Determinantal Barlow surfaces and phantom categories

Böhning, Christian; Bothmer, Hans-Christian Graf von; Katzarkov, Ludmil; Sosna, Pawel

doi:10.4171/jems/539

Cited by 31 publications

(30 citation statements)

References 18 publications

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“…By [44,Theorem 4.1], it follows that the integral Chow motive of S is a direct sum of Lefschetz motives. On the other hand, Böhning, Graf von Bothmer, Katzarkov, and Sosna [8] have exhibited a complex Barlow surface S (a determinantal Barlow surface) with an exceptional collection whose orthogonal complement is a phantom category, that is, a nontrivial strictly full triangulated category with vanishing K 0 . Of course this does not say that the Barlow surface S does not admit a full exceptional collection, but it looks like a possibility that it won't.…”

Section: Exceptional Collections Of Maximal Length and Chow Motivesmentioning

confidence: 99%

“…Numerical constraints. Although the problem of classifying smooth projective complex surfaces that admit a full exceptional collection seems out of reach at present (it is conjectured that only the surfaces that are rational have a full exceptional collection), a fair amount of work [1,7,8,16] has been carried out in order to construct exceptional collections of maximal length on complex surfaces with p g = q = 0. (As usual, for a smooth projective surface S, the geometric genus is p g := h 0 (Ω 2 S ) = h 2 (O S ) and the irregularity is q := h 1 (O S ).)…”

Section: Introductionmentioning

confidence: 99%

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Exceptional collections, and the Néron–Severi lattice for surfaces

Vial

2017

Advances in Mathematics

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We work out properties of smooth projective varieties X over a (not necessarily algebraically closed) field k that admit collections of objects in the bounded derived category of coherent sheaves D b (X) that are either full exceptional, or numerically exceptional of maximal length. Our main result gives a necessary and sufficient condition on the Néron-Severi lattice for a smooth projective surface S with χ(O S ) = 1 to admit a numerically exceptional collection of maximal length, consisting of line-bundles. As a consequence we determine exactly which complex surfaces with pg = q = 0 admit a numerically exceptional collection of maximal length. Another consequence is that a minimal geometrically rational surface with a numerically exceptional collection of maximal length is rational.As is apparent, the class of varieties admitting full exceptional collections is rather restricted. Perhaps the simplest constraint for a smooth projective variety to admit a full exceptional collection is the following. If we have a semi-orthogonal decomposition T = A, B , then. , E r is a full exceptional collection, then K 0 (X) = Z r . As will be explained in the introduction of Section 2, this implies that the Chow motive of X with rational coefficients is a direct sum of Lefschetz motives. Such a constraint was originally obtained via the theory of non-commutative motives by Marcolli and Tabuada [34].Chow motives and Lefschetz motives. Let R be a ring. The category of Chow motives with R-coefficients over k is constructed as follows. First one linearizes the category of smooth projective varieties over k by declaring that Hom(X, Y ) = CH d (X × Y ) ⊗ Z R, that is, by declaring that the morphism between X and Y are given by correspondences with R-coefficients modulo rational equivalence. Here, X is assumed to be of pure dimension d (otherwise one works component-wise) and the composition law is given byHere, e is the dimension of Y , and p XZ , p XY , p Y Z are the projections from X × Y × Z onto X × Z, X × Y, Y × Z, respectively. This R-linear category is far from being abelian, so that one formally adds to this R-linear category the images of idempotents. This is called taking the pseudo-abelian, or Karoubi, envelope. This new category is called the category of effective Chow motives, and objects are pairs (X, p), where X is a smooth projective variety of dimension d and p ∈ CH d (X × X) ⊗ Z R is an idempotent. When p is the class of the diagonal ∆ X in CH d (X × X), we write h(X) for (X, ∆ X ). In general, the object (X, p) should be thought of as the image of p acting on the motive h(X) of X. For example, in the category of effective Chow motives, the motive h(P 1 ) of the projective line becomes isomorphic to (P 1 , p) ⊕ (P 1 , q), where p := {0} × P 1 and q := P 1 × {0} are idempotents in Hom(h(P 1 ), h(P 1 )) := CH 1 (P 1 × P 1 ). The object (P 1 , p) is isomorphic to 1 := h(Spec k), and the motive (P 1 , p) is called the Lefschetz motive and is written 1(−1). The fiber product of two smooth projective varieties induces a...

