Positivity Geometry: Canonical Forms & Applications

• Positivity geometry is an interdisciplinary field that characterizes geometric positivity through canonical forms, semi-algebraic subsets, and recursive boundary axioms.
• Its framework integrates methods from algebraic geometry, combinatorics, and physics to construct positive cones, bases, and explicit recursive formulas applicable in effective field theories and moduli spaces.
• Recent developments extend the theory to tropical, stringy, and foliated settings, offering new insights into convex bodies, adjoint forms, and representation-theoretic positivity.

Positivity geometry is an emerging interdisciplinary field at the interface of algebraic geometry, combinatorics, analysis, tropical geometry, and mathematical physics. In its working characterization, a positive geometry is a tuple consisting of a complex algebraic variety $X$, a semi-algebraic subset $X_{\ge 0}\subset X(\mathbb R)$, and a meromorphic top-form $\Omega(X_{\ge 0})$, subject to recursive axioms prescribed by physics (Ranestad et al., 18 Feb 2025). The expression is also used in closely related senses: for geometric positivity in cohomology rings, for positive loci on representation-theoretic varieties, for convex bodies encoding asymptotic ampleness, and for closed convex cones governing effective field theory constraints (Goldin, 2023).

1. Definitions and conceptual scope

The original Arkani-Hamed–Bai–Lam definition starts with an irreducible complex projective variety $X$ of dimension $d$ and a $d$-dimensional closed semi-algebraic subset $X_{\ge 0}\subset X(\mathbb R)$ such that $X_{\ge 0}=\overline{\operatorname{int}(X_{\ge 0})}$ and $\operatorname{int}(X_{\ge 0})$ is an oriented real manifold. Its Euclidean boundary determines an algebraic boundary $Y\subset X$, and the canonical form is the unique nonzero rational $X_{\ge 0}\subset X(\mathbb R)$0-form with simple poles along the irreducible components of $X_{\ge 0}\subset X(\mathbb R)$1, no other poles on $X_{\ge 0}\subset X(\mathbb R)$2, and residues equal to the canonical forms of the lower-dimensional boundary positive geometries (Telen, 5 Jun 2025).

A more recent Brown–Dupont perspective replaces the semi-algebraic input by a genus zero pair $X_{\ge 0}\subset X(\mathbb R)$3, where $X_{\ge 0}\subset X(\mathbb R)$4 is a projective variety and $X_{\ge 0}\subset X(\mathbb R)$5 is a divisor containing the singular locus. In that setting there is a canonical map

$X_{\ge 0}\subset X(\mathbb R)$6

suggesting a Hodge-theoretic mechanism for canonical forms and a possible refinement of the definition (Ranestad et al., 18 Feb 2025).

A distinct but related usage appears in Schubert and Hessenberg geometry. There, positivity geometry refers to a phenomenon in which a cohomology ring carries a basis that is geometric, because it is dual to homology classes of actual invariant subvarieties, and positive, because multiplication in that basis has nonnegative structure constants. For the Peterson variety this is explicitly ring-theoretic and geometric positivity in cohomology, not positivity in the sense of ample divisors, total positivity, or positivity of forms (Goldin, 2023).

These definitions suggest that contemporary usage is not exhausted by a single formalism. One branch centers canonical forms and recursive boundary structures; another centers positive bases, positive structure constants, and semialgebraic positive loci. The common theme is that positivity is encoded by distinguished geometric data rather than by arbitrary inequalities.

2. Canonical forms, algebraic boundaries, and recursion

The canonical form is the central invariant of the canonical-form branch of positivity geometry. Its poles are controlled by the algebraic boundary, and its defining feature is recursive residue behavior on boundary components (Ranestad et al., 18 Feb 2025). For the interval $X_{\ge 0}\subset X(\mathbb R)$7,

$X_{\ge 0}\subset X(\mathbb R)$8

with residues $X_{\ge 0}\subset X(\mathbb R)$9 and $\Omega(X_{\ge 0})$0 at the endpoints (Telen, 5 Jun 2025). For the standard simplex in an affine chart,

$\Omega(X_{\ge 0})$1

and residues along the facet hyperplanes reproduce the lower-dimensional simplex forms (Telen, 5 Jun 2025).

