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作者：Goncharov, Alexander; Shen, Linhui
作者单位：Yale University
摘要：A decorated surface is an oriented surface with boundary and a finite, possibly empty, set of special points on the boundary, considered modulo isotopy. Let be a split reductive group over . A pair gives rise to a moduli space , closely related to the moduli space of -local systems on . It is equipped with a positive structure (Fock and Goncharov, Publ Math IHES 103:1-212, 2006). So a set of its integral tropical points is defined. We introduce a rational positive function on the space , calle...
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作者：Szekelyhidi, Gabor
作者单位：University of Notre Dame
摘要：This is a continuation of the work of Arezzo-Pacard-Singer and the author on blowups of extremal Kahler manifolds. We prove the conjecture stated in Sz,kelyhidi (Duke Math J 161(8):1411-1453, 2012), and we relate this result to the K-stability of blown up manifolds. As an application we prove that if a Kahler manifold of dimension 2 admits a constant scalar curvature (cscK) metric, then the blowup of at a point admits a cscK metric if and only if it is K-stable, as long as the exceptional divi...
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作者：Chung, Nhan-Phu; Li, Hanfeng
作者单位：State University of New York (SUNY) System; University at Buffalo, SUNY; Chongqing University
摘要：We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of p-expansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is 1-expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms, and show that the IE group determines the Pinsker factor for such actions. For an ...
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作者：Hochman, Michael; Shmerkin, Pablo
作者单位：University of Surrey
摘要：We give a fractal-geometric condition for a measure on to be supported on points that are normal in base , i.e. such that equidistributes modulo 1. This condition is robust under coordinate changes, and it applies also when is a Pisot number rather than an integer. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host's theorem to multiplicatively independent inte...
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作者：Freitas, Nuno; Le Hung, Bao V.; Siksek, Samir
作者单位：University of Bayreuth; Harvard University; University of Warwick
摘要：We prove that all elliptic curves defined over real quadratic fields are modular.
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作者：McKinnon, David; Roth, Mike
作者单位：University of Waterloo; Queens University - Canada
摘要：In this paper, we associate an invariant to an algebraic point on an algebraic variety with an ample line bundle . The invariant measures how well can be approximated by rational points on , with respect to the height function associated to . We show that this invariant is closely related to the Seshadri constant measuring local positivity of at , and in particular that Roth's theorem on generalizes as an inequality between these two invariants valid for arbitrary projective varieties.
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作者：Sheridan, Nick
作者单位：Princeton University; Institute for Advanced Study - USA
摘要：We prove Homological Mirror Symmetry for a smooth -dimensional Calabi-Yau hypersurface in projective space, for any (for example, is the quintic threefold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the '-dimensional pair of pants'; the introduction of the 'relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an 'orbifold' Fukaya ca...
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作者：Straube, Emil J.; Zeytuncu, Yunus E.
作者单位：Texas A&M University System; Texas A&M University College Station; University of Michigan System; University of Michigan
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作者：Adiprasito, Karim A.; Ziegler, Guenter M.
作者单位：Universite Paris Saclay; Free University of Berlin
摘要：We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to 96. This in particular settles a problem going back to Legendre and Steinitz: whether and how the dimension of the realization space of a polytope is determined/bounded by its f-vector. From this, we derive an infinite family of combinatorially distinct 69-dimensional polytopes whose realization is unique up to projective transformation. This answers a problem posed by Perles and Shephard...
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作者：Bauer, Ingrid; Catanese, Fabrizio; Grunewald, Fritz
作者单位：University of Bayreuth
摘要：We use a canonical procedure associating to an algebraic number a first a hyperelliptic curve C-a, and then a triangle curve (D-a, G(a)) obtained through the normal closure of an associated Belyi function. In this way we show that the absolute Galois group Gal((Q) over bar /Q) acts faithfully on the set of isomorphism classes of marked triangle curves, and on the set of connected components of marked moduli spaces of surfaces isogenous to a higher product (these are the free quotients of a pro...