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Section: Exceptional Collections Of Maximal Length and Chow Motivesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exceptional collections, and the Néron–Severi lattice for surfaces

Vial

2017

Advances in Mathematics

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“…After initiated by the work of Böhning, Graf von Bothmer, and Sosna [5], there also have come numerous results on the surfaces of general type (e.g. [1,4,7,9,10,17,23]). For surfaces with Kodaira dimension one, such exceptional collections have not been shown to exist, thus it is a natural attempt to find an exceptional collection in D b (S).…”

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confidence: 99%

Exceptional collections on Dolgachev surfaces associated with degenerations

Cho

Lee

2018

Advances in Mathematics

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Abstract. Dolgachev surfaces are simply connected minimal elliptic surfaces with pg = q = 0 and of Kodaira dimension 1. These surfaces are constructed by logarithmic transformations of rational elliptic surfaces. In this paper, we explain the construction of Dolgachev surfaces via Q-Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities. This construction is based on the paper [25]. Also, some exceptional bundles on Dolgachev surfaces associated with Q-Gorenstein smoothing have been constructed based on the idea of Hacking [12]. In the case if Dolgachev surfaces were of type (2, 3), we describe the Picard group and present an exceptional collection of maximal length. Finally, we prove that the presented exceptional collection is not full, hence there exists a nontrivial phantom category in the derived category.

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“…The starting point of this work was a question asked by the authors of [10]: They needed to know that the Bloch conjecture holds for a simply connected surface with p g = 0 (eg a Barlow surface [3]) and furthermore, they needed it for a general deformation of this surface. The Bloch conjecture was proved by Barlow [4] for some Barlow surfaces admitting an extra group action allowing to play on group theoretic arguments as in [16], but it was not known for the general Barlow surface.…”

mentioning

confidence: 99%

“…[11], [25], [10]). Catanese surfaces can be constructed starting from a 5 × 5 symmetric matrix M (a), a ∈ P 11 , of linear forms on P 3 satisfying certain conditions (cf.…”

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confidence: 99%

Bloch's conjecture for Catanese and Barlow surfaces

Voisin¹

2014

J. Differential Geom.

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Catanese surfaces are regular surfaces of general type with pg = 0. They specialize to double covers of Barlow surfaces. We prove that the CH0 group of a Catanese surface is equal to Z, which implies the same result for the Barlow surfaces. IntroductionIn this paper, we establish an improved version of the main theorem of [27] and use it in order to prove the Bloch conjecture for Catanese surfaces. We will also give a conditional application (more precisely, assuming the variational Hodge conjecture) of the same method to the Chow motive of low degree K3 surfaces.Bloch's conjecture for 0-cycles on surfaces states the following:, where Y is smooth projective and X is a smooth projective surface. Assume that [Γ]* :vanishes. Here [Γ]∈ H 4 (X × X, Q) is the cohomology class of Γ andParticular cases concern the case where Γ is the diagonal of a surface X with q = p g = 0. Then Γ * = Id CH0(X) alb , so that the conjecture predicts CH 0 (X) alb = 0. This statement is known to hold for surfaces which are not of general type by [9], and for surfaces of general type, it is known to hold by Kimura [17] if the surface X is furthermore rationally dominated by a product of curves (cf.[5] for many such examples). Furthermore, for several other families of surfaces of general type with q = p g = 0, it is known to hold either for the general member of the family (eg.

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Determinantal Barlow surfaces and phantom categories

Cited by 31 publications

References 18 publications

Exceptional collections, and the Néron–Severi lattice for surfaces

Exceptional collections, and the Néron–Severi lattice for surfaces

Exceptional collections on Dolgachev surfaces associated with degenerations

Bloch's conjecture for Catanese and Barlow surfaces

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