The $m=2$ amplituhedron is a positive geometry

Introduction

The amplituhedron $\mathcal{A}_{n,k,m}$, introduced in [2], is a semi-algebraic subset of the Grassmannian. It plays an important role in particle physics, where it is used to calculate scattering amplitudes in $\mathcal{N} = 4$ Super Yang–Mills quantum field theory. It is also a central object in Positive Geometry [RST25], a new field at the intersection of mathematics and theoretical physics. The key notion in this field is that of a positive geometry [1]. This is a pair $(X,S)$ consisting of a real projective variety $X$ and a semi-algebraic subset $S$ such that there exists a unique rational differential top form on $X$ with simple poles along the boundary of $S$ satisfying a number of additional technical properties. Known examples of positive geometries include polytopes, positive parts of toric varieties and the totally non-negative Grassmannian. It is, however, difficult to prove that a given pair $(X,S)$ is a positive geometry in general. In particular, a central conjecture [1 Conjecture 6.3] that has remained open since the field was established is that amplituhedra are positive geometries.

In this article we settle this conjecture for the $m = 2$ amplituhedron. The cases $m = 2$ and $m = 4$ are of immediate relevance for physics. Our approach is motivated by an exciting recent development in the theory of positive geometries. In [4] Brown and Dupont recast this notion in the language of Deligne’s mixed Hodge theory, providing a systematic framework to study positive geometries. In this framework, positive geometries arise from pairs of complex varieties for which the genus, a certain Hodge-theoretic invariant, vanishes. The new framework, however, is not equivalent to the original one proposed in [1]. At the heart of the latter are semi-algebraic sets, whereas the former is purely about the complex geometry and topology of a pair of varieties. In this article, we establish a conceptual link by providing a sufficient mixed Hodge-theoretic criterion for being a positive geometry. We will then apply it to the $m = 2$ amplituhedron. The result can be phrased as follows, and the necessary terminology and notation are introduced in § 3.

The mathematical structure of amplituhedra has been studied extensively over the last decade. The description of the $m = 1$ amplituhedron was given in [31], and in this case it is a linear object. For the case of $m = 2$ the boundary structure was described in [35] and [33]. Tilings (i.e. subdivisions with special properties) of the $m = 2$ amplituhedron are one of the central subjects of [5], [34], and [39]. Tilings in the $m = 4$ case are studied in [ELT25], [ELP+23] and subsequent work. Other interesting developments include, for instance, [11, 12, 13], and the full body of work is too large to describe here. Despite this vast amount of deep work, the central conjecture [1 Conjecture 6.3] that amplituhedra are positive geometries has only been settled in a few special cases.

• $k = 1$: The amplituhedron is a cyclic polytope [44], and polytopes are positive geometries [1 Section 6].

• $m = 1$: The amplituhedron is the union of bounded cells of a cyclic hyperplane arrangement [31], so again a polyhedral object.

• $k + m = n$: In this case the amplituhedron is isomorphic to $\operatorname{Gr}(k,n)_{\geq0}$, which gives rise to a positive geometry by [32 Theorem 8].

• $k = m = 2$: This is proven in [RST24].

Our mixed Hodge-theoretic criterion in Theorem A, combined with combinatorial results mentioned above, implies that the $m = 2$ amplituhedron is a positive geometry. This is, however, not at all straightforward. A natural way to go from a pair $(X,S)$ with a semi-algebraic set $S$ to a pair of varieties $(X,D)$ with a divisor $D \subset X$ is to set $D$ to be the algebraic boundary $\partial_{\mathrm{a}}S$ of $S$, i.e. the Zariski closure of its boundary in the analytic topology on $X$. Setting $X := \operatorname{Gr}(k,k+2)$ and $D := \partial_{\mathrm{a}}\mathcal{A}_{n,k,2}(\mathbb{Z})$, however, does not provide a genus zero pair. Indeed, the main result in [29] states that for $k \geq3$ and large enough $n$ the pair $(X,D)$ has strictly positive genus. We will therefore have to alter this pair to reduce its genus in a way that locally preserves the geometry of the amplituhedron. It is not clear a priori that this is possible at all. We now briefly comment on our approach.

