Unobstructed deformations for singular Calabi-Yau varieties

1. Problem Statement

The central problem addressed is the unobstructedness of deformations for singular Calabi-Yau varieties.

• Context: For a smooth compact complex manifold $M$, the Bogomolov-Tian-Todorov theorem guarantees that deformations are unobstructed if $M$ is Calabi-Yau (trivial canonical bundle). This relies on the degeneration of the Hodge-de Rham spectral sequence and the $T^1$-lifting property.
• Challenge: When $M$ is replaced by a singular analytic space $Y$, obstructions can arise from both global cohomology and local singularities.
• Specific Gap: Previous results (e.g., by Kawamata, Ran, Tian, Namikawa) established unobstructedness for singular Calabi-Yau varieties under restrictive conditions:
  • Singularities must be Ordinary Double Points (ODP) or Rational Local Complete Intersection (lci).
  • The variety often required a Kähler resolution.
  • Recent work by Y. Imagi extended this to weighted homogeneous rational lci singularities but relied on constructing specific model metrics and harmonic analysis, which is difficult to generalize.
• Goal: Friedman aims to generalize these results to:
  1. Du Bois singularities (a broader class including rational singularities and log canonical singularities), removing the requirement for them to be weighted homogeneous.
  2. Replace the Kähler assumption with the existence of a resolution satisfying the $\bar{\partial}\partial$-lemma.
  3. Handle both lci and non-lci cases.

2. Methodology

The paper employs a synthesis of Hodge theory, deformation theory, and resolutions of singularities.

A. Hodge Theory and Relative Cohomology

• Resolution Setup: Let $\pi: \tilde{Y} \to Y$ be a good resolution of singularities with exceptional set $E = \pi^{-1}(Z)$, where $Z$ is the singular locus.
• $\bar{\partial}\partial$-Lemma: The author assumes $\tilde{Y}$ satisfies the $\bar{\partial}\partial$-lemma. This ensures the degeneration of the Hodge-de Rham spectral sequence for relative cohomology $H^*(\tilde{Y}, E; \mathbb{C}) \cong H^*(Y, Z; \mathbb{C})$.
• Du Bois Condition: The singularities are assumed to be Du Bois. This implies that the natural map $R^i\pi_*\mathcal{O}_{\tilde{Y}}(-E) \to 0$ for $i>0$, and allows the identification of certain cohomology groups with those of the resolution.
• Key Insight: Instead of requiring $H^i(Y, \Omega^1_{Y/A})$ to be free for all $i$ (which is too strong), the author focuses on the surjectivity of specific maps in degree $n-2$ (where $n = \dim Y$).
  • lci case: Surjectivity of $H^{n-2}(Y, \Omega^1_{Y/A}) \to H^{n-2}(Y, \Omega^1_Y)$.
  • non-lci case: Surjectivity of $H^{n-2}(Y, \Omega^1_{Y/A}) \to H^{n-2}(Y, \Omega^1_Y) / \text{Im}(H^{n-2}_Z(Y, \Omega^1_Y))$.

B. Deformation Theory and $T^1$-Lifting

• Kuranishi Space: Deformations are governed by a Kuranishi map $\Psi: T^1_Y \to T^2_Y$. Unobstructedness means $\Psi = 0$.
• $T^1$-Lifting Criterion: Following Ran, Kawamata, and Gross, unobstructedness is proven by showing the $T^1$-lifting property: the map $T^1(Y_k/A_k) \to T^1(Y_{k-1}/A_{k-1})$ is surjective for any Artin extension.
• Generalized $T^1$-Lifting: For non-lci singularities, Friedman utilizes a criterion from Mark Gross. This involves the generalized $T^1$-lifting property, which requires the restriction of a natural map $\ell: T^2_Y \to H^0(Y, T^2_Y)$ to the image of the obstruction map $\delta$ to be injective.
• Serre Duality: The proof leverages Serre duality and the Calabi-Yau condition ($\omega_Y \cong \mathcal{O}_Y$) to relate obstruction spaces to cohomology groups of differential forms.

3. Key Contributions and Results

Main Theorem (Theorem 1.6)

Let $Y$ be a singular Calabi-Yau variety of dimension $n \geq 3$ with:

1. A resolution $\tilde{Y}$ satisfying the $\bar{\partial}\partial$-lemma.
2. Isolated, lci, and Du Bois singularities.
3. $H^1(Y, \mathcal{O}_Y) = 0$.

Conclusion: The deformations of $Y$ are unobstructed.

• Note: This generalizes Namikawa's result (which required rational singularities) to Du Bois singularities and removes the Kähler requirement in favor of the $\bar{\partial}\partial$-lemma.

Generalization to Non-lci Singularities (Theorem 3.10)

If the lci assumption is dropped (but singularities remain isolated and Du Bois), the generalized $T^1$-lifting property holds.

• If the deformation functor is prorepresentable (automatic if singularities are rational), the deformation space is smooth over the base of the local deformation of the singularities.

Log Calabi-Yau Pairs (Theorem 3.15)

The results extend to log Calabi-Yau pairs $(Y, D)$, where $D$ is a simple normal crossing divisor in the smooth locus and $\omega_Y \cong \mathcal{O}_Y(-D)$.

• Under similar conditions (resolution with $\bar{\partial}\partial$-lemma, isolated lci Du Bois singularities, $H^1(Y, \mathcal{O}_Y)=0$), the log deformation functor is unobstructed.

Fano Case (Theorem 3.17)

For a compact analytic space $Y$ with isolated lci rational singularities where $\omega_Y^{-1}$ is nef and big:

• The deformation functor $\text{Def}_Y$ is unobstructed.
• Method: The proof uses a cyclic cover construction to reduce the problem to the log Calabi-Yau case, utilizing Kawamata-Viehweg vanishing.

4. Technical Significance

1. Relaxation of Singularity Conditions: The paper significantly broadens the class of singularities for which unobstructedness is known. By moving from "weighted homogeneous" or "rational" to Du Bois, it encompasses log canonical singularities (in the Gorenstein case), which are crucial in the Minimal Model Program (MMP).
2. Topological vs. Metric Assumptions: By replacing the Kähler condition with the $\bar{\partial}\partial$-lemma, the results apply to a wider range of complex analytic spaces, including those that may not be projective or Kähler but still possess the necessary cohomological properties.
3. Refinement of Lifting Criteria: The paper clarifies the necessary ingredients for $T^1$-lifting. It demonstrates that the full strength of the $T^1$-lifting property (free modules for all degrees) is not required; rather, the surjectivity of specific cohomology maps in degree $n-2$ is sufficient. This insight simplifies the verification of unobstructedness.
4. Handling Non-lci Singularities: The explicit treatment of the non-lci case via the cokernel of local cohomology maps ($H^{n-2}_Z$) provides a robust framework for studying deformations of varieties with more complex singularities, bridging the gap between local and global deformation theory.

5. Conclusion

Robert Friedman's paper provides a powerful generalization of the Bogomolov-Tian-Todorov theorem to the singular setting. By combining the geometric insights of Imagi (regarding the sufficiency of specific cohomological surjectivity) with the Hodge-theoretic tools of Namikawa and Gross, the author establishes that singular Calabi-Yau varieties with Du Bois singularities and a $\bar{\partial}\partial$-lemma resolution have unobstructed deformations. This result is foundational for constructing smoothings of singular Calabi-Yau varieties and understanding the moduli spaces of such objects in higher dimensions.