TY - JOUR

T1 - Some new categorical invariants

AU - Dimitrov, George

AU - Katzarkov, Ludmil

N1 - Funding Information:
Open access funding provided by University of Vienna. The authors wish to express their gratitude to Dhyan Aranha, Aleksej Bondal, Bogdan Georgiev and Maxim Kontsevich for their interest. The authors are very thankful to Denis Auroux for his help on the last Sect. 13 . The first author was supported by FWF Projects P 29178-N35, P 27784. Parts of this work was carried out during the stay 01.07.15-30.06.16 of the first author at the Max-Planck-Institute für Mathematik Bonn and his stay 01.07.16-30.06.17 at International Centre for Theoretical Physics Trieste and G. Dimitrov gratefully acknowledges the support and the excellent conditions at these institutes. The second author was supported by Simons research Grant, NSF DMS 150908, ERC Gemis, DMS-1265230, DMS-1201475 OISE-1242272 PASI. Simons collaborative Grant-HMS. HSE-Grant, HMS and automorphic forms. The second author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government Grant, ag. No 14.641.31.0001. Connection with our preprint arXiv:1602.09117 : Following advice by Auroux, Pantev, Kontsevich and one referee we split the paper arXiv:1602.09117 into two parts. The present paper is the second part.

PY - 2019/8/1

Y1 - 2019/8/1

N2 - In this paper we introduce new categorical notions and give many examples. In an earlier paper we proved that the Bridgeland stability space on the derived category of representations of K(l), the l-Kronecker quiver, is biholomorphic to C× H for l≥ 3. In the present paper we define a new notion of norm, which distinguishes {Db(K(l))}l≥2. More precisely, to a triangulated category T which has property of a phase gap we attach a non-negative real number ∥ T∥ ε. Natural assumptions on a SOD T= ⟨ T1, T2⟩ imply ∥⟨T1,T2⟩∥ε≤min{∥T1∥ε,∥T2∥ε}. Using the norm we define a topology on the set of equivalence classes of proper triangulated categories with a phase gap, such that the set of discrete derived categories is a discrete subset, whereas the rationality of a smooth surface S ensures that [Db(point)] ∈ Cl ([Db(S)]). Categories in a neighborhood of Db(K(l)) have the property that Db(K(l)) is embedded in each of them. We view such embeddings as non-commutative curves in the ambient category and introduce categorical invariants based on counting them. Examples show that the idea of non-commutative curve-counting opens directions to new categorical structures and connections to number theory and classical geometry. We give a definition, which specializes to the non-commutative curve-counting invariants. In an example arising on the A side we specialize our definition to non-commutative Calabi–Yau curve-counting, where the entities we count are a Calabi–Yau modification of Db(K(l)). In the end we speculate that one might consider a holomorphic family of categories, introduced by Kontsevich, as a non-commutative extension with the norm, introduced here, playing a role similar to the classical notion of degree of an extension in Galois theory.

AB - In this paper we introduce new categorical notions and give many examples. In an earlier paper we proved that the Bridgeland stability space on the derived category of representations of K(l), the l-Kronecker quiver, is biholomorphic to C× H for l≥ 3. In the present paper we define a new notion of norm, which distinguishes {Db(K(l))}l≥2. More precisely, to a triangulated category T which has property of a phase gap we attach a non-negative real number ∥ T∥ ε. Natural assumptions on a SOD T= ⟨ T1, T2⟩ imply ∥⟨T1,T2⟩∥ε≤min{∥T1∥ε,∥T2∥ε}. Using the norm we define a topology on the set of equivalence classes of proper triangulated categories with a phase gap, such that the set of discrete derived categories is a discrete subset, whereas the rationality of a smooth surface S ensures that [Db(point)] ∈ Cl ([Db(S)]). Categories in a neighborhood of Db(K(l)) have the property that Db(K(l)) is embedded in each of them. We view such embeddings as non-commutative curves in the ambient category and introduce categorical invariants based on counting them. Examples show that the idea of non-commutative curve-counting opens directions to new categorical structures and connections to number theory and classical geometry. We give a definition, which specializes to the non-commutative curve-counting invariants. In an example arising on the A side we specialize our definition to non-commutative Calabi–Yau curve-counting, where the entities we count are a Calabi–Yau modification of Db(K(l)). In the end we speculate that one might consider a holomorphic family of categories, introduced by Kontsevich, as a non-commutative extension with the norm, introduced here, playing a role similar to the classical notion of degree of an extension in Galois theory.

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U2 - 10.1007/s00029-019-0493-8

DO - 10.1007/s00029-019-0493-8

M3 - Article

AN - SCOPUS:85069173114

VL - 25

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

SN - 1022-1824

IS - 3

M1 - 45

ER -