[Paper Review] Semiorthogonal decomposition for algebraic varieties

Alexey Bondal, Dmitri Olegovich Orlov|ArXiv.org|Jun 19, 1995

Algebraic structures and combinatorial models6 references377 citations

TL;DR

This paper establishes a criterion for fully faithful functors between derived categories of coherent sheaves on smooth algebraic varieties, applies it to derive a semiorthogonal decomposition for the derived category of the intersection of two even-dimensional quadrics, and proves that a smooth projective variety with ample or anti-ample canonical bundle is uniquely determined by its derived category of coherent sheaves. The key contribution is a reconstruction theorem and evidence for derived equivalence under flops.

ABSTRACT

A criterion for a functor between derived categories of coherent sheaves to be full and faithful is given. A semiorthogonal decomposition for the derived category of coherent sheaves on the intersection of two even dimensional quadrics is obtained. The behaviour of derived categories with respect to birational transformations is investigated. A theorem about reconstruction of a variety from the derived category of coherent sheaves is proved.

Motivation & Objective

• To establish a criterion for a functor between derived categories of coherent sheaves to be fully faithful.
• To describe a semiorthogonal decomposition for the derived category of the intersection of two even-dimensional quadrics.
• To investigate the behavior of derived categories under birational transformations, particularly flips and flops.
• To prove that a smooth projective variety with ample or anti-ample canonical bundle is uniquely determined by its derived category of coherent sheaves.
• To characterize the group of auto-equivalences of the derived category in the case of ample or anti-ample canonical class.

Proposed method

• Introduce a criterion for full faithfulness of a functor $\Phi_E: D^b_{coh}(M) \to D^b_{coh}(X)$ via its action on skyscraper sheaves and their shifts.
• Construct a functor $\Phi_E$ using a kernel $E$ in $D^b_{coh}(X \times M)$ via the formula $\Phi_E(\mathcal{F}) = \pi_*(E \otimes p^*\mathcal{F})$, where $p$ and \pi are projections from $X \times M$.
• Apply the criterion to the case where $X$ is the smooth intersection of two even-dimensional quadrics, identifying a full subcategory equivalent to $D^b_{coh}(C)$, where $C$ is the associated hyperelliptic curve.
• Show that the orthogonal complement in $D^b_{coh}(X)$ to $D^b_{coh}(C)$ admits an exceptional collection of line bundles.
• Use the derived category framework to analyze birational maps, particularly flops, and prove that $D^b_{coh}(X^+)$ admits a full and faithful embedding into $D^b_{coh}(X)$ for certain flips.
• Prove the reconstruction theorem by identifying the variety via the structure of the derived category: the canonical ring and the set of point objects $\mathcal{O}_x$ recover the variety up to isomorphism.

Experimental results

Research questions

• RQ1 Under what conditions is a functor between derived categories of coherent sheaves fully faithful?
• RQ2 Can the derived category of the intersection of two even-dimensional quadrics be decomposed into simpler components?
• RQ3 How do derived categories behave under birational transformations such as flips and flops?
• RQ4 To what extent is a smooth projective variety determined by its derived category of coherent sheaves?
• RQ5 What is the structure of the group of auto-equivalences of $D^b_{coh}(X)$ when the canonical or anticanonical bundle is ample?

Key findings

• A functor $\Phi_E: D^b_{coh}(M) \to D^b_{coh}(X)$ is fully faithful if and only if it induces isomorphisms on $\operatorname{Hom}$-spaces between the images of skyscraper sheaves and their shifts.
• For $X$ the smooth intersection of two even-dimensional quadrics, there is a semiorthogonal decomposition $D^b_{coh}(X) = \langle D^b_{coh}(C), \mathcal{O}_X, \mathcal{O}_X(1), \dots, \mathcal{O}_X(n-1) \rangle$, where $C$ is the associated hyperelliptic curve.
• The derived category of the intersection of quadrics admits a full and faithful embedding of $D^b_{coh}(C)$, and the orthogonal complement is generated by an exceptional collection of line bundles.
• For a flop between varieties $X$ and $X^+$, there is a full and faithful embedding $D^b_{coh}(X^+) \hookrightarrow D^b_{coh}(X)$, supporting the conjecture that their derived categories are equivalent.
• If $X$ is a smooth projective variety with ample or anti-ample canonical bundle, then any variety $X'$ with $D^b_{coh}(X) \simeq D^b_{coh}(X')$ must be isomorphic to $X$, proving a derived reconstruction theorem.
• The group of exact auto-equivalences of $D^b_{coh}(X)$ for such $X$ is generated by automorphisms of the variety, tensoring with line bundles, and shifts (translations).