[Paper Review] The Derived Category of the Intersection of Four Quadrics

Nicolas Addington|ArXiv.org|Apr 11, 2009

Algebraic Geometry and Number Theory19 references20 citations

TL;DR

This paper establishes a semi-orthogonal decomposition of the derived category of a general complete intersection of four quadrics in $\mathbb{P}^{2n-1}$, showing it splits as $\langle\mathcal{O}(-2n+9),\dotsc,\mathcal{O}(-1),\mathcal{O},\mathcal{D}\rangle$, where $\mathcal{D}$ is the derived category of twisted sheaves on a non-algebraic complex 3-fold arising from a moduli problem. For $n=4$, this yields a derived equivalence between Calabi-Yau 3-folds, confirming a prediction by Gross using a geometric construction that avoids non-commutative geometry.

ABSTRACT

The derived category of a general complete intersection of four quadrics in P^{2n-1} has a semi-orthogonal decomposition < O(-2n+9), ..., O(-1), O, D >, where D is the derived category of twisted sheaves on a certain non-algebraic complex 3-fold coming from a moduli problem. In particular, when n=4 we obtain a (twisted) derived equivalence of Calabi-Yau 3-folds predicted by Gross. This differs from Kuznetsov's result in that our construction is geometric and avoids non-commutative varieties.

Motivation & Objective

• To provide a geometric construction of a semi-orthogonal decomposition for the derived category of a complete intersection of four quadrics in $\mathbb{P}^{2n-1}$, avoiding non-commutative varieties.
• To identify the residual category $\mathcal{D}$ as the derived category of twisted sheaves on a non-algebraic complex 3-fold arising from a moduli problem.
• To establish a derived equivalence between Calabi-Yau 3-folds in the case $n=4$, confirming a prediction by Gross.
• To generalize the framework of Bondal-Orlov and Kapranov by extending the use of spinor bundles and Clifford algebras to higher-dimensional intersections.

Proposed method

• Constructing the double cover $\mathcal{M}$ of the line $L$ spanned by the four quadrics, branched over the singular locus of quadrics, which serves as a moduli space for vector bundles on the intersection $X$.
• Using the geometry of the singular locus $\Delta_2$ in the space of quadrics to show that $\mathcal{M}$ is singular when $n > 3$, necessitating the use of twisted sheaves.
• Defining a sheaf of algebras $\mathcal{A}$ on $L$ via a generalized Clifford algebra structure, and relating its derived category to $D(X)$ via pushforward and duality.
• Showing that the pushforward of the sheaf $\mathcal{H}$ from the resolution $\tilde{\mathcal{M}}$ to $\mathcal{M}$ is a vector bundle with no higher cohomology, implying $\pi_*\mathcal{H}$ is locally free.
• Establishing an isomorphism $\mathcal{J}_{2n}^* \cong \pi_*\mathcal{I}_{2n}^*$ via a dualized exact sequence on the $\mathbb{P}^1$-bundle $P$, which identifies the structure of the ideal sheaves.
• Using the isomorphism $\mathcal{H}om(\mathcal{I}_k, \mathcal{I}_{k+j}) \cong \mathcal{A}_{2n}(j/2 - n) \otimes \pi^*\mathcal{O}_L(j/2 - n)$ to control global sections and cohomology, ensuring the decomposition is semi-orthogonal.

Experimental results

Research questions

• RQ1 How can the derived category of a general complete intersection of four quadrics in $\mathbb{P}^{2n-1}$ be decomposed into simpler components?
• RQ2 What is the nature of the residual category $\mathcal{D}$ in such a decomposition, and how is it related to moduli spaces of bundles?
• RQ3 Can a geometric construction yield a derived equivalence between Calabi-Yau 3-folds without relying on non-commutative resolutions?
• RQ4 How does the presence of corank 2 quadrics in the span of four quadrics affect the geometry of the associated moduli space $\mathcal{M}$?

Key findings

• The derived category of a general complete intersection of four quadrics in $\mathbb{P}^{2n-1}$ admits a semi-orthogonal decomposition $\langle\mathcal{O}(-2n+9),\dotsc,\mathcal{O}(-1),\mathcal{O},\mathcal{D}\rangle$, where $\mathcal{D}$ is the derived category of twisted sheaves on a non-algebraic complex 3-fold.
• The moduli space $\mathcal{M}$ of spinor bundles on $X$ is singular when $n > 3$, due to the line $L$ intersecting the corank 2 locus $\Delta_2$, necessitating the use of twisted sheaves.
• For $n=4$, the construction yields a (twisted) derived equivalence between Calabi-Yau 3-folds, confirming a prediction by Gross.
• The pushforward $\pi_*\mathcal{H}$ is a vector bundle with no higher cohomology, which follows from the vanishing of $H^1(\mathcal{H}|_\ell)$ and the rank equality $\dim H^0(\mathcal{H}|_\ell) = \operatorname{rank}\mathcal{H}$.
• The isomorphism $\mathcal{J}_{2n}^* \cong \pi_*\mathcal{I}_{2n}^*$ is established via dualizing a sequence on the $\mathbb{P}^1$-bundle $P$, showing that $\mathcal{I}_{2n}$ is the image of a natural map from $\mathcal{O}_P \otimes \pi^*\mathcal{J}_{2n}$.
• The cohomological control $\mathcal{H}om(\mathcal{I}_k, \mathcal{I}_{k+j}) \cong \mathcal{A}_{2n}(j/2 - n) \otimes \pi^*\mathcal{O}_L(j/2 - n)$ ensures no middle cohomology and vanishing top cohomology for $j/2 - n \geq -3$, which supports the semi-orthogonality of the decomposition.