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Jacobian rings and the infinitesimal Torelli Theorem

Julius Giesler

Abstract

In this article we deal with jacobian rings and identify a mixed Hodge component of a nondegenerate hypersurface in the torus with a lattice geometric quotient vector space. We introduce a period map, study its differential and compute the kernel of the differential much explicitly via certain Laurent polynomials. As an application we deal with the infinitesimal Torelli theorem.

Jacobian rings and the infinitesimal Torelli Theorem

Abstract

In this article we deal with jacobian rings and identify a mixed Hodge component of a nondegenerate hypersurface in the torus with a lattice geometric quotient vector space. We introduce a period map, study its differential and compute the kernel of the differential much explicitly via certain Laurent polynomials. As an application we deal with the infinitesimal Torelli theorem.
Paper Structure (8 sections, 18 theorems, 138 equations, 1 figure)

This paper contains 8 sections, 18 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Jacobian ring of Batyrev
  3. Construction of the components $R_{Int,f}^k$
  4. Hodge and mixed Hodge structures
  5. The Hodge components of $Z_f$ and $Y_f$
  6. Properties of the period map and its differential
  7. Smooth and stable points of $\mathcal{M}(\Delta)$
  8. Explicit description of $d \phi_{f}$

Key Result

Proposition 1.1

Let $\Delta$ be an $n$-dimensional lattice polytope with $l^*(\Delta) > 0$ and a given $f$. Given $\Gamma_1,...,\Gamma_{n+1} \leq \Delta$ with $n_{\Gamma_1},...,n_{\Gamma_{n+1}}$ affine linear independent. Then where $U_{f,k}$ denotes the vector space over $\mathbb{C}$ spanned by with certain Laurent polynomials $g_{\Gamma}(f)$. If $k=2$ these polynomials are linearly independent. Here $\mathop{

Figures (1)

  • Figure 1: Illustration of the construction of $R_{Int,f}^2$ for a simplex $\Delta = \langle v_1,v_2,v_3 \rangle$ in dimension $2$. The shaded regions do not belong to $R_{Int,f}^2$, there are $4$ points left in $R_{Int,f}^2$.

Theorems & Definitions (54)

  • Proposition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 3.1
  • Proposition 3.3
  • ...and 44 more