As we show in [29], the obvious obstacles for the pair $(\operatorname{Gr}(k,k+2),\partial_a\mathcal{A}_{n,k,2}(Z))$ to have genus zero come from the residual arrangement of the amplituhedron, see Definition 2.10. These are strata of $\partial_a\mathcal{A}_{n,k,2}(Z)$ that are disjoint from the semi-algebraic set $\mathcal{A}_{n,k,2}(Z)$. In this article we will show that all obstacles come from the residual arrangement, i.e. that each face of the amplituhedron is supported on a variety of genus zero (Proposition 4.9). We will then remove the residual arrangement from each of the varieties in the original pair, thus removing positive contributions to the genus. In order to show that the new pair satisfies the assumptions of Theorem A, we will have to deal with subtleties of genera and differential forms on singular and open varieties. After this, our main result follows from Theorem A.

This article is organized as follows. In § 2 we introduce the necessary notions and recall basic results that will be needed in later parts of the article. In § 3 we recall the framework of [4], explain how to make it applicable for non-compact varieties, and prove Theorem A. § 4 is devoted to the algebro-geometric properties of the faces of $\mathcal{A}_{n,k,2}(Z)$: we show their Zariski closures are unirational varieties with rational singularities by relying on the results of [33]. We also show that any such variety (and any Zariski open subset thereof) has genus zero. Finally, in § 5 we prove Theorem B.

Preliminaries

In this section we introduce the basic notions used in this article. We recall the definitions of the totally non-negative Grassmannian, positroid varieties, the amplituhedron, and positive geometries. We also define the algebraic boundary stratification of a codimension one subvariety and a semi-algebraic set.

Denote by $\operatorname{Gr}(k,n)$ the Grassmannian of $k$-dimensional linear subspaces of $\mathbb{C}^{n}$. We usually consider the Grassmannian in its Plücker embedding into $\mathbb{P}^{N}$ with $N=\binom{n}{k}-1$. If it is clear from context, we will write $\Lambda$ both for the point in $\operatorname{Gr}(k,n)$ and a full rank $k \times n$ matrix representing it. The Plücker coordinates $p_I$ with $I \subset[n] := \{1,\ldots,n\}$ of size $k$ are given by the $k \times k$ minors of $\Lambda$ indexed by the columns in $I$.

The totally non-negative Grassmannian admits a stratification into positroid cells, see, e.g., [37]. The boundary structure of the $m=2$ amplituhedron, which is the main object of this article, is closely related to the positroid stratification [33]. We thus now review the basics and fix notation. For a subset $\mathcal{M} \subset\binom{[n]}{k}$ define

If $\Pi_{\mathcal{M}}^{>0} \neq\varnothing$, this set is called a positroid cell. In this situation $\mathcal{M}$ is a matroid, called a positroid. Replacing strict inequalities with non-strict ones, we get a space $\Pi_{\mathcal{M}}^{\geq0}$. The complex Zariski closure of $\Pi_{\mathcal{M}}^{>0}$ in $\operatorname{Gr}(k,n)$ is called a positroid variety, and is denoted by

The quasi-projective open positroid varieties $\mathring{\Pi}_{\mathcal{M}}$ are defined as

By seminal work of Postnikov [37], the totally non-negative Grassmannian admits a decomposition

into positroid cells, each of which is an open ball. Moreover, in [PSW09] Postnikov, Speyer, and Williams prove that this decomposition equips $\operatorname{Gr}(k,n)_{\geq0}$ with the structure of a CW-complex. We furthermore have $\mathring{\Pi}_{\mathcal{M}}\cap\operatorname{Gr}(k,n)_{\geq0}=\Pi_{\mathcal{M}}^{>0}$.

We now give a definition of the central object of this article.

Due to the positivity assumption on $Z$, the map $\widetilde{Z}$ is regular on $\operatorname{Gr}(k,n)_{\geq0}$, by the Cauchy–Binet formula for the maximal minors of $\Lambda Z$. So the amplituhedron is a compact semi-algebraic subset of $\operatorname{Gr}(k,k+m)$. Henceforth, we will often suppress the dependence on $Z$ and write $\mathcal{A}_{n,k,m}$ for $\mathcal{A}_{n,k,m}(Z)$. Our next goal is to recall the definition of positive geometries in the sense of [1], see also [32]. We start with auxiliary definitions.

We are now ready to recall the definition of positive geometries in the sense of [1].

