Jacobian rings and the infinitesimal Torelli Theorem

1. Problem Statement

The paper addresses the Infinitesimal Torelli Theorem (ITT) for nondegenerate hypersurfaces $Z_f = \{f=0\}$ inside an algebraic torus $T \cong (\mathbb{C}^*)^n$. The ITT asks whether the period map, which associates a hypersurface to its Hodge structure, is locally injective. Specifically, the author investigates the injectivity of the differential of the period map restricted to the image of the Kodaira-Spencer map.

The core challenge lies in explicitly computing the kernel of the differential of the period map ($dP_{B,f}$) and determining whether this kernel vanishes when restricted to the image of the Kodaira-Spencer map ($\kappa_f$). While the ITT is well-understood for smooth projective hypersurfaces in $\mathbb{P}^n$, the case for toric hypersurfaces (defined by Laurent polynomials with specific Newton polytopes) requires a more nuanced approach involving mixed Hodge structures and lattice geometry.

2. Methodology

The author employs a combination of toric geometry, combinatorial commutative algebra, and Hodge theory. The methodology proceeds through the following steps:

• Jacobian Rings and Interior Modules: The paper utilizes Batyrev's construction of the Jacobian ring $R_f = S_\Delta / J_{\Delta, f}$, where $S_\Delta$ is a graded algebra associated with the Newton polytope $\Delta$ and $J_{\Delta, f}$ is the Jacobian ideal generated by logarithmic derivatives of the defining polynomial. The author focuses on the interior module $R_{Int, f}$, which corresponds to the minimal weight mixed Hodge components of the cohomology of the hypersurface.
• Explicit Generators: A key methodological innovation is the construction of explicit generators for the Jacobian ideal using facet-dependent Laurent polynomials $g_\Gamma(f)$. Instead of relying solely on the standard partial derivatives, the author defines $g_\Gamma(f)$ based on the facets $\Gamma$ of the polytope $\Delta$. This allows for a more granular description of the relations in the quotient ring.
• Kernel Computation via Lattice Geometry: The differential of the period map is analyzed by factoring it through the Kodaira-Spencer map $\kappa_f$ and a cup-product map $\Phi_f$. The author explicitly computes the kernel of the differential of the period map ($\ker(d\phi_f)$) by identifying specific Laurent polynomials $g_\Gamma(f) \cdot x^w$ that vanish in the quotient. This involves analyzing the support of these polynomials relative to the interior of the polytope and its dilations.
• Combinatorial Contradictions (White's Theorem): To prove injectivity (i.e., that the kernel is trivial), the author uses a proof by contradiction involving empty simplices. If a non-trivial element exists in the kernel, it implies the existence of a specific geometric configuration of lattice points (a polytope $Q$) that violates known theorems regarding the number of lattice points in empty 3-simplices (specifically White's Theorem).

3. Key Contributions

A. Representation of Interior Modules

The paper provides a new representation of the graded components of the interior module $R^k_{Int, f}$ as a quotient vector space.

• Proposition 1.1: It establishes that $R^k_{Int, f} \cong L^*(k \cdot \Delta) / U_{f,k}$, where $U_{f,k}$ is spanned by specific products of the facet polynomials $g_\Gamma(f)$ and monomials $x^v$. This offers a concrete computational tool for determining the dimensions of Hodge components.

B. Explicit Description of the Kernel

The central result is Theorem 1.2 (Theorem 8.4), which gives a precise description of the kernel of the differential of the period map modulo the Lie algebra of the torus action:
$$\ker(dP^k_{B,f}) \cong \langle g_\Gamma(f) \cdot x^w \mid \Gamma \leq \Delta \text{ is a facet}, \dots \rangle$$
The theorem characterizes the kernel in terms of the inner facet normals $n_\Gamma$ and the lattice points $w$. Specifically, it distinguishes cases based on the pairing $\langle w, n_\Gamma \rangle$:

• $0$: The element is zero.
• $-1$: The element lies in the kernel of the Kodaira-Spencer map ($\ker \kappa_f$).
• $\leq -2$: These are "exceptional cases" that could potentially cause the failure of the ITT.

C. Proof of the Infinitesimal Torelli Theorem

The paper proves that for nondegenerate hypersurfaces satisfying specific lattice conditions (where the interior lattice points are not contained in a hyperplane), the map $\Phi_f$ restricted to the image of $\kappa_f$ is injective.

• Corollary 1.3 / Corollary 8.16: $\ker(\Phi_f|_{\text{Im } \kappa_f}) = \{0\}$.
• The proof relies on showing that the "exceptional case" ($\langle w, n_\Gamma \rangle \leq -2$) leads to a contradiction. By constructing a 3-dimensional polytope $Q$ from the lattice points involved, the author shows that $Q$ must contain at least 7 lattice points (by White's Theorem), whereas the assumption of a non-trivial kernel implies $Q$ is empty or has fewer points. This contradiction forces the kernel to be trivial.

D. New Proof for Projective Hypersurfaces

As a byproduct, the author provides a new proof of the ITT for smooth projective hypersurfaces $Y_f \subset \mathbb{P}^n$ of degree $d \geq n+2$ (where $n \geq 3$), utilizing the surjectivity of the Kodaira-Spencer map in this context.

4. Key Results

1. Factorization of the Differential: The differential of the period map is explicitly factored through the Kodaira-Spencer map and a map $\Phi_f$ acting on cohomology groups, allowing for a direct comparison of their kernels.
2. Kernel Equality: Under the theorem's assumptions, $\ker(dP^k_{B,f}) = \ker(dP^1_{B,f})$ for all $k \geq 1$. This simplifies the analysis of higher-order deformations.
3. Injectivity Condition: The ITT holds for the family of toric hypersurfaces $X \to B$ if the Newton polytope $\Delta$ satisfies the condition that $\text{Int}(\Delta) \cap M$ is not contained in a hyperplane.
4. Counterexamples Context: The paper notes that the dimension of $\ker(dP_{B,f})$ can depend on the specific polynomial $f$ (unlike $\ker \kappa_f$), and references known counterexamples (Kanev and Todorov surfaces) where the ITT fails due to specific lattice configurations (where $\text{Int}(\Delta) \cap M = \{(0,0,0)\}$ but the boundary dimension is high).

5. Significance

• Computational Explicitness: The paper moves beyond abstract existence proofs to provide explicit formulas for the generators of the Jacobian ideal and the kernel of the period map differential. This makes the theory computationally accessible for specific toric varieties.
• Bridge between Combinatorics and Hodge Theory: It deepens the connection between the combinatorial geometry of lattice polytopes (specifically the distribution of interior points and facets) and the analytic properties of the period map.
• Resolution of the ITT for Toric Hypersurfaces: It establishes a robust criterion for when the Infinitesimal Torelli Theorem holds for nondegenerate hypersurfaces in tori, extending the classical results from projective space to the toric setting.
• Foundation for Generic Torelli: The author suggests that these results on the infinitesimal level can be extended to the Generic Torelli Theorem by considering mixed period maps on the full mixed Hodge structure, not just the minimal weight components.

In summary, Giesler's work provides a rigorous, combinatorially grounded framework for analyzing the variation of Hodge structures on toric hypersurfaces, successfully proving the Infinitesimal Torelli Theorem under broad geometric conditions by leveraging lattice point counting arguments.