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**\(D\)-equivalence and \(K\)-equivalence.**
*(English)*
Zbl 1056.14021

For a smooth projective variety \(X\), \(D(X):=D^b( \text{Coh} (X))\) denotes the derived category of bounded complexes of coherent sheaves on \(X\). Smooth projective varieties \(X\) and \(Y\) are said to be \(D\)-equivalent if \(D(X)\) and \(D(Y)\) are equivalent as triangulated categories, i.e. if there exists an equivalence \(D(X) \to D(Y)\) of categories which commutes with translations and which maps any triangle in \(D(X)\) to a triangle in \(D(Y)\). The notion of \(K\)-equivalence is defined by the following conditions: \(X\) and \(Y\) are birationally equivalent and there exists a smooth projective variety \(Z\) together with birational morphisms \(Z\to X\) and \(Z \to Y\) such that \(f^* (K_X)\) and \(g^* (K_Y)\) are linearly equivalent. The author states the following conjecture: Both notions coincide for birationally equivalent smooth projective varieties, i.e. in this case \(X\) and \(Y\) are \(K\)-equivalent iff they are \(D\)-equivalent. The question to reconstruct a variety from its derived category was studied by A. Bondal and D. Orlov [Compos. Math. 125, 327–344 (2001; Zbl 0994.18007)]. The author proves the following generalization of their theorem:

If \(X\) and \(Y\) are \(D\)-equivalent, then the following holds:

0. \(\dim (X) = \dim (Y) \;\;(=:n)\).

1. If \(K_X\) (resp. \(-K_X\)) is nef, then \(K_Y\) (resp. \(-K_Y\)) is also nef and \(\nu (X) = \nu (Y)\) (resp. \(\nu (X,-K_X) = \nu (Y,-K_Y) \)).

2. If \(\kappa (X)=n\) or \(\kappa (X,-K_X)=n\) then \(X\) and \(Y\) are \(K\)-equivalent.

Conjecture 1.5 states that for a given smooth projective variety \(X\) there exist (up to isomorphism) only finitely many smooth projective varieties \(Y\) which are \(D\)-equivalent to \(X\). An affirmative answer (extending a result of T. Bridgeland and A. Maciocia [Math. Z. 236, 677–697 (2001; Zbl 1081.14023)]) is given for the case of surfaces. The last section of the paper is dealing with singular varieties \(X\), especially for quotient or hypersurface singularities; those are (in the given context) sufficiently close to the nonsingular ones. The question whether \(K\)-equivalence implies \(D\)-equivalence is answered for the following case:

Theorem. Let \(X\) and \(Y\) be 3-dimensional normal projective varieties having only \(\mathbb{Q}\)-factorial terminal singularities. Denote by \(\mathcal X\), \(\mathcal Y\) their canonical covering stacks. Now assume that \(X\) and \(Y\) are \(K\)-equivalent. Then the bounded derived categories of coherent orbifold sheaves \(D({\mathcal X})\) and \(D({\mathcal Y})\) are equivalent as triangulated categories.

If \(X\) and \(Y\) are \(D\)-equivalent, then the following holds:

0. \(\dim (X) = \dim (Y) \;\;(=:n)\).

1. If \(K_X\) (resp. \(-K_X\)) is nef, then \(K_Y\) (resp. \(-K_Y\)) is also nef and \(\nu (X) = \nu (Y)\) (resp. \(\nu (X,-K_X) = \nu (Y,-K_Y) \)).

2. If \(\kappa (X)=n\) or \(\kappa (X,-K_X)=n\) then \(X\) and \(Y\) are \(K\)-equivalent.

Conjecture 1.5 states that for a given smooth projective variety \(X\) there exist (up to isomorphism) only finitely many smooth projective varieties \(Y\) which are \(D\)-equivalent to \(X\). An affirmative answer (extending a result of T. Bridgeland and A. Maciocia [Math. Z. 236, 677–697 (2001; Zbl 1081.14023)]) is given for the case of surfaces. The last section of the paper is dealing with singular varieties \(X\), especially for quotient or hypersurface singularities; those are (in the given context) sufficiently close to the nonsingular ones. The question whether \(K\)-equivalence implies \(D\)-equivalence is answered for the following case:

Theorem. Let \(X\) and \(Y\) be 3-dimensional normal projective varieties having only \(\mathbb{Q}\)-factorial terminal singularities. Denote by \(\mathcal X\), \(\mathcal Y\) their canonical covering stacks. Now assume that \(X\) and \(Y\) are \(K\)-equivalent. Then the bounded derived categories of coherent orbifold sheaves \(D({\mathcal X})\) and \(D({\mathcal Y})\) are equivalent as triangulated categories.

Reviewer: Marko Roczen (Berlin)