IVHS: Infinitesimal Variation of Hodge Structure

• IVHS is a framework that describes the first-order deformation of Hodge structures, offering a linear-algebraic view of period maps and moduli spaces.
• It employs the Kodaira–Spencer map and cup product to connect tangent directions with differential forms, enabling precise residue computations at singularities.
• Recent advances extend IVHS via logarithmic geometry and Jacobian ring techniques to study equisingular families and to achieve maximal variation criteria.

The infinitesimal variation of Hodge structure (IVHS) encodes the first-order deformation of the Hodge filtration on the cohomology of an algebraic variety as the variety moves in a family. Originating in the work of Griffiths, IVHS provides a linear-algebraic description of how the Hodge structure varies infinitesimally under deformations of complex structures, and plays a central role in understanding period maps, Torelli theorems, and the local geometry of moduli spaces. For singular varieties, IVHS requires a careful analysis of normalization, deformation theory, and residue computations, with sharp criteria determining when maximal variation occurs. Recent advances extend the scope of IVHS far beyond smooth curves and hypersurfaces, providing residue- and logarithmic-geometry-based frameworks applicable to families with isolated singularities and to equisingular settings.

1. Formal Definition of IVHS in Smooth and Singular Settings

Let $\pi : \mathcal{X} \to \mathcal{B}$ be a flat family of reduced projective curves with only isolated singularities, and denote by $\nu_b : C_b \to \mathcal{C}_b$ the normalization of a fiber. Over the smooth locus, the local system $\{H^1(C_b, \mathbb{C})\}$ carries a polarized variation of Hodge structures (VHS) of weight 1, with Hodge filtration $F^1_b = H^0(C_b,\omega_{C_b})$.

The IVHS at a point $b \in \mathcal{B}$ is the differential of the period map: $$\Phi : T_b\mathcal{B} \longrightarrow \operatorname{Hom}(F^1_b, H^1(C_b,\mathcal{O}_{C_b})) \cong H^1(C_b,\mathcal{O}_{C_b}) \otimes (H^0(C_b, \omega_{C_b}))^\vee.$$ Alternatively, this can be phrased via the Kodaira–Spencer map: $$\delta_b : T_b\mathcal{B} \to H^1(C_b, T_{C_b}), \quad \varphi \mapsto (\text{first order deformation class}),$$ and the cup product

$$H^1(C_b, T_{C_b}) \otimes H^0(C_b, \omega_{C_b}) \longrightarrow H^1(C_b, \mathcal{O}_{C_b}).$$

The composite yields the IVHS map at $b$ (Nisse, 7 Jan 2026).

For a morphism $\varphi_b = i \circ \nu_b : C_b \to P^2$ (or a smooth surface), the relevant deformation space becomes $H^0(C_b, N_{\varphi_b})$ for the normal sheaf, with

$$0 \to T_{C_b} \to \varphi_b^* T_{P^2} \to N_{\varphi_b} \to 0$$

and the induced Kodaira–Spencer map used as above (Nisse, 7 Jan 2026).

2. Local Contributions of Singularities and Rank Bounds

The IVHS decomposes into local and global contributions. At each singular point $p\in \operatorname{Sing}(\mathcal{C})$ (with delta invariant $\delta(p)$) one has an exact sequence: $$0 \to \text{IVHS}_p^{\mathrm{jet}} \to \text{IVHS}_p \to \text{IVHS}_p^{\mathrm{res}} \to 0$$ with:

• $\dim \text{IVHS}_p^{\mathrm{res}} = 1$ for nodes (A$_1$),
• $\dim \text{IVHS}_p^{\mathrm{res}} = 0$ for cusps (A$_2$) and higher ADE singularities,
• $\dim \text{IVHS}_p^{\mathrm{jet}} \leq \delta(p)-1$ controlled by higher jets of differentials at $p$. The global rank satisfies

$$\operatorname{rank} \Phi \leq \sum_{p \in \text{Sing}} (\delta(p)-1) + (\# \text{nodes})$$

illustrating that only nodes contribute "nontrivial" residue data to the infinitesimal period map; more complicated singularities decrease the room for IVHS (Nisse, 7 Jan 2026).

3. Maximal Variation Criterion: Numerical Invariants and Nodal Curves

A family exhibits maximal IVHS if the period map is surjective at the infinitesimal level. For nodal and cuspidal singularities, the residue functionals $$r_p(\omega) = \operatorname{Res}_{\text{branch}}(\omega)$$ for the nodes $p$ produce rank-one contributions. Maximal variation occurs exactly when

$\delta \geq g$

where $\delta$ is the number of nodes and $g$ is the genus of the normalization, implying that the IVHS achieves maximal rank $g^2$ (Nisse, 7 Jan 2026).

