[Paper Review] Hyperplane sections of Calabi-Yau varieties

Jonathan Wahl|ArXiv.org|Apr 17, 2001

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes that a smooth divisor $X$ on a complex projective variety $W$ with $h^1(\mathcal{O}_W) = 0$ cannot be a hyperplane section of a Calabi-Yau variety unless $W$ itself is Calabi-Yau. The method constructs universal extensions of $X$ via the Kodaira-Spencer map and analyzes singularities of the total space; the key result is a precise degree bound: a smooth hypersurface of degree $d$ in $\mathbb{P}^n$ is a hyperplane section of a Calabi-Yau iff $n+1 \leq d \leq 2n+2$. This resolves the hyperplane section problem for Calabi-Yau varieties in terms of canonical embeddings and extension theory.

ABSTRACT

Theorem: If W is a smooth complex projective variety with h^1 (O-script_W) = 0, then a sufficiently ample smooth divisor X on W cannot be a hyperplane section of a Calabi-Yau variety, unless W is itself a Calabi-Yau. Corollary: A smooth hypersurface of degree d in P^n (n >= 2) is a hyperplane section of a Calabi-Yau variety iff n+2 <= d <= 2n+2. The method is to construct out of the variety W a universal family of all varieties Z for which X is a hyperplane section with normal bundle K_X, and examine the "bad" singularities of such Z. A motivation is to show many curves cannot be divisors on a K-3 surface.

Motivation & Objective

• To determine which smooth projective varieties can arise as hyperplane sections of Calabi-Yau varieties.
• To characterize the conditions under which a smooth divisor $X$ on a variety $W$ with $h^1(\mathcal{O}_W) = 0$ can be a hyperplane section of a Calabi-Yau threefold or higher-dimensional Calabi-Yau.
• To develop a general method for constructing all possible extensions of a canonical embedding $X \subset \mathbb{P}^{g-1}$ via deformation theory and normal bundle sequences.
• To prove that for sufficiently ample divisors $X$ on $W$, the only way $X$ can be a hyperplane section of a Calabi-Yau is if $W$ itself is Calabi-Yau.

Proposed method

• Construct a universal family of varieties $Z$ for which $X$ is a hyperplane section with normal bundle $K_X$, using extension theory.
• Use the Kodaira-Spencer map associated to the normal bundle sequence to analyze the infinitesimal deformations of the embedding $X \subset Z$.
• Apply the coboundary map $\gamma: \Gamma(N_{X/Z}(-1)) \to H^1(\Theta_X(-1))$ to determine universality of the extension.
• Relate the Gaussian-Wahl map $\Phi_K$ to the non-surjectivity of $\gamma$, especially in the case of curves.
• Generalize the construction via blow-ups and double covers to build Calabi-Yau varieties containing $X$ as a hyperplane section.
• Use cohomological vanishing and the condition $h^1(\mathcal{O}_Z) = 0$ to restrict possible extensions and eliminate non-rational singularities.

Experimental results

Research questions

• RQ1 Under what conditions can a smooth divisor $X$ on a variety $W$ with $h^1(\mathcal{O}_W) = 0$ be a hyperplane section of a Calabi-Yau variety?
• RQ2 When is the canonical embedding of a smooth variety $X$ extendable to a hyperplane section of a Calabi-Yau threefold or higher-dimensional Calabi-Yau?
• RQ3 What role do singularities of the total space play in determining whether $X$ can be a hyperplane section of a Calabi-Yau?
• RQ4 How does the Kodaira-Spencer map of the normal bundle sequence detect universality of extensions?
• RQ5 What is the precise degree range for smooth hypersurfaces in $\mathbb{P}^n$ that can be hyperplane sections of Calabi-Yau varieties?

Key findings

• A smooth divisor $X$ on a variety $W$ with $h^1(\mathcal{O}_W) = 0$ cannot be a hyperplane section of a Calabi-Yau variety unless $W$ is itself Calabi-Yau.
• For a smooth hypersurface of degree $d$ in $\mathbb{P}^n$, it is a hyperplane section of a Calabi-Yau variety if and only if $n+1 \leq d \leq 2n+2$.
• Plane curves of degree $\geq 7$ cannot lie on a $K$-3 surface, even though their Gaussian-Wahl map has corank 10.
• For complete intersections of dimension $r \geq 2$, if $\kappa = \sum d_i - (n+1) > d$, then no extension exists, so $X$ cannot be a hyperplane section of a Calabi-Yau.
• When $d > \kappa > d/2$ and $\kappa > d_{n-r-1}$, non-trivial extensions exist but all have non-rational singularities, so no Calabi-Yau extension exists.
• A double cover construction via blow-ups along $X \cap X'$ with $d' = 2n+2-d$ yields a singular Calabi-Yau $Y^*$ in which $X$ is a hyperplane section, proving the sufficiency of the degree bound.