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Real Toric Varieties: Interactions between their Geometry and their Topology

Jules Chenal, Matilde Manzaroli

TL;DR

This work studies the topology of real loci of real toric varieties with non-split tori, tying together geometric and topological properties through canonical fibrations and unwinding. It develops a robust framework based on the twist class and winding group to decompose real loci as twisted products and to reduce questions to split tori via an unwinding isogeny, yielding a precise description of the real part as a quotient of a fiber by a finite group. The authors prove that, for smooth toric embeddings with compact real loci, the cohomology is totally algebraic, compute Betti numbers via a universal isogeny-invariant polynomial $e[X]$, and establish orientability criteria. They further classify low-dimensional real loci (curves, surfaces, threefolds), describe prime decompositions and lens-space realizations, and provide a detailed framework connecting toric invariants to topological types, with explicit formulas and combinatorial data.

Abstract

In the present article, we investigate the topology of real toric varieties, especially those whose torus is not split over the field of real numbers. We describe some canonical fibrations associated to their real loci. Then, we establish various properties of their cohomology provided that their real loci are compact and smooth. For instance, we compute their Betti numbers, show that their cohomology is totally algebraic, and extend a criterion of orientability. In addition, we provide the topological classification of equivariant embeddings of non-split tridimensional tori.

Real Toric Varieties: Interactions between their Geometry and their Topology

TL;DR

This work studies the topology of real loci of real toric varieties with non-split tori, tying together geometric and topological properties through canonical fibrations and unwinding. It develops a robust framework based on the twist class and winding group to decompose real loci as twisted products and to reduce questions to split tori via an unwinding isogeny, yielding a precise description of the real part as a quotient of a fiber by a finite group. The authors prove that, for smooth toric embeddings with compact real loci, the cohomology is totally algebraic, compute Betti numbers via a universal isogeny-invariant polynomial , and establish orientability criteria. They further classify low-dimensional real loci (curves, surfaces, threefolds), describe prime decompositions and lens-space realizations, and provide a detailed framework connecting toric invariants to topological types, with explicit formulas and combinatorial data.

Abstract

In the present article, we investigate the topology of real toric varieties, especially those whose torus is not split over the field of real numbers. We describe some canonical fibrations associated to their real loci. Then, we establish various properties of their cohomology provided that their real loci are compact and smooth. For instance, we compute their Betti numbers, show that their cohomology is totally algebraic, and extend a criterion of orientability. In addition, we provide the topological classification of equivariant embeddings of non-split tridimensional tori.

Paper Structure

This paper contains 48 sections, 75 theorems, 207 equations, 10 figures, 6 tables.

Table of Contents

  1. Toric Real Structures.
  2. Structure of Affine Varieties.
  3. Canonical Fibration and Isogeny.
  4. Cycles and Cohomology
  5. Topological Types in Low Dimension.
  6. Group Cohomology.
  7. Monoid Algebra.
  8. Homeomorphism.
  9. Varieties.
  10. Cycle Class Map.
  11. Affine Geometry.
  12. Toric Real Structures
  13. Real Tori
  14. Closed Subgroups.
  15. 2-Torsion.
  16. ...and 33 more sections

Key Result

Proposition 1.5

Let $N$ be a lattice endowed with an involution $\tau$. It splits, as a module over $\mathbb{Z}[\tau]$, into a direct sum of the three factors $\mathbb{Z}[1]$, $\mathbb{Z}[-1]$, and $\mathbb{Z}[\tau]$.

Figures (10)

  • Figure 1: The orbital lattice of $\mathbb{P}^1_\mathbb{C}\times_\mathbb{C}\mathbb{P}^1_\mathbb{C}$.
  • Figure 2: The graded pieces of the toric real forms of $\mathbb{P}^1_\mathbb{C}$.
  • Figure 3: The fan and graded pieces of $\textnormal{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{P}^1_\mathbb{C}$.
  • Figure 4: The fan of a complete real surface with disconnected real locus.
  • Figure 5: The Possible Sets of Invariant Cones of Equivariant Embeddings of $\textnormal{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{\textnormal{m},\mathbb{C}}$ with Compact Real Locus.
  • ...and 5 more figures

Theorems & Definitions (194)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: Character and Cocharacter Lattices
  • Proposition 1.5: Theorem 2 in cass08
  • Corollary 1.6: Theorem 2 in cass08
  • Definition 1.7
  • Lemma 1.8
  • proof
  • Lemma 1.9
  • proof
  • ...and 184 more