[Paper Review] The Deformation Space of Calabi-Yau n-folds with canonical singularities can be obstructed

Mark Gross|ArXiv.org|Feb 25, 1994

Geometry and complex manifolds2 references16 citations

TL;DR

This paper constructs a counterexample showing that the deformation space of Calabi-Yau $n$-folds with canonical singularities can be obstructed, by exhibiting a Calabi-Yau $n$-fold $X$ lying in the intersection of two distinct families—one of smooth Calabi-Yaus and another of Calabi-Yaus with terminal singularities—proving the Kuranishi space at $X$ is singular, hence deformation theory is obstructed.

ABSTRACT

This paper gives a simple example of a family of Calabi-Yaus of any dimension with canonical singularities of dimension one, whose Kuranishi space is singular. Thus the Bogomolov-Tian-Todorov unobstructedness theorem is not true for Calabi-Yaus with canonical singularities.

Motivation & Objective

• To address whether the deformation space of Calabi-Yau $n$-folds with canonical singularities is unobstructed, as is known for milder singularities.
• To provide a counterexample to the conjecture that canonical singularities always yield unobstructed deformation theory.
• To demonstrate that the moduli space of such Calabi-Yaus can be reducible and singular at certain points.
• To establish that the Kuranishi space of a Calabi-Yau $n$-fold with canonical singularities can be singular, implying obstructed deformation theory.

Proposed method

• Construct two distinct families of Calabi-Yau $n$-folds using projective bundles $P_1 = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}^{\oplus(n+1)})$ and $P_2 = \mathbb{P}(\mathcal{E})$, where $\mathcal{E} = \mathcal{O}_{\mathbb{P}^1}(-1) \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus(n-1)} \oplus \mathcal{O}_{\mathbb{P}^1}(1)$.
• Define the Calabi-Yau $n$-folds as zero loci of sections in $H^0(\omega_{P_i}^{-1})$, with $X_1(s)$ smooth for general $s$, and $X_2(s)$ having canonical singularities along a curve $C \subset P_2$.
• Use a filtration on $S^{n+1}\mathcal{E}$ induced by a short exact sequence to analyze the sheaf cohomology and construct a subspace $V \subset H^0(\omega_{P_2}^{-1})$ of sections yielding canonical singularities.
• Construct a flat family $\mathcal{X} \to \mathbb{A}^1$ via a universal extension bundle $\mathcal{F}$ on $T = \mathbb{A}^1 \times \mathbb{P}^1$, such that $\mathcal{X}_1$ is smooth and $\mathcal{X}_0 = X_2(s)$ for $s \in V$.
• Apply Grauert’s Theorem and analyze the restriction map $\phi_x$ to show that $\phi_0$ has image exactly $V$, ensuring $X_2(s)$ is deformation equivalent to a smooth Calabi-Yau.
• Conclude via Lemma 2.1 that the Kuranishi space is singular at $X_2(s)$, proving obstructed deformation theory.

Experimental results

Research questions

• RQ1 Is the deformation space of Calabi-Yau $n$-folds with canonical singularities unobstructed?
• RQ2 Can a Calabi-Yau $n$-fold with canonical singularities lie in the intersection of two distinct families of Calabi-Yau $n$-folds?
• RQ3 Does the existence of such a point in the moduli space imply obstructed deformation theory?
• RQ4 Can the Kuranishi space of a Calabi-Yau $n$-fold with canonical singularities be singular?
• RQ5 Is there a deformation family connecting a Calabi-Yau with canonical singularities to a smooth one, while preserving the Calabi-Yau condition?

Key findings

• The Kuranishi space of a Calabi-Yau $n$-fold $X_2(s)$ with canonical singularities along a curve $C \subset P_2$ is singular at $X_2(s)$, proving obstructed deformation theory.
• For $n \geq 3$, there exists a Calabi-Yau $n$-fold $X_2(s)$ with canonical singularities that is deformation equivalent to a smooth Calabi-Yau $n$-fold $X_1(s')$ via a flat family over $\mathbb{A}^1$.
• The family $\mathcal{X} \to \mathbb{A}^1$ constructed via the universal extension bundle $\mathcal{F}$ satisfies $\mathcal{X}_1 = X_1(s')$ (smooth) and $\mathcal{X}_0 = X_2(s)$ (canonical singularities).
• The restriction map $\phi_0$ has image exactly $V = H^0(G^{n-1}(2))$, which corresponds to sections yielding canonical singularities, and this ensures the family connects the two types.
• The moduli space of Calabi-Yau $n$-folds is reducible at $X_2(s)$, as it lies in the intersection of two distinct components: one of smooth Calabi-Yaus and one of Calabi-Yaus with terminal singularities.
• For $n=3$, $X_2(s)$ is non-singular, but for $n>3$, $X_2(s)$ has singularities of multiplicity $\lfloor n/2 \rfloor$ along $C$, yet still has canonical singularities when $s \in V$.