In algebraic geometry, a tropical compactification is a compactification (projective completion) of a subvariety of an algebraic torus, introduced by Jenia Tevelev. Given an algebraic torus and a connected closed subvariety of that torus, a compactification of the subvariety is defined as a closure of it in a toric variety of the original torus. The concept of a tropical compactification arises when trying to make compactifications as "nice" as possible. For a torus T {\displaystyle T} and a toric variety P {\displaystyle \mathbb {P} }, the compactification X ¯ {\displaystyle {\bar {X}}} is tropical when the map

Φ : T × X ¯ → P , ( t , x ) → t x {\displaystyle \Phi :T\times {\bar {X}}\to \mathbb {P} ,\ (t,x)\to tx}

is faithfully flat and X ¯ {\displaystyle {\bar {X}}} is proper.

See also

From left: Hannah Markwig, Aaron Bertram, and Renzo Cavalieri, 2012 at the MFO
  • Cavalieri, Renzo; Markwig, Hannah; Ranganathan, Dhruv (2017). "Tropical compactification and the Gromov–Witten theory of P 1 {\displaystyle \mathbb {P} ^{1}}". Selecta Mathematica. 23: 1027–1060. arXiv:. Bibcode:. doi:.