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The DK hypothesis: K-equivalence implies derived equivalence

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Let XX and YY be smooth projective birational varieties, and choose a common resolution

Z→pX,Z→qY.Z\xrightarrow{p}X,\qquad Z\xrightarrow{q}Y.

They are K-equivalent when p∗KX=q∗KYp^*K_X=q^*K_Y.

DK hypothesis. K-equivalence should imply an equivalence

DbCoh⁡(X)≃DbCoh⁡(Y).D^b\operatorname{Coh}(X)\simeq D^b\operatorname{Coh}(Y).

More generally, if p∗KX≥q∗KYp^*K_X\geq q^*K_Y, one expects a fully faithful embedding from DbCoh⁡(Y)D^b\operatorname{Coh}(Y) into DbCoh⁡(X)D^b\operatorname{Coh}(X).

This is the categorical form of the minimal model program: flops should preserve the derived category, while flips and divisorial contractions should create semiorthogonal pieces. The flop conjecture is known in dimension three, but remains widely open for general flops in dimension at least four.

Equivalent formulations 2Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The DK conjecture: K-equivalence implies derived equivalence

    Let X+X_+ and X−X_- be smooth varieties. They are K-equivalent if there exist a smooth variety X~\widetilde{X} and projective birational morphisms

    ϕ± ⁣:X~→X±\phi_{\pm}\colon \widetilde{X}\to X_{\pm}

    such that

    ϕ+∗KX+∼ϕ−∗KX−,\phi_+^*K_{X_+}\sim\phi_-^*K_{X_-},

    where KX±K_{X_{\pm}} are the canonical divisors. They are D-equivalent if there is an exact equivalence of triangulated categories

    D⁡b(coh⁡X+)≃D⁡b(coh⁡X−).\operatorname{D}^{\mathrm{b}}(\operatorname{coh}X_+)\simeq\operatorname{D}^{\mathrm{b}}(\operatorname{coh}X_-).

    DK conjecture. If X+X_+ and X−X_- are K-equivalent, then they are D-equivalent. This conjecture proposes a parallel between derived categories and birational geometry. It is attributed in the source to Bondal–Orlov and Kawamata; the paper studies derived equivalence for a particular simple flop of type G2†G_2^{\dagger}, rather than resolving the conjecture in general.

    source: Wahei Hara, “Derived equivalence for the simple flop of type G_2^ via tilting bundles”, arXiv:2412.14314 (2026).

  2. Kawamata's K-equivalence and D-equivalence conjecture

    Let XX and YY be smooth projective varieties. They are K-equivalent if there is a smooth variety ZZ with a birational correspondence

    X←πXZ→πYYX \xleftarrow{\pi_X} Z \xrightarrow{\pi_Y} Y

    such that

    πX∗KX≅πY∗KY.\pi_X^*K_X\cong\pi_Y^*K_Y.

    They are D-equivalent if their bounded derived categories of coherent sheaves are equivalent, namely if Db(Coh⁡(X))≃Db(Coh⁡(Y))D^b(\operatorname{Coh}(X))\simeq D^b(\operatorname{Coh}(Y)). Kawamata's conjecture. If XX and YY are smooth projective varieties, then K-equivalence implies D-equivalence. This conjecture is known for smooth Calabi–Yau threefolds and toroidal varieties, but remains open in general.

    source: Zhan Li, “On the birationality of complete intersections associated to nef-partitions”, arXiv:1310.2310 (2016).

birational-geometryderived-categoriesflopsk-equivalence
Posted by Vihaan Dheer Posted

Progress summary

Refreshed
Open

The conjecture remains open in general, with proofs only for three-dimensional flops and several more specialized cases.

The DK hypothesis predicts that KK-equivalent smooth projective varieties have equivalent bounded derived categories, while a KK-inequality should induce a fully faithful functor. No proposer is identified in the retrieved sources.

Known results

  • Bridgeland proved the derived equivalence for three-dimensional flops (2002).
  • Kawamata established cases for orbifolds, canonical covering stacks, and symplectic projective fourfolds (2002).
  • Fully faithful functors are known for standard flips, with semiorthogonal decompositions in the expected direction.
  • Grassmannian flip examples satisfy the conjectural fully faithful embedding, and in relevant cases derived equivalence (2019-era work).

No recent settlement found (August 2026)

The 2017 survey and the retrieved arXiv material continue to present the general implication as conjectural. They report no general proof, counterexample, claimed solution, or AI-generated settlement; the unrestricted converse is known to fail for rational elliptic surfaces.

Current status (as of August 2026): The DK hypothesis is proved for three-dimensional flops and several special classes, but the general implication for higher-dimensional flops and arbitrary KK-equivalent smooth projective varieties remains open.

Sources

Solutions 0

No solutions have been posted yet.