Quotient categories, stability conditions, and birational geometry

Meinhardt, Sven; Partsch, Holger

doi:10.1007/s10711-013-9947-x

Cited by 10 publications

(11 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. We will define mutually inverse maps, following [10,Lemma 3.6]. We start by constructing a map from right to left.…”

Section: John Calabrese and Roberto Pirisimentioning

confidence: 99%

“…Diverging a little from [10], finding the precise relationship between the derived category D(X) and the birational geometry of X has been an active area of research for quite some time (see [5] for an excellent overview). We wonder if the study of the derived category of the quotients D(C c (X)), or some intermediate version of them, might help make some progress in the matter.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Gabriel’s theorem and birational geometry

Calabrese¹,

Pirisi

2020

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Extending work of Meinhardt and Partsch, we prove that two varieties are isomorphic away from a subset of a given dimension if and only if certain quotients of their categories of coherent sheaves are equivalent. This result interpolates between Gabriel's reconstruction theorem and the fact that two varieties are birational if and only if they have the same function field. Contents 1. Quotient categories 909 2. A locally ringed space 913 3. Gabriel's theorem 917 4. The group of autoequivalences 919 Acknowledgments 921 References 921It is a well-known fact that two varieties (i.e., irreducible and reduced schemes of finite-type over a field k) X and Y are birational if and only if their function fields are isomorphic. At the same time, a theorem of Gabriel says that X and Y are isomorphic if and only if their categories of coherent sheaves are equivalent (as k-linear categories). In this article we show that these results are actually related: they are the two extreme cases of our main theorem.Before giving a precise statement, we introduce some notation. For an integer k, we write Coh ≤k (X) ⊂ Coh(X) for the subcategory of sheaves supported in dimension at most k. There is a robust theory of quotients of abelian categories, and we define C k (X) ∶= Coh(X)/ Coh ≤k−1 (X). It is often convenient to re-index these categories by codimension, defining C c (X) ∶= C dim X−c (X). We have C dim X (X) = Coh(X), and one shows that C 0 (X) is equivalent to finite-dimensional vector spaces over the function field of X. Finally, recall that two schemes X and Y of finite-type over a field are isomorphic in codimension c (resp., outside of dimension k − 1) if there exist open subsets U ⊂ X, V ⊂ Y such that U is isomorphic to V , and the codimensions of X ∖ U and Y ∖ V are at least c + 1 (resp., dimension at most k − 1). In particular, two varieties X and Y are birational if and only if they are isomorphic in codimension zero.

show abstract

“…Proof. We will define mutually inverse maps, following [10,Lemma 3.6]. We start by constructing a map from right to left.…”

Section: John Calabrese and Roberto Pirisimentioning

confidence: 99%

mentioning

confidence: 99%

Gabriel’s theorem and birational geometry

Calabrese¹,

Pirisi

2020

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…(1) (X) = D b (X)/S, where S is the subcategory of complexes with cohomology supported in codimension at least two (see [40]). On the other hand, φ has a well-defined dynamical degree λ(φ).…”

Section: 2mentioning

confidence: 99%

Dynamical Systems and Categories

Dimitrov¹,

Haiden²,

Kontsevich

et al. 2014

The Influence of Solomon Lefschetz in Geometry and Topology

151

View full text Add to dashboard Cite

We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A∞-categories. First, entropy is defined for exact endofunctors and computed in a variety of examples. In particular, the classical entropy of a pseudo-Anosov map is recovered from the induced functor on the Fukaya category. Second, the density of the set of phases of a Bridgeland stability condition is studied and a complete answer is given in the case of bounded derived categories of quivers. Certain exceptional pairs in triangulated categories, which we call Kronecker pairs, are used to construct stability conditions with density of phases. Some open questions and further directions are outlined as well.

show abstract

“…We consider the triangulated category D b Coh (1) (X) defined as the quotient of the bounded derived category D b Coh(X) by the full subcategory consisting of complexes whose support has codimension at least 1 [32, Definition 3.1] (see Section 5.2). Meinhardt-Partsch [32] observed that the triangulated category D b Coh (1) (X) is closely related to the birational geometry of X and showed that the space of stability conditions on D b Coh (1) (X) behaves roughly like that on a curve. Using their results and Theorem 1.7, we can show the following.…”

mentioning

confidence: 99%

On Gromov-Yomdin type theorems and a categorical interpretation of holomorphicity

Federico¹,

Kim²

2021

Preprint

View full text Add to dashboard Cite

In topological dynamics, the Gromov-Yomdin theorem states that the topological entropy of a holomorphic automorphism f of a smooth projective variety is equal to the logarithm of the spectral radius of the induced map f * . In order to establish a categorical analogue of the Gromov-Yomdin theorem, one first needs to find a categorical analogue of a holomorphic automorphism. In this paper, we propose a categorical analogue of a holomorphic automorphism and prove that the Gromov-Yomdin type theorem holds for them.

show abstract

Quotient categories, stability conditions, and birational geometry

Cited by 10 publications

References 40 publications

Gabriel’s theorem and birational geometry

Gabriel’s theorem and birational geometry

Dynamical Systems and Categories

On Gromov-Yomdin type theorems and a categorical interpretation of holomorphicity

Contact Info

Product

Resources

About