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Non-commutative curve-counting invariants. (arXiv:1805.00294v2 [math.CT] UPDATED)

来源于:arXiv
In our previous paper, viewing $D^b(K(l))$ as a non-commutative curve, we observed that it is reasonable to count non-commutative curves in certain triangulated categories, where $K(l)$ is the Kronecker quiver with $l$-arrows. We gave a general definition, which specializes to the non-commutative curve-counting invariants. Roughly it defines the set of subcategories of a fixed type in a given category modulo conditions. Here, after recalling the definition, we focus on examples. We compute the non-commutative curve-counting invariants in $D^b(Q)$ for two affine quivers, in $ D^b(A_n)$, and in $D^b(D_4)$. We estimate these numbers for $D^b({\mathbb P}^2)$, in particular we prove finiteness and that the exact determining of these numbers for $D^b({\mathbb P}^2)$ leads to proving (or disproving) of Markov conjecture. Via homological mirror symmetry this gives a new approach to this conjecture. In the present paper are derived also formulas for counting of the subcategories of type $D^b(A_ 查看全文>>