For convex polytopes $\Omega(X_{\ge 0})$2, one of the cleanest formulas is

$\Omega(X_{\ge 0})$3

where $\Omega(X_{\ge 0})$4 is the polar dual of the translated polytope. In this description the poles are simple and occur exactly along the hyperplanes supporting the facets of $\Omega(X_{\ge 0})$5, while the zeros lie on the adjoint hypersurface of $\Omega(X_{\ge 0})$6 (Ranestad et al., 18 Feb 2025). The notes on polytopes and polypols make the recursive structure equally explicit by triangulation: if $\Omega(X_{\ge 0})$7 is a triangulation into simplices with disjoint interiors, then

$\Omega(X_{\ge 0})$8

so canonical forms add under decomposition (Telen, 5 Jun 2025).

The boundary calculus is logarithmic. If locally

$\Omega(X_{\ge 0})$9

then the residue along $X$0 is $X$1. This local model underlies the global recursion and explains why simple poles, rather than higher-order singularities, are structurally distinguished (Telen, 5 Jun 2025).

3. Polytopes, polypols, and wondertopes

For simple polytopes the canonical form admits a vertex expansion. If

$X$2

is a $X$3-dimensional simple polytope, then

$X$4

For general polytopes the same canonical form is obtained by meromorphic continuation of the dual-volume function $X$5 (Telen, 5 Jun 2025).

The planar theory extends from polygons to rational polypols, where the boundary components are rational plane curves rather than lines. For a nodal quasi-regular rational polypol with boundary equations $X$6 and adjoint curve equation $X$7, the canonical form is

$X$8

or, in affine coordinates,

$X$9

Here the adjoint curve is uniquely determined in the nodal rational case and cancels unwanted poles at residual singular points (Telen, 5 Jun 2025).

Wondertopes arise by log resolution. Starting from a full-dimensional polytope $d$0 and a building set $d$1 of proper linear subvarieties such that each $d$2 meets $d$3 in a face and each facet hyperplane belongs to $d$4, one forms the wonderful compactification

$d$5

If $d$6 is the Euclidean closure of the strict transform of $d$7, then

$d$8

is a positive geometry with canonical form

$d$9

A familiar wondertope is the curvy associahedron, which tiles the moduli space of pointed stable rational curves (Brauner et al., 2024).

4. Grassmannians, amplituhedra, positivity sectors, and stringy extensions

The positive Grassmannian $d$0 is one of the central examples of positive geometry. For a matrix $d$1 with all ordered $d$2 minors positive, the tree-level amplituhedron is

$d$3

$d$4

When $d$5, the amplituhedron is a cyclic polytope; when $d$6, it is isomorphic to the positive Grassmannian; for general $d$7, it is conjectured to be a positive geometry; loop amplituhedra are known not to be positive geometries in the original sense (Ranestad et al., 18 Feb 2025).

At $d$8, $d$9, and loop order $X_{\ge 0}\subset X(\mathbb R)$0, the ambient space

$X_{\ge 0}\subset X(\mathbb R)$1

is segmented into positivity sectors by fixing the signs of the pairwise $X_{\ge 0}\subset X(\mathbb R)$2 minors $X_{\ge 0}\subset X(\mathbb R)$3. There are $X_{\ge 0}\subset X(\mathbb R)$4 sectors, and at three loops the sector $X_{\ge 0}\subset X(\mathbb R)$5 is the amplituhedron $X_{\ge 0}\subset X(\mathbb R)$6. The paper on positivity sectors develops a recursive gluing algorithm showing how the intricate internal $X_{\ge 0}\subset X(\mathbb R)$7 boundaries of individual sectors cancel under assembly; when all eight three-loop sectors are assembled into $X_{\ge 0}\subset X(\mathbb R)$8, the resulting space has Euler number $X_{\ge 0}\subset X(\mathbb R)$9 (Galloni, 2016).

String-theoretic extensions require geometries with infinitely many resonance poles. The associahedral grid is defined by

$X_{\ge 0}=\overline{\operatorname{int}(X_{\ge 0})}$0

an infinite periodic union of copies of the ABHY associahedron. Its canonical form is

$X_{\ge 0}=\overline{\operatorname{int}(X_{\ge 0})}$1

where $X_{\ge 0}=\overline{\operatorname{int}(X_{\ge 0})}$2 is the inverse string-theory KLT kernel. At four points,

$X_{\ge 0}=\overline{\operatorname{int}(X_{\ge 0})}$3

so positivity geometry extends beyond rational canonical forms to trigonometric and stringy analytic structure (Bartsch et al., 27 Aug 2025).