Remark 2.6. The uniqueness of the canonical form is essentially equivalent to the statement that $X$ has no non-zero holomorphic top forms if $X$ is a normal variety: adding such a form to a canonical form would produce another canonical form, and the difference between two canonical forms is a holomorphic form, since they have the same poles and residues along these poles. See the proof of Theorem 3.19 for more details.

The amplituhedron is a top-dimensional regular semi-algebraic set in $\operatorname{Gr}(k,k+m)$. The central goal of this article is to prove that the $m=2$ amplituhedron is a positive geometry for arbitrary values of $k$ and $n$ and generic choices of $Z$. Definition 2.5 suggests that to do this, it is essential to understand the algebraic boundary $\partial_a\mathcal{A}_{n,k,2}(Z)$. We now introduce the notion of the algebraic boundary stratification of a semi-algebraic set.

If $S$ is a regular semi-algebraic subset of a projective variety $X$, with algebraic boundary $\partial_aS$, then we define the algebraic boundary stratification of $S$ as $\mathcal{L}(S):=\mathcal{L}(\partial_aS)$. We call the elements of $\mathcal{L}(S)$ boundary strata of $S$. These objects are defined analogously inside a variety of the form $X\setminus R$, where $R$ is a closed subvariety of $X$ disjoint from $S$.

Example 2.8. When $S$ is a polytope in $\mathbb{P}^n$, the algebraic boundary stratification $\mathcal{L}(S)$ is isomorphic to the poset of flats of the facet hyperplane arrangement of $S$.

Example 2.9. The algebraic boundary stratification of $\operatorname{Gr}(k,n)_{\geq0}$ is its positroid stratification (and is isomorphic to the cell decomposition in [37], see also [45 Section 2.3]). The case $\operatorname{Gr}(2,4)_{\geq0}$ already shows the importance of taking the irreducible components of the intersections of boundary divisors. The algebraic boundary of $\operatorname{Gr}(2,4)_{\geq0}$ consists of four irreducible components, given inside $\operatorname{Gr}(2,4)$ by the equations $p_{12}=0$, $p_{14}=0$, $p_{23}=0$ and $p_{34}=0$, where $p_{ij}$ are the Plücker coordinates. However, the intersection $\{p_{12}=p_{23}=0\}$ in $\operatorname{Gr}(2,4)$ is reducible: it consists of two planes, each containing a positroid cell. Each of these two planes is an element of $\mathcal{L}(\operatorname{Gr}(2,4)_{\geq0})$.

We conclude this section by defining faces, the residual arrangement, corners, and vertices of a semi-algebraic set.

Remark 2.11. Note that, by construction, all faces of a regular semi-algebraic set $S$ are regular semi-algebraic sets again.

Remark 2.13. Suppose we have a full-dimensional semi-algebraic subset $S \subset X$ such that its face stratification $\mathcal{F}(S)$ is a graded poset of rank $n := \dim X$. Then, $S$ having a vertex implies that each face of $S$ has a vertex. Indeed, the Zariski closure of every face is contained in a maximal chain of $\mathcal{F}(S)$ and, since all such chains in $\mathcal{F}(S)$ have maximal length, must contain a zero-dimensional element.

Example 2.14. We continue with the totally non-negative Grassmannian $\operatorname{Gr}(2,4)_{\geq0}$ from Example 2.9, where we saw that we only obtain all positroid strata when also allowing for intersecting the irreducible components of the intersections of the four algebraic boundary components. This allows us to find zero-dimensional strata in $\mathcal{L}(\operatorname{Gr}(2,4)_{\geq0})$: there are six corners corresponding to the points in $\operatorname{Gr}(2,4)$ representing the lines $\operatorname{span}_{\mathbb{C}}(e_i,e_j)$, where $e_i$ are the standard basis vectors of $\mathbb{C}^4$. All of these are vertices of $\operatorname{Gr}(2,4)_{\geq0}$ since their Plücker coordinates are all obviously non-negative.

Let us also demonstrate an example in which not all corners are vertices.

