Affine Structures and Non-Archimedean Analytic Spaces

By M. Kontsevich, Y. Soibelman. 2004

427 citations recorded by the source corpus.

In this paper we propose a way to construct an analytic space over a non-archimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction is motivated by the junction of the Homological Mirror conjecture and the geometric Strominger-Yau-Zaslow conjecture. In particular, we glue from “flat pieces” an analytic K3 surface. As a byproduct of our approach we obtain an action of an arithmetic subgroup of the group SO(1, 18) by piecewise-linear transformations on the two-dimensional sphere S 2 equipped with naturally defined singular affine structure.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fd30a305858c005da02078ed65c6751268c75350

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Citations in the corpus

The tropical vertex (2009, methodology): “The following result was obtained by Kontsevich and Soibelman in [13] with a number of variants proved in [10].”
Potential Theory and Dynamics on the Berkovich Projective Line (2010, methodology): “As another example, Kontsevich and Soibelman [68] have recently used Berkovich’s theory over fields such as the completion of an algebraic closure of C((T )) to study homological mirror symmetry.”
Valuations and plurisubharmonic singularities (2007, methodology): “Section 1 contains basic definitions concerning the space of divisors in X, and a thorough discussion of the geometry of V, mostly borrowed from [KKMS] and [KS] with a few adaptations to our local context.”
Singular semipositive metrics in non-Archimedean geometry (2011, methodology): “The first important fact is that any dual complex ∆X comes equipped with a natural affine structure [KS06] such that any θ-psh function is convex on the faces of ∆X .”
Constructible sheaves and the Fukaya category (2006, methodology): “Fourth, the work of Kontsevich-Soibelman [ 26 ] and Gross-Siebert [14] paints the large complex structure limit of a Calabi-Yau n-fold as a collapse into a real n-fold with integral affine structure and a Monge-Amp`ere metric.”
Mirror Symmetry via Logarithmic Degeneration Data I (2003, background): “This has already been discussed in many places, see for example [25],[10],[ 24 ],[6].”
Quivers, curves, and the tropical vertex (2009, methodology): “The tropical vertex group (iii) first arose in the work of Kontsevich and Soibelman [12] and plays a significant role in the program of [9].”
Family Floer cohomology and mirror symmetry (2014, methodology): “These corrections are expected to be expressible in terms of the geometry of the base via wall-crossing formulae [23, 19].”
Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory (2008, background): “• One of the inspirations for the Kontsevich-Soibelman WCF was their earlier work [17], in which they gave a construction of the sheaf of holomorphic functions on a K3 surface, by “correcting” the sheaf of functions on the semiflat K3.”
Valuations and asymptotic invariants for sequences of ideals (2010, background): “More surprisingly, they have recently seen a number of applications to problems over the complex numbers [4, 33, 21, 20, 9, 10, 29, 30].”
The wall-crossing formula and Lagrangian mutations (2017, methodology): “The statement from the last item is the wall-crossing formula which has been predicted by Auroux [3], and in a slightly different context by Kontsevich and Soibelman [23].”