A further Grassmannian-derived example is the ABCT variety

$X_{\ge 0}=\overline{\operatorname{int}(X_{\ge 0})}$4

the image closure of the quadratic Veronese map. It is the locus of $X_{\ge 0}=\overline{\operatorname{int}(X_{\ge 0})}$5 labeled points in $X_{\ge 0}=\overline{\operatorname{int}(X_{\ge 0})}$6 lying on a conic, possibly degenerate, and the paper proves that

$X_{\ge 0}=\overline{\operatorname{int}(X_{\ge 0})}$7

is a positive geometry with canonical form

$X_{\ge 0}=\overline{\operatorname{int}(X_{\ge 0})}$8

Its iterated boundaries are explicitly classified in terms of collinear and colliding point configurations (Shen et al., 10 Mar 2026).

5. Representation-theoretic and Schubert-theoretic positivity

A different but closely related strand studies positive loci on representation-theoretic varieties. Let $X_{\ge 0}=\overline{\operatorname{int}(X_{\ge 0})}$9 be a simple, simply-connected complex algebraic group and $\operatorname{int}(X_{\ge 0})$0 the stabilizer of a principal nilpotent $\operatorname{int}(X_{\ge 0})$1 in the Langlands dual group. Ginzburg and Peterson identified

$\operatorname{int}(X_{\ge 0})$2

so affine Schubert classes $\operatorname{int}(X_{\ge 0})$3 become regular functions on $\operatorname{int}(X_{\ge 0})$4. On this space Lam and Rietsch compare affine Schubert positivity, Mirković–Vilonen positivity, and Lusztig total positivity, proving

$\operatorname{int}(X_{\ge 0})$5

and constructing a global homeomorphism

$\operatorname{int}(X_{\ge 0})$6

This identifies three a priori different positive structures and gives the totally nonnegative part a Euclidean orthant parametrization (Lam et al., 2012).

The Peterson variety furnishes a cohomological version of positivity geometry. For $\operatorname{int}(X_{\ge 0})$7, one has an $\operatorname{int}(X_{\ge 0})$8-invariant affine paving by Peterson cells

$\operatorname{int}(X_{\ge 0})$9

indexed by subsets $Y\subset X$0. The duality theorem identifies the cohomology basis $Y\subset X$1 with the dual basis to the geometric homology classes $Y\subset X$2, up to multiplicities: $Y\subset X$3 The product

$Y\subset X$4

then satisfies

$Y\subset X$5

Here positivity is equivariant polynomial positivity in one variable, inherited from Schubert positivity on $Y\subset X$6 but mediated by the inclusion $Y\subset X$7 rather than by a direct $Y\subset X$8-equivariant argument on $Y\subset X$9 itself (Goldin, 2023).

These two developments use the same expression, “positivity geometry,” in different but compatible senses. One concerns semialgebraic positive loci and parametrizations; the other concerns geometric bases in cohomology with nonnegative structure constants.

6. Convex, tropical, and foliated formulations

In asymptotic algebraic geometry, positivity can be encoded by convex regions in angle space. For a line bundle $X_{\ge 0}\subset X(\mathbb R)$00 and divisor $X_{\ge 0}\subset X(\mathbb R)$01, Rubinstein defines

$X_{\ge 0}\subset X(\mathbb R)$02

and the body of ample angles

$X_{\ge 0}\subset X(\mathbb R)$03

When nonempty, $X_{\ge 0}\subset X(\mathbb R)$04 is an open convex body in $X_{\ge 0}\subset X(\mathbb R)$05. Under tail blow-ups of surface boundary chains, the ambient angle space grows from $X_{\ge 0}\subset X(\mathbb R)$06 to $X_{\ge 0}\subset X(\mathbb R)$07, and asymptotically log Fano positivity reduces to the threshold condition

$X_{\ge 0}\subset X(\mathbb R)$08

together with feasibility of a linear program $X_{\ge 0}\subset X(\mathbb R)$09 (Rubinstein, 2019).

For smooth complex toric varieties, positivity is compared across complex and tropical geometry. If $X_{\ge 0}\subset X(\mathbb R)$10 is the complex analytic toric variety, $X_{\ge 0}\subset X(\mathbb R)$11 its Kajiwara–Payne tropicalization, and $X_{\ge 0}\subset X(\mathbb R)$12 with $X_{\ge 0}\subset X(\mathbb R)$13, then the correspondence theorem identifies the cone of $X_{\ge 0}\subset X(\mathbb R)$14- and $X_{\ge 0}\subset X(\mathbb R)$15-invariant closed positive complex currents on $X_{\ge 0}\subset X(\mathbb R)$16 with the cone of closed positive Lagerberg currents on $X_{\ge 0}\subset X(\mathbb R)$17: $X_{\ge 0}\subset X(\mathbb R)$18 This places positivity of currents in a real/complex/tropical correspondence rather than in a single category (Gil et al., 2020).

