Symplectic perspectives on tropical geometry aka Some tropical aspects of mirror symmetry II.5 PDF
Preview Symplectic perspectives on tropical geometry aka Some tropical aspects of mirror symmetry II.5
Symplectic perspectives on tropical geometry a.k.a. Some tropical aspects of mirror symmetry II.5 Mohammed Abouzaid Clay/MIT August 27, 2009 Homological Mirror Symmetry through tropical geometry Tropical Manifold B Submanifold V Homological Mirror Symmetry through tropical geometry Complex Tropical Symplectic Manifold BC B Bω Submanifold VC V VL Lagrangian Homological Mirror Symmetry through tropical geometry Complex Tropical Symplectic Manifold BC B Bω Submanifold VC V VL Lagrangian Baby case TB/Λ= (C∗)n B = Rn T∗B/Λ∨ = (C∗)n ∗ Subspaces VC = TV/Λ V integral VL = T L/Λ Homological Mirror Symmetry through tropical geometry Complex Tropical Symplectic Manifold BC B Bω Submanifold VC V VL Lagrangian Baby case TB/Λ= (C∗)n B = Rn T∗B/Λ∨ = (C∗)n ∗ Subspaces VC = TV/Λ V integral VL = T L/Λ Categorification Coherent Sheaves Fukaya Category Morphisms Ext∗(OV ,OV ) HF∗(VL,VL) C C This dream builds upon work of Strominger-Yau-Zaslow. Recent contributions include work of Fukaya, Kontsevich-Soibelman, Gross-Siebert, A., Auroux, ... We will focus on B tropicalisation of a hypersurface in (C∗)n. Computing a category Categories are categorifications of rings, in particular they have generators and relations. In general, 1 If you maximize the number of generators, you get many relations all of a very simple type. 2 Minimize the number of generators, getting few “complicated relations.” But there are special groups in which we can choose few generators and still have no relations. Computing a category Categories are categorifications of rings, in particular they have generators and relations. In general, 1 If you maximize the number of generators, you get many relations all of a very simple type. 2 Minimize the number of generators, getting few “complicated relations.” But there are special groups in which we can choose few generators and still have no relations. Two approaches to passing from tropical geometry to Fukaya categories 1 Associate a Lagrangian to each “tropical cell” (progress by N. Sheridan). 2 Find a minimal number of Lagrangians. In our setting, expect the second approach to yield few relations. Geometrically, this should come from a Lagrange skeleton. Lagrange Skeleta All Stein manifolds (e.g. affine varieties) have “isotropic skeleta,” i.e. a CW complex which can be “thickened” to a symplectic manifold. Conjecture Tropical geometry produces skeleta consisting of smooth Lagrangians meeting cleanly. Today, I’ll explain the ideas for curves and CY hypersurfaces. Lagrange Skeleta All Stein manifolds (e.g. affine varieties) have “isotropic skeleta,” i.e. a CW complex which can be “thickened” to a symplectic manifold. Conjecture Tropical geometry produces skeleta consisting of smooth Lagrangians meeting cleanly. Today, I’ll explain the ideas for curves and CY hypersurfaces. Corollary (arXiv:0904.1474 +ǫ) The Fukaya categories of Bω is modelled by a finite dimensional differential graded algebra. R and the tropical line Tropical Symplectic R Tropical line
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