Tropical Lagrangian Hypersurfaces are Unobstructed

Abstract

We produce for each tropical hypersurface $V(\varphi )\subset Q={\mathbb{R}}^n$ a Lagrangian $L(\varphi )\subset ({\mathbb{C}}^{\ast }{)}^n$ whose moment map projection is a tropical amoeba of $V(\varphi )$. When these Lagrangians are admissible in the Fukaya-Seidel category, we show that they are unobstructed objects of the Fukaya category, and mirror to sheaves supported on complex hypersurfaces in a toric mirror.

arXiv comment48 Pages. First portion of arXiv:1904.06005 which was split in two. This version has been accepted for publication by the Journal of Topology

References in corpus (10)

• Mirror Symmetry is T-Duality
• D-Branes And Mirror Symmetry
• Enumerative tropical algebraic geometry in R2
• Family Floer cohomology and mirror symmetry
• Lagrangian pairs of pants
• Floer theory and flips
• Affine manifolds, log structures, and mirror symmetry
• Tropically constructed Lagrangians in mirror quintic threefolds
• The Lagrangian cobordism group of $T^2$
• Refined disk potentials for immersed Lagrangian surfaces

Cited by in corpus (9)

• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• Tropically constructed Lagrangians in mirror quintic threefolds
• Aspects of functoriality in homological mirror symmetry for toric varieties
• Lagrangian submanifolds from tropical hypersurfaces
• Tropical Lagrangians in toric del-Pezzo surfaces
• Lagrangian Cobordisms in Liouville manifolds
• Some statistics about Tropical Sandpile Model
• Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds
• Toric vector bundles, non-abelianization, and spectral networks