By Ricardo Castano-Bernard, Fabrizio Catanese, Maxim Kontsevich, Tony Pantev, Yan Soibelman, Ilia Zharkov

The courting among Tropical Geometry and reflect Symmetry is going again to the paintings of Kontsevich and Y. Soibelman (2000), who utilized equipment of non-archimedean geometry (in specific, tropical curves) to Homological reflect Symmetry. together with the following paintings of Mikhalkin at the “tropical” method of Gromov-Witten thought and the paintings of Gross and Siebert, Tropical Geometry has now develop into a robust instrument. Homological reflect Symmetry is the world of arithmetic centred round a number of specific equivalences connecting symplectic and holomorphic (or algebraic) geometry. The imperative principles first seemed within the paintings of Maxim Kontsevich (1993). approximately conversing, the topic could be approached in methods: both one makes use of Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow photo, extra built through Kontsevich and Soibelman) or one makes use of Lefschetz fibrations of symplectic manifolds (suggested via Kontsevich and extra built by means of Seidel). Tropical Geometry experiences piecewise-linear items which seem as “degenerations” of the corresponding algebro-geometric objects.

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Kuznetsov, Derived categories of cubic fourfolds, in Cohomological and Geometric Approaches to Rationality Problems. Progress in Mathematics, vol. 282 (Birkhäuser, Boston, 2010), pp. 219–243 20. C. V. 2/-bundles and the differentiable structure of Barlow’s surface. Invent. Math. 95(3), 601–614 (1989) 21. D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc. Steklov Inst. Math. 246(3), 227–248 (2004) 22. D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, in Algebra, Arithmetic, and Geometry: In Honor of Yu.

KŒŒx0 ; : : : ; xm ; x0nC1 C x12 C : : : C xm (20) An Orbit Construction of Phantoms, Orlov Spectra, and Knörrer Periodicity 37 with m odd. D/ are given by Vi W kŒŒx x nC1 i xi ! kŒŒx ! kŒŒx (22) for 1 Ä i Ä n, see for example [21]. 2/ via the shift functor. Remark 1. kŒŒX ; x nC1 / denote the dg-category of graded matrix factorizations of the An -singularity. kŒŒX ; x nC1 / by a general result from [16]. The category A itself coincides with the bounded derived category of finite-dimensional representations of an An quiver, as was show in [22].

TL M ! L TM ! TL M ! 0; M T M ! T L ! 0: One identifies M to TM M , the zero-section of T M . One sets TP M WD T M n TM M and one denotes by P M W TP M ! M the projection. Let f W M ! N be a morphism of real manifolds. M N / ! M N / and M N T N . S1 ; S2 /, is the closed cone of TM defined as follows. xI v/ denote the associated coordinate system on TM. xn yn / ! v0 . TM if and only if there exists a sequence n n S2 RC such that xn ! x0 , yn ! L; S /. S /. xI / on T M by ˛M D h ; dxi: The antipodal map aM is defined by: aM W T M !