Versal dg deformation of Calabi--Yau manifolds

1. Problem Statement

The paper addresses the relationship between the deformation theory of a higher-dimensional Calabi–Yau (CY) manifold $X_0$ and the deformation theory of its derived category of coherent sheaves (specifically, the dg category of perfect complexes, $\text{Perf}_{dg}(X_0)$).

While it is a known philosophical consensus that deforming a variety and deforming its derived category are "essentially the same" (due to the isomorphism of their Hochschild cohomologies, which control deformations), a rigorous framework to prove this equivalence for linear and abelian categories was lacking until recent developments (e.g., Lowen–Van den Bergh, Dinh–Liu–Lowen).

The specific problems tackled are:

1. Equivalence of Deformation Functors: Proving that the deformation functor of the CY manifold $X_0$ is naturally isomorphic to the deformation functor of its dg category of perfect complexes (specifically, Morita deformations).
2. Uniqueness of Versal Morita Deformations: Establishing that the versal Morita deformation of $\text{Perf}_{dg}(X_0)$ is unique up to quasi-equivalence, even when $X_0$ is replaced by a derived-equivalent CY manifold $X'_0$.
3. Generic Fiber Construction: Defining and analyzing the "generic fiber" of the versal Morita deformation as a Drinfeld quotient and relating it to the derived category of the generic fiber of the geometric deformation.

2. Methodology

The author employs a synthesis of deformation theory for schemes, Hochschild cohomology, and the deformation theory of dg categories.

• Hochschild Cohomology & Gerstenhaber–Shack Complexes: The paper utilizes the isomorphism between the relative Hochschild cohomology of a scheme $X$, $HH^*(X/R)$, and the cohomology of the Gerstenhaber–Shack complex of its restricted structure sheaf. This links geometric deformations (controlled by $H^1(T_X)$) to algebraic deformations of categories.
• Toda's Construction: The author adapts Toda's construction of twisted coherent sheaves on noncommutative schemes to the setting of relatively smooth proper schemes. This establishes an equivalence between the category of quasi-coherent sheaves on a geometric deformation $X_\phi$ and the flat abelian deformation of the category $\text{Qch}(X)$ along a Hochschild cocycle $\phi$.
• Curved DG and $A_\infty$-Deformations: The paper relies heavily on the work of Lowen regarding curved dg deformations. It uses the characteristic morphism $\chi: HH^2 \to Z^2(K^+)$ to identify obstructions to lifting objects.
  • A key technical step involves showing that for a CY manifold, the curvature obstruction vanishes for perfect complexes, allowing the construction of maximal partial dg deformations.
• Morita Equivalence: The deformation of the dg category is treated in the Morita model structure (where weak equivalences are derived equivalences). The author uses the fact that for triangulated dg categories, Morita equivalence coincides with quasi-equivalence.
• Fourier–Mukai Kernels: To handle derived-equivalent manifolds, the paper utilizes the deformation theory of Fourier–Mukai kernels. It constructs a system of deformations of the kernel $P_0$ connecting $X_0$ and $X'_0$ to induce equivalences between their deformed categories.

3. Key Contributions and Results

A. Equivalence of Deformation Functors (Theorem 1.1 / Theorem 7.1)

The central result is the construction of a natural isomorphism of deformation functors:
$$\zeta: \text{Def}_{X_0} \xrightarrow{\sim} \text{Def}^{mo}_{\text{Perf}_{dg}(X_0)}$$

• Mechanism: The map sends an $A$-deformation of the manifold $X_A$ to the Morita $A$-deformation of the dg category $\text{Perf}_{dg}(X_A)$.
• Significance: This proves that the deformation theory of the manifold is entirely captured by the deformation theory of its derived category. It implies that the Kodaira–Spencer dg Lie algebra controls the Morita deformations of $\text{Perf}_{dg}(X_0)$.

B. Algebraization and Versality (Corollary 1.4 / Corollary 7.4)

The paper proves that any effective universal formal family for the Morita deformation functor is algebraizable.

• Specifically, if $(\text{Spec } S, s, X_S)$ is a versal deformation of the CY manifold $X_0$, then $\text{Perf}_{dg}(X_S)$ provides a versal Morita deformation of $\text{Perf}_{dg}(X_0)$.
• This bridges the gap between formal deformation theory (pro-represented by a complete local ring) and algebraic geometry (schemes over affine varieties).

C. The Generic Fiber as a Drinfeld Quotient (Definition 1.5 / Proposition 1.6)

The author introduces the dg categorical generic fiber of a family $\text{Perf}_{dg}(X_S)$ as the Drinfeld quotient:
$$\text{Perf}_{dg}(X_S) / \text{Perf}_{dg}(X_S)_0$$
where $\text{Perf}_{dg}(X_S)_0$ is the subcategory of perfect complexes with $S$-torsion cohomology.

• Result: This quotient is quasi-equivalent to $\text{Perf}_{dg}(X_{Q(S)})$, where $X_{Q(S)}$ is the geometric generic fiber. This provides a natural dg enhancement of the categorical generic fiber concept introduced by Huybrechts–Macrì–Stellari.

D. Uniqueness for Derived-Equivalent Manifolds (Theorem 1.7 / Corollary 1.8)

This is a significant extension of previous uniqueness results.

• Context: Let $X_0$ and $X'_0$ be derived-equivalent CY manifolds.
• Result: There exists a sufficiently large index $j$ in the filtered inductive system approximating the universal deformation ring such that the algebraic deformations $X_{R_j}$ and $X'_{R_j}$ are derived-equivalent.
• Consequence: The versal Morita deformations $\text{Perf}_{dg}(X_S)$ and $\text{Perf}_{dg}(X'_S)$ are quasi-equivalent.
• Implication: The versal Morita deformation is independent of the choice of geometric realization (up to Morita equivalence) and depends only on the derived equivalence class of the original manifold. This allows extending derived equivalences from special fibers to general fibers (and vice versa) without shrinking the base.

4. Significance

1. Rigorous Foundation for Homological Mirror Symmetry (HMS): The paper provides a rigorous algebraic framework for the intuition that deforming a CY manifold is equivalent to deforming its derived category. This is crucial for HMS, where deformations of the complex structure (geometric) correspond to deformations of the Fukaya category (symplectic/derived).
2. Unification of Geometric and Categorical Deformations: By proving the isomorphism of deformation functors, the paper unifies the study of moduli spaces of varieties with the moduli spaces of dg categories.
3. Resolution of Ambiguity in Versal Deformations: The result that versal Morita deformations of derived-equivalent manifolds are quasi-equivalent resolves a potential ambiguity: it shows that the "derived moduli space" is well-defined regardless of which specific geometric representative (from a derived equivalence class) one starts with.
4. Technical Advancement in DG Categories: The paper successfully navigates the complexities of curved dg deformations, characteristic morphisms, and Drinfeld quotients to establish concrete links between geometric objects and their categorical counterparts.

In summary, Morimura's work establishes that for higher-dimensional Calabi–Yau manifolds, the "derived category" is a robust invariant that fully encodes the deformation theory of the manifold, and this encoding behaves consistently under derived equivalences.