We construct a canonical basis of two-cycles, on a $K3$ surface, in which the intersection form takes the canonical form $2E_8(-1) oplus 3H$. The basic elements are realized by formal sums of smooth submanifolds.
In this article we study a conjecture of Geiss-Leclerc-Schr{o}er, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type $A_2$ quiver. In general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic.
Let $k$ be a number field. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of $k$ to that of zero-cycles over $k$ for Kummer varieties over $k$. For example, for any Kummer variety $X$ over $k$, we show that if the Brauer-Manin obstruction is the only obstruction to the Hasse principle for rational points on $X$ over all finite extensions of $k$, then the ($2$-primary) Brauer-Manin obstruction is the only obstruction to the Hasse principle for zero-cycles of any given odd degree on $X$ over $k$. We also obtain similar results for products of Kummer varieties, K3 surfaces and rationally connected varieties.
This paper aims at comparing two coupling approaches as basic layers for building clustering criteria, suited for modularizing and clustering very large networks. We briefly use optimal transport theory as a starting point, and a way as well, to derive two canonical couplings: statistical independence and logical indetermination. A symmetric list of properties is provided and notably the so called Monges properties, applied to contingency matrices, and justifying the $otimes$ versus $oplus$ notation. A study is proposed, highlighting logical indetermination, because it is, by far, lesser known. Eventually we estimate the average difference between both couplings as the key explanation of their usually close results in network clustering.
The convolution ring $K^{GL_n(mathcal{O})rtimesmathbb{C}^times}(mathrm{Gr}_{GL_n})$ was identified with a quantum unipotent cell of the loop group $LSL_2$ in [Cautis-Williams, J. Amer. Math. Soc. 32 (2019), pp. 709-778]. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell.
We show that the K-moduli spaces of log Fano pairs $(mathbb{P}^1timesmathbb{P}^1, cC)$ where $C$ is a $(4,4)$-curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ complete intersection curves in $mathbb{P}^3$. This, together with recent results by Laza-OGrady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$-curves on $mathbb{P}^1timesmathbb{P}^1$ and the Baily-Borel compactification of moduli of quartic hyperelliptic K3 surfaces.