The torus action on a regular affine chart is holomorphic

The coordinate-free complex torus ComplexTorus N acts on the complex points of the affine toric scheme of a toric cone σ: a torus point t multiplies the value of a complex point on the monomial of m in the dual semigroup by the character value t m. For a regular cone, the torus carries the complex structure of a free presentation of its character lattice and the affine chart carries the complex structure of an extending basis. This file proves that the action map ComplexTorus N × U_σ → U_σ is holomorphic for these structures.

Since a map into the affine chart is holomorphic exactly when its values on all monomials are, it suffices that (t, x) ↦ t m * x(m) is holomorphic, which holds because character evaluation is holomorphic on the torus and monomials are holomorphic on the chart.

Main declarations

• TauCeti.Toric.contMDiffSMul_complexTorus_coneChartedSpace: the torus acts holomorphically on the complex points of a regular cone.

References

• W. Fulton, Introduction to Toric Varieties, §§1.3 and 2.1.
• D. Cox, J. Little and H. Schenck, Toric Varieties, §§1.3 and 3.1.

theorem TauCeti.Toric.contMDiffSMul_complexTorus_coneChartedSpace {N : Type u_1} {V : Type u_2} {ι : Type u_3} [AddCommGroup N] [AddCommGroup V] [Module ℝ V] [Fintype ι] {i : N →+ V} {σ : PointedCone ℝ V} {s k l : ℕ} (hi : IsIntegralLattice i) (hσ : IsToricCone i σ) {B : Module.Basis (ToricRay σ ⊕ Fin l) ℤ N} (hB : ∀ (ρ : ToricRay σ), IsPrimitiveGenerator i ρ (B (Sum.inl ρ))) (κ : ToricRay σ ≃ Fin k) (g : AddGeneratingFamily (↥(dualSemigroup hi σ)) s) (e : IntegralCharacter N ≃+ (ι →₀ ℤ)) (n : WithTop ℕ∞) : have x := complexTorusChartedSpace e; let x_1 := affinePointTopology g; have x_2 := coneChartedSpace hi hσ hB κ g; ContMDiffSMul (modelWithCornersSelf ℂ (ι → ℂ)) (modelWithCornersSelf ℂ ((Fin k → ℂ) × (Fin l → ℂ))) n (ComplexTorus N) (AffineSemigroupComplexPoint ↥(dualSemigroup hi σ))

The coordinate-free complex torus acts holomorphically on the complex points of the affine toric scheme of a regular cone. The torus carries the complex structure of a free presentation of its character lattice, and the affine chart that of an integral basis extending the primitive ray generators.