Example 2.15. Consider the non-convex polygon $X_{\geq0}$ in $\mathbb{R}^2$ given by the union of the two triangles $\operatorname{Conv}\{(0,0),(-1,0),(0,2)\}$ and $\operatorname{Conv}\{(0,0),(1,0),(0,1)\}$, see Figure 1. This configuration has six corners, namely the five vertices of the two triangles $((-1,0),(0,0),(1,0),$ $(0,1)$, $(0,2)$ and the intersection point $(-1/3,4/3)$ of the two lines spanned by $(-1,0)$ and $(0,2)$, and $(0,1)$ and $(1,0)$, respectively. Among these six points, the four vertices of the non-convex polygon are the vertices according to our definition. The corner $(0,0)$, for example, is not a vertex, because it is an interior point of the one-dimensional face, which is the line segment connecting the points $(-1,0)$ and $(1,0)$. It is therefore a residual stratum. The numerator of the canonical form of this set is a linear polynomial that vanishes at the residual points, i.e., at the corners that are not vertices. ◇

A mixed Hodge theory criterion for positive geometries

As mentioned in the introduction, the framework for positive geometries suggested by Brown and Dupont [4] based on mixed Hodge theory is not equivalent to that proposed originally in [1]. Therefore, a natural goal is to find a clear criterion describing when the mixed Hodge theory approach gives rise to positive geometries as in Definition 2.5. In this section we make this comment more precise and give such a criterion, which is the main conceptual result of this article.

Recalling the basics

We start by recalling the basic notions used in [4].

We say $H$ is a pure Hodge structure of weight $k$ if there exists a decomposition $H_{\mathbb{C}} = \bigoplus_{p+q=k} H_{\mathbb{C}}^{p,q}$ with $\overline{H_{\mathbb{C}}^{q,p}} = H_{\mathbb{C}}^{p,q}$, and such that

for all integers $a$. Let $W_{\bullet}$ be an increasing filtration on $H$ called weight filtration, i.e.

For any non-negative integer $p$ write

We say $H$ is a mixed Hodge structure if for every integer $w$, the filtration induced by $F^{\bullet}$ on $(\operatorname{gr}_{w}^{W}H)_{\mathbb{C}}$ defines a pure Hodge structure of weight $w$. For $w=p+q$ one defines

The corresponding mixed Hodge numbers are the complex vector space dimensions

Remark 3.2. See [6] for the construction of these two filtrations. In our setting, we will exclusively deal with mixed Hodge structures on the $k$-th relative homology group with rational coefficients of a pair of complex varieties $(X,D)$, where $X$ is of dimension $n$ and $D$ is a closed subvariety of $X$. Then, for the Hodge numbers we explicitly write

The framework suggested in [4] relies on the introduction of two Hodge-theoretic invariants of the mixed Hodge structure of $H_{n}(X,D)$. The first is the genus $g(X,D)$ of the pair which is defined as the sum of the “outer” Hodge numbers

When $D = \varnothing$, we write $g(X) := g(X,\varnothing)$.

The second invariant is the combinatorial rank which is defined as

Similarly, if $D = \varnothing$, we write $\operatorname{cr}(X) := \operatorname{cr}(X,\varnothing)$. The main theorem in the work of Brown and Dupont states that for any pair $(X,D)$ of genus zero such that $X \setminus D$ is smooth and $X$ is compact, there is a well-defined map, essentially induced by Poincaré duality,

associating with each relative homology cycle $\sigma\in H_n(X,D)$ a holomorphic $n$-form on $X \setminus D$ with logarithmic poles along $D$ (the target space $\Omega_{\log}^{n}(X \setminus D)$ is the space of such forms). This form is called the canonical form of $\sigma$. Canonical forms defined in this way satisfy a number of nice properties, including compatibility with taking residues, additivity and functoriality. If $g(X,D)=0$ and if $\sigma\in H_n(X,D)$ is the homology class of a semi-algebraic set $X_{\geq0} \subset X(\mathbb{R})$ with boundary contained in $D$ and connected interior such that $(X,X_{\geq0})$ is a positive geometry with canonical form $\Omega_{(X,X_{\geq0})}$, then by [21 Lemma 2.12] $\Omega(\sigma)=\Omega_{(X,X_{\geq0})}$ is the canonical form of this positive geometry in the original sense of [1]. In the most natural case we have $D=\partial_a X_{\geq0}$.