A transverse version of positivity geometry appears on foliated manifolds. For a closed oriented taut transversely Kähler foliation $X_{\ge 0}\subset X(\mathbb R)$19, positivity is defined in basic Bott–Chern cohomology through the cones

$X_{\ge 0}\subset X(\mathbb R)$20

corresponding to transverse Kähler, nef, big, and pseudoeffective classes. The main theorem is a foliated Wu–Yau type statement: nonpositive transverse holomorphic sectional curvature implies that $X_{\ge 0}\subset X(\mathbb R)$21 is transverse nef, while negative transverse holomorphic sectional curvature implies the existence of a transverse Kähler–Einstein metric $X_{\ge 0}\subset X(\mathbb R)$22 with

$X_{\ge 0}\subset X(\mathbb R)$23

This extends the standard positivity package for $X_{\ge 0}\subset X(\mathbb R)$24-classes to the basic category (Zhang et al., 2021).

7. Positivity cones, cycle-theoretic positivity, and current directions

In multifield effective field theory, positivity geometry becomes a problem in convex algebraic geometry. For flavor space $X_{\ge 0}\subset X(\mathbb R)$25, the forward-limit positivity cone is

$X_{\ge 0}\subset X(\mathbb R)$26

where

$X_{\ge 0}\subset X(\mathbb R)$27

Classifying extremal rays of $X_{\ge 0}\subset X(\mathbb R)$28 yields the full system of positivity bounds. This classification is complete up to three flavors. For $X_{\ge 0}\subset X(\mathbb R)$29, all extremals are elastic; for $X_{\ge 0}\subset X(\mathbb R)$30, there is an additional inelastic family, Type III, characterized intrinsically by

$X_{\ge 0}\subset X(\mathbb R)$31

In the $X_{\ge 0}\subset X(\mathbb R)$32-, $X_{\ge 0}\subset X(\mathbb R)$33-, and $X_{\ge 0}\subset X(\mathbb R)$34-invariant sectors considered in the paper, elastic bounds are sufficient to imply full positivity (Bonnefoy et al., 25 Aug 2025).

A cycle-theoretic analogue studies the positivity of the diagonal $X_{\ge 0}\subset X(\mathbb R)$35. If $X_{\ge 0}\subset X(\mathbb R)$36 is homologically big, then

$X_{\ge 0}\subset X(\mathbb R)$37

For surfaces, the only smooth projective examples with big and nef diagonal are

$X_{\ge 0}\subset X(\mathbb R)$38

This places positivity of the diagonal among the strongest geometric constraints currently known for higher-codimension cycle classes (Lehmann et al., 2017).

Hodge modules supply a broad positivity package that underlies many of these applications. If $X_{\ge 0}\subset X(\mathbb R)$39 underlies a pure Hodge module, then the lowest nonzero filtration piece

$X_{\ge 0}\subset X(\mathbb R)$40

is weakly positive and torsion-free, and for every $X_{\ge 0}\subset X(\mathbb R)$41 the dual of the Kodaira–Spencer kernel

$X_{\ge 0}\subset X(\mathbb R)$42

is weakly positive. These statements sit on top of Kodaira–Saito and Kawamata–Viehweg-type vanishing for Hodge modules and generalize classical semipositivity results for Hodge bundles and direct images (Popa, 2016).

Open problems reflect the breadth of the subject. The amplituhedron is conjectured to be a positive geometry in general at tree level, loop amplituhedra suggest a generalized notion such as weighted positive geometry, and Brown–Dupont theory may broaden the foundational definition (Ranestad et al., 18 Feb 2025). For Hessenberg varieties, the Peterson case raises the question whether a general affine paving plus a dual basis can produce a positive Hessenberg Schubert calculus (Goldin, 2023). For ABCT varieties, the $X_{\ge 0}\subset X(\mathbb R)$43 case is now settled, while the cases

$X_{\ge 0}\subset X(\mathbb R)$44

remain open (Shen et al., 10 Mar 2026). These directions indicate that positivity geometry is still expanding simultaneously as a theory of canonical forms, a theory of positive bases and cones, and a bridge between representation theory, moduli spaces, tropical geometry, and quantum field theory.