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Regular version of the site

Article

Some new categorical invariants

Selecta Mathematica, New Series. 2019. Vol. 25:45. P. 1-60.
Dimitrov G., Katzarkov L.

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), thel-Kronecker quiver, is biholomorphic toC×Hfor 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, T2ε ≤ min{T1ε , T2ε}. 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 newcategorical 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 curvecounting,
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.