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Geometric phantom categories. (English) Zbl 1285.14018

For a smooth, projective variety \(X\) one can define a quasiphantom to be an admissible subcategory of the bounded derived category of coherent sheaves \(\mathrm{D}^b(X)\), with trivial Hochschild homology (and finite abelian Grothendieck group). For a quasiphantom subcategory to be a phantom subcategory one requires that it also has a trivial Grothendieck group.
The authors construct phantom triangulated categories by considering the tensor product of quasiphantoms for which the orders of their Grothendieck groups are coprime. They not only prove that such products are phantoms, but that they are also universal phantoms (which they show to be equivalent to the vanishing of their \(K\)-motive). Last, but not least, they prove that, over \(\mathbb C\), the vanishing of \(K_0\)-groups of an admissible subcategory imply the vanishing of its Hochschild homology.
In the paper, they first consider Chow motives of surfaces of general type with \(q=p_g=0\) which satisfy Bloch’s conjecture for \(0\)-cycles (i.e. \(\mathrm{CH}^2(S)\cong\mathbb Z\)) and prove that for such an \(S\), there exists a decomposition of its Chow motive \(M(S)\cong 1\oplus\mathbb L^{\oplus r}\oplus\mathbb L^2\oplus M\), where \(1\) is the unit object, \(\mathbb L\) is the Lefschetz motive and M is a Chow motive over \(\mathbb C\). Then, they go on to show that, for S and S’ surfaces as above, the tensor product of the corresponding \(M\) and \(M'\) is trivial, whereas the external product map \(\mathrm{CH}^\bullet(S)\otimes\mathrm{CH}^\bullet(S')\rightarrow\mathrm{CH}^\bullet(S\times S')\) is an isomorphism. This allows them to prove that, assuming the derived categories \(\mathrm{D}^b(S)\) and \(\mathrm{D}^b(S')\) have exceptional collections of maximal length and the orders of \(\mathrm{Pic}(S)_{\mathrm{tors}}\) and \(\mathrm{Pic}(S)_{\mathrm{tors}}\) are coprime, the tensor product of the orthogonals to these exceptional collections is a phantom. Applying this to the Burniat surface with \(c_1^2=6\) and the classical Godeaux surface (or a Beauville surface), they construct a phantom category.
The second part of the paper deals with \(K\)-motives and the proof that the above constructed phantoms are actually universal phantoms. We should remark here that a universal phantom is indeed a phantom, however the proofs that the corresponding tensor product is a phantom (respectively a universal phantom) are different and independently interesting.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
18F30 Grothendieck groups (category-theoretic aspects)
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
14F42 Motivic cohomology; motivic homotopy theory
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