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Periods of Limiting Mixed Hodge Structures of Projective Hypersurfaces

Masanori Asakura, Saiei-Jaeyeong Matsubara-Heo

Abstract

For a generic one-parameter degeneration of projective hypersurfaces, we show that the periods of the limiting mixed Hodge structure are generated by certain special values of logarithm, Gamma and Dirichlet $L$-functions. Our proof is based on the analytic continuation of solutions to the GKZ system.

Periods of Limiting Mixed Hodge Structures of Projective Hypersurfaces

Abstract

For a generic one-parameter degeneration of projective hypersurfaces, we show that the periods of the limiting mixed Hodge structure are generated by certain special values of logarithm, Gamma and Dirichlet -functions. Our proof is based on the analytic continuation of solutions to the GKZ system.
Paper Structure (18 sections, 30 theorems, 190 equations, 1 figure)

This paper contains 18 sections, 30 theorems, 190 equations, 1 figure.

Table of Contents

  1. Introduction
  2. GKZ systems
  3. GKZ systems and secondary fans
  4. Series solutions of GKZ systems
  5. GKZ-systems and Periods of Projective Hypersurfaces
  6. Periods of Fermat deformations
  7. Periods of Fermat varieties
  8. Preliminaries on Fermat deformations
  9. Periods of Fermat deformations
  10. Limiting Periods of Degeneration of Hypersurfaces
  11. Review of limiting mixed Hodge structures
  12. Main Theorem
  13. Proof of Theorem \ref{['thm:main']}
  14. Step 1
  15. Step 2
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose $\boldsymbol{w}_f\not\in \mathrm{Sk}({\mathrm{Fan}}(A))$. Then, the limiting periods of $H^{n-2}_\infty(X_f/\Delta)_K$ lie in the $\overline{K}$-subalgebra of $\mathbb{C}$ generated by special values where $j$ and $k$ run over integers such that $0<j<N_A$ and $2\leq k$, and $\chi$ runs over Dirichlet characters of conductors dividing $N_A$.

Figures (1)

  • Figure 1: Regular triangulations $T({\rm Fer})$ and $T(\boldsymbol{a}_1)$.

Theorems & Definitions (59)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 49 more