# Tropical compactification

Last updated

In algebraic geometry, a tropical compactification is a compactification (projective completion) of a subvariety of an algebraic torus, introduced by Jenia Tevelev. [1] [2] Given an algebraic torus and a connected closed subvariety of that torus, a compatification of the subvariety is defined as a closure of it in a toric variety of the original torus. The concept of a tropical compatification arises when trying to make compactifications as "nice" as possible. For a torus ${\displaystyle T}$, a toric variety ${\displaystyle \mathbb {P} }$, the compatification ${\displaystyle {\bar {X}}}$ is tropical when the map

${\displaystyle \Phi :T\times {\bar {X}}\to \mathbb {P} ,\ (t,x)\to tx}$

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

## Related Research Articles

In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by Candelas et al. (1985), after Eugenio Calabi who first conjectured that such surfaces might exist, and Shing-Tung Yau (1978) who proved the Calabi conjecture.

In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special, but still quite general class of toric varieties, this information is also encoded in a polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space.

In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition:

In mathematics, the Teichmüller space of a (real) topological surface , is a space that parametrizes complex structures on up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from to itself.

In mathematics, more particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to all algebraic cycles of a given projective space of given dimension and degree. In other words, it is a moduli space with a variety structure parametrizing all -dimensional and algebraic cycles of degree in .

In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space , the quotient of a nilpotent Lie group N modulo a closed subgroup H. This notion was introduced by Anatoly Mal'cev in 1951.

In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus.

In mathematics, a toric manifold is a topological analogue of toric variety in algebraic geometry. It is an even-dimensional manifold with an effective smooth action of an -dimensional compact torus which is locally standard with the orbit space a simple convex polytope.

In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.

In mathematics, the Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over is a direct sum of holomorphic line bundles. The theorem was proved by Alexander Grothendieck, and is more or less equivalent to Birkhoff factorization introduced by George David Birkhoff (1909).

In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category of equivariant coherent sheaves on an algebraic scheme X with action of a linear algebraic group G, via Quillen's Q-construction; thus, by definition,

This is a glossary of algebraic geometry.

In algebraic group theory, a wonderful compactification of a variety acted on by an algebraic group is a -equivariant compactification such that the closure of each orbit is smooth. Corrado de Concini and Claudio Procesi (1983) constructed a wonderful compactification of any symmetric variety given by a quotient of an algebraic group by the subgroup fixed by some involution of over the complex numbers, sometimes called the De Concini–Procesi compactification, and Elisabetta Strickland (1987) generalized this construction to arbitrary characteristic. In particular, by writing a group itself as a symmetric homogeneous space, , this gives a wonderful compactification of the group itself.

In geometry, Hironaka's example is a non-Kähler complex manifold that is a deformation of Kähler manifolds found by Heisuke Hironaka. Hironaka's example can be used to show that several other plausible statements holding for smooth varieties of dimension at most 2 fail for smooth varieties of dimension at least 3.

In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme with an action by a group scheme G is the affine scheme , the prime spectrum of the ring of invariants of A, and is denoted by . A GIT quotient is a categorical quotient: any invariant morphism uniquely factors through it.

In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus T is called a T-variety. In differential geometry, one considers an action of a real or complex torus on a manifold.

In algebraic geometry, a toroidal embedding is an open embedding of algebraic varieties that locally looks like the embedding of the open torus into a toric variety. The notion was introduced by Mumford to prove the existence of semistable reductions of algebraic varieties over one-dimensional bases.

In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack , the Chow group of X is the same as the G-equivariant Chow group of Y.

In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces in quantum cohomology.These moduli spaces are smooth orbifolds whenever the target space is convex. A variety is called convex if the pullback of the tangent bundle to a stable rational curve has globally generated sections. Geometrically this implies the curve is free to move around infinitesimally without any obstruction. Convexity is generally phrased as the technical condition

Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural moduli space over characteristic 0 constructed as a quotient of the upper-half plane by the action of , there is an analogous construction for abelian varieties using the Siegel upper half-space and the Symplectic group .

## References

1. Tevelev, Jenia (2007-08-07). "Compactifications of subvarieties of tori". American Journal of Mathematics . 129 (4): 1087–1104. arXiv:. doi:10.1353/ajm.2007.0029. ISSN   1080-6377.
2. Brugallé, Erwan; Shaw, Kristin (2014). "A Bit of Tropical Geometry". The American Mathematical Monthly . 121 (7): 563–589. arXiv:. doi:10.4169/amer.math.monthly.121.07.563. JSTOR   10.4169/amer.math.monthly.121.07.563.
• Cavalieri, Renzo; Markwig, Hannah; Ranganathan, Dhruv (2017). "Tropical compactification and the GromovWitten theory of ${\displaystyle \mathbb {P} ^{1}}$". Selecta Mathematica. 23: 1027–1060. arXiv:. Bibcode:2014arXiv1410.2837C.