By adjunction on nodal plane curves,

$H^0(C, \omega_C) \cong H^0(P^2, I_\Delta(d-3))$

where $\Delta$ is the node scheme. A Cayley–Bacharach criterion then shows the residue functionals span the full dual space if $\delta \geq g$, yielding full surjectivity for the infinitesimal period map (Nisse, 7 Jan 2026). The same logic governs curves on other projective surfaces, where similar adjunction and residue/Cayley–Bacharach arguments apply.

4. Deformation and Residue-Theoretic Realizations

The period map is tightly linked to explicit deformation and residue computations. A first-order deformation $v \in H^0(C,N_\varphi)$ gives a class $\xi_v \in H^1(C,T_C)$. The cup product

$$\xi_v \,\lrcorner\, \omega_1 \in H^1(C,\mathcal{O}_C)$$

paired with another holomorphic form $\omega_2$ under Serre duality is computed via local residues: $$(\omega_1, \omega_2) \mapsto \sum_{q\in \nu^{-1}(p)} \operatorname{Res}_q\left((\xi_v\,\lrcorner\,\omega_1)\cdot \omega_2\right)$$ at all branches over each singularity. Nodes contribute nontrivially via this sum, while higher singularities force vanishing of residues or restrict to higher jets, reflecting the underlying local geometry (Nisse, 7 Jan 2026).

The description of $$\omega_{\mathcal{C}} \cong \nu_*[\omega_C(\sum m(p)\nu^{-1}(p))]$$ with $m(p)=\delta(p)$ controls possible pole orders and ensures that only certain singularities can affect the infinitesimal period map (Nisse, 7 Jan 2026).

5. Relation to Logarithmic Geometry and the Jacobian Ring Formalism

Logarithmic geometry extends IVHS analysis to families with prescribed singularities by encoding equisingular deformations as logarithmic tangent vectors, which act trivially on the Hodge structure. The effective IVHS arises from actual (non-logarithmic) deformation directions and their residue contributions. The cotangent sheaf $\Omega^1_X(\log D)$ and logarithmic vector fields $T_X(-\log D)$ provide a natural home for these computations (Nisse, 20 Jan 2026).

The Jacobian ring $J(f) = \mathcal{O}_X/(f_x,\ldots, f_n)$, familiar from Griffiths' work on hypersurface cohomology, gives an explicit algebraic model for IVHS: deformations in degree $d$ multiply with classes corresponding to $H^{p,q}$ in the Jacobian ring, providing a direct link between algebraic multiplication and the infinitesimal period map (Nisse, 20 Jan 2026, Eyssidieux et al., 2013).

6. Extensions to Higher Dimensions, Severi Varieties, and Equisingular Families

For hypersurfaces with simple (ADE) singularities, IVHS admits a description in terms of the Jacobian ring even on isosingular strata: the period map’s differential is modeled by multiplication in $R_f$ with appropriate degree shifts. The injectivity criterion for infinitesimal Torelli in the singular setting boils down to the Macaulay duality and generation properties of the Jacobian and Tjurina ideals (Eyssidieux et al., 2013).

Severi and equisingular families of curves (e.g., the locus of plane curves with prescribed nodes or cusps) admit a uniform IVHS description: tangent directions for equisingular deformations correspond to ring elements vanishing doubly at singular points, and the IVHS is again given by multiplication in the Jacobian ring, provided cohomological vanishing (i.e., unobstructedness) and Lefschetz-type conditions are satisfied (Nisse, 19 Jan 2026, Nisse, 20 Jan 2026). The residue description via the Poincaré map—taking derivatives of algebraic forms with respect to a family parameter—gives explicit computational control (Nisse, 19 Jan 2026).

7. Degenerations, Obstructions, and Failure of Maximality

On very general Picard rank-one surfaces, nodal-cuspidal curves with $\delta \geq g$ maintain maximal IVHS, but the presence of higher ADE singularities introduces equisingular directions annihilating all residues, causing the image of the period map to lie in a proper subspace and precluding maximal rank for large genus (Nisse, 7 Jan 2026).

This phenomenon is intimately linked with the local geometry of singularities: only nodes (A$_1$) provide true rank-one residue operators, while higher singularities—cusps (A$_2$) and more complex ADE types—are “invisible” or obstructive at the infinitesimal level (Nisse, 7 Jan 2026). The implications extend beyond pure Hodge theory; they affect moduli of curves, the scope of global Torelli theorems, and the structure of Hodge loci determined by determinantal conditions on the IVHS image (Movasati, 2014).

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References:

• "Infinitesimal Variations of Hodge Structure for Singular Curves I" (Nisse, 7 Jan 2026)
• "Logarithmic geometry and Infinitesimal Hodge Theory" (Nisse, 20 Jan 2026)
• "Residues and Infinitesimal Torelli for Equisingular Curves" (Nisse, 19 Jan 2026)
• "Sur l'application des périodes d'une Variation de Structure de Hodge attachée aux familles d'hypersurfaces á singularités simples" (Eyssidieux et al., 2013)