobian ring provides the algebraic mechanism ensuring --maximal variation. In particular, we prove that smooth hypersurfaces of degree and smooth complete intersections with exhibit --maximal variation. We further establish that for complete intersections of general type the infinitesimal Torelli property is equivalent to the nondegeneracy of the Yukawa coupling. Finally, we analyze degenerations and show that the failure of the Strong Lefschetz property leads to degeneration of the Yukawa coupling and the loss of --maximal variation. These results identify the Lefschetz property of the Jacobian ring as the fundamental algebraic structure governing maximal variation of Hodge structure.
Paper Structure (14 sections, 236 equations)

This paper contains 14 sections, 236 equations.

Table of Contents

  1. Introduction
  2. Generalizing $I$--Maximal Variation from Curves to Surfaces
  3. I--Maximal Variation for Smooth Surfaces in $\mathbb{P}^3$
  4. Strong Lefschetz Property for the Jacobian Ring and Maximal Variation for Surfaces in $\mathbb{P}^3$
  5. Strong Lefschetz Property and Maximal Yukawa Coupling for Smooth Hypersurfaces
  6. Higher--Dimensional Variation and Yukawa Couplings for Smooth Hypersurfaces
  7. $I$--Maximal Variation for Complete Intersections
  8. Infinitesimal Torelli and $I$--Maximal Variation for Complete Intersections
  9. The Calabi--Yau Case and Degeneration of $I$--Maximal Variation
  10. The Calabi--Yau Case and Degeneration of $I$--Maximal Variation
  11. Failure of the Strong Lefschetz Property and Degeneration of Yukawa Couplings
  12. Degenerations of Hypersurfaces and Rank Drop of the Yukawa Coupling
  13. Milnor Algebras and Rank Drop of the Yukawa Coupling
  14. Limiting Mixed Hodge Structures and Degeneration of the Yukawa Coupling

Theorems & Definitions (36)

  • proof
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  • Definition 6.1
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  • Definition 7.1
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  • Definition 8.1
  • Definition 8.2
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  • ...and 26 more