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Register a new userToward finite generation of higher rational rank valuations
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Chenyang Xu
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We propose a finite generation conjecture for the valuation which computes the stability threshold of a log Fano pair. We also initiate a degeneration strategy of attacking the conjecture.
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We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $frac{n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies: (a) a log Fano pair is uniformly K-stable (resp. reduced uniformly K-stable) if and only if it is K-stable (resp. K-polystable); (b) the K-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of Kahler-Einstein metric and reduced uniform K-stability proved by the variational approach, (c) the Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano pairs.
We study higher rank Donaldson-Thomas invariants of a Calabi-Yau 3-fold using Joyce-Songs wall-crossing formula. We construct quivers whose counting invariants coincide with the Donaldson-Thomas invariants. As a corollary, we prove the integrality and a certain symmetry for the higher rank invariants.
In this paper we define the notion of monic representation for the $C^*$-algebras of finite higher-rank graphs with no sources, and undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative $C^*$-algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the $Lambda$-semibranching representations previously studied by Farsi, Gillaspy, Kang, and Packer, and also provide a universal representation model for nonnegative monic representations.
We construct higher-dimensional Calabi-Yau varieties defined over a given number field with Zariski dense sets of rational points. We give two elementary constructions in arbitrary dimensions as well as another construction in dimension three which involves certain Calabi-Yau threefolds containing an Enriques surface. The constructions also show that potential density holds for (sufficiently) general members of the families.
Whereas matrix rank is additive under direct sum, in 1981 Schonhage showed that one of its generalizations to the tensor setting, tensor border rank, can be strictly subadditive for tensors of order three. Whether border rank is additive for higher order tensors has remained open. In this work, we settle this problem by providing analogs of Schonhages construction for tensors of order four and higher. Schonhages work was motivated by the study of the computational complexity of matrix multiplication; we discuss implications of our results for the asymptotic rank of higher order generalizations of the matrix multiplication tensor.
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