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作者：Blum, Harold; Xu, Chenyang
摘要：We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable Q-Fano variety is finite.
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作者：Gao, Ziyang; Habegger, Philipp
摘要：On an abelian scheme over a smooth curve over (Q) over bar a symmetric relatively ample line bundle defines a fiberwise Neron-Tate height. If the base curve is inside a projective space, we also have a height on its (Q) over bar -points that serves as a measure of each fiber, an abelian variety. Silverman proved an asymptotic equality between these two heights on a curve in the abelian scheme. In this paper we prove an inequality between these heights on a subvariety of any dimension of the ab...
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作者：Moreira, Joel; Richter, Florian K.; Robertson, Donald
摘要：In this paper we show that every set A subset of N with positive density contains B +C for some pair B, C of infinite subsets of N, settling a conjecture of Erdos. The proof features two different decompositions of an arbitrary bounded sequence into a structured component and a pseudo-random component. Our methods are quite general, allowing us to prove a version of this conjecture for countable amenable groups.
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作者：Shmerkin, Pablo
摘要：We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the L-q dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of xp- and xq-invariant se...
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作者：Larsen, Michael; Shalev, Aner; Tiep, Pham Huu
摘要：The probabilistic Waring problem for finite simple groups asks whether every word of the form w(1)w(2) , where w(1) and w(2) are non-trivial words in disjoint sets of variables, induces almost uniform distributions on finite simple groups with respect to the L-1 norm. Our first main result provides a positive solution to this problem. We also provide a geometric characterization of words inducing almost uniform distributions on finite simple groups of Lie type of bounded rank, and study relate...
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作者：Brendle, Simon
摘要：We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short). In the first part of this paper, we prove new curvature pinching estimates that ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension n >= 12, we show that blow-up limits are weakly PIC2. This can be viewed as a higher-dimensional version of the fundamental Hamilton-Ivey pinching estimate in dimension 3. In the second part, we develop a theory of ancient solu...
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作者：Hyde, James
摘要：A left-order on a group G is a total order < on G such that for any f, g and h in G we have f < g double left right arrow hf < hg. We construct a finitely generated subgroup H of Homeo(I-2; delta I-2), the group of those homeomorphisms of the disc that fix the boundary pointwise, and show H does not admit a left-order. Since any left-order on Homeo(I-2; delta I-2) would restrict to a left-order on H, this shows that Homeo(I-2; delta I-2) does not admit a left-order. Since Homeo(I; delta I) adm...
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作者：Yun, Zhiwei; Zhang, Wei
摘要：For arithmetic applications, we extend and refine our previously published results to allow ramifications in a minimal way. Starting with a possibly ramified quadratic extension F'/F of function fields over a finite field in odd characteristic, and a finite set of places Sigma of F that are unramified in F', we define a collection of Heegner-Drinfeld cycles on the moduli stack of PGL(2)-Shtukas with r-modifications and Iwahori level structures at places of Sigma. For a cuspidal automorphic rep...
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作者：Shitov, Yaroslav
摘要：The chromatic number of G x H can be less than the minimum of the chromatic numbers of finite simple graphs G and H.