Remark 3.3. While Brown and Dupont define canonical forms, they do not give a definition of a positive geometry in the language of mixed Hodge theory. Such a definition is presented in [43] and is based on the notion of positive arrangements. This definition is not equivalent to the one in [1], as the notion of positive arrangements is not recursive in nature. One can show that the $m=2$ amplituhedron is a positive geometry of the positive arrangement obtained from $(\operatorname{Gr}(k,k+2),\partial_a\mathcal{A}_{n,k,2}(Z))$ by blowing up all desingularized residual strata in increasing dimension. Such a sequence of blowups keeps the ambient variety smooth but introduces new singularities to the face strata of the amplituhedron. To show that the $m=2$ amplituhedron is a positive geometry in the original sense of [1] we will thus pursue a different strategy based on removing the residual arrangement rather than blowing it up.

We now comment on the inequivalence of the frameworks of [1] and [4]. On the one hand, the framework of [4] is purely complex. There exist pairs $(X,D)$ of genus zero for which the variety $D$ does not have real points and thus cannot contain the boundary of any semi-algebraic set. This is the case e.g. when $X$ is a rational curve and $D$ is a complex point on it. On the other hand, as we demonstrate in our previous article [29 Section 6], there exist semi-algebraic sets $X_{\geq0} \subset X$ such that $(X,X_{\geq0})$ is a positive geometry, but $X_{\geq0}$ gives rise to a homology cycle in $H_n(X,D)$ with $g(X,D)>0$, where $D$ is the algebraic boundary of $X_{\geq0}$. However, in many important cases (for instance, for convex polytopes) these frameworks are in perfect agreement.

In view of this, it is desirable to have a criterion in the language of mixed Hodge theory for being a positive geometry in the sense of [1]. The main goal of this section is to present such a criterion, see Theorem 3.19.

The first step towards Theorem 3.19 is the definition of recursive genus zero pairs. Note that in the definition below the varieties are not assumed to be compact.

by $D_W \subset W$ the union of all codimension one subvarieties of $W$ that are elements of $\mathcal{L}(D)$. We say $(X,D)$ is a recursive genus zero pair if $g(W,D_W)=0$ for all $W\in\mathcal{L}(D)$.

Remark 3.5. As demonstrated in [4 §2.4.3], having genus zero is not a recursive condition. Thus, not every genus zero pair is a recursive genus zero pair.

Remark 3.6. By [4 Proposition 3.9], the genus of a pair is insensitive to codimension two subvarieties. Therefore, in Definition 3.4 we can require $D_W$ to have codimension one in $W$ without any loss of generality. See Example 3.21 for a semi-algebraic set giving rise to a recursive genus zero pair whose face stratification is not graded. This example will elucidate the importance of the notion of graded face posets in the context of positive geometries.

Canonical forms from non-compact varieties

The input of the framework of [4] is a pair of compact complex varieties $(X,D)$ of genus zero such that $X\setminus D$ is smooth. It is pointed out in the introduction to that article that the formalism of canonical forms should still hold for non-compact pairs. We would like to consider the amplituhedron $\mathcal{A}_{n,k,2}$ inside a pair of non-compact varieties to remedy the fact that the genus of the pair $(\operatorname{Gr}(k,k+2),\partial_a\mathcal{A}_{n,k,2})$ is in general non-zero. Thus, in this subsection we make this claim precise. This is done largely by adapting the statements in [4] to the non-compact case.

Remark 3.7. Canonical forms in the non-compact setting for hyperplane arrangements were studied in recent work [36], where they are defined without relying on compactifying the pair. Our construction uses a compactification with special properties but is independent of the particular choice, see Lemma 3.13.

We start by recalling a few more essential notions from [4].

We also recall the notion of good compactifications.

The latter means that analytically locally the triple has the form

for complex analytic pairs $(A,D_A)$ and $(C,B_C)$.

In what follows, we will slightly abuse notation and for two sets $A$ and $B$ not contained in each other write $A \setminus B$ for $A \setminus(A \cap B)$.

Remark 3.10. Let $\mathcal{X}$ be smooth and compact, and let $\mathcal{B},D \subset\mathcal{X}$ be divisors. If $\mathcal{B} \cup D$ is a simple normal crossing divisor with no common component, then $(\mathcal{X},D)$ is a good compactification of $(\mathcal{X} \setminus\mathcal{B},D \setminus\mathcal{B})$.

We fix the following setup for the rest of the section. Let $X$ be a projective irreducible complex variety of dimension $n$, and let $D$ be a closed codimension one subvariety as usual. Further, let $R$ be a closed subvariety in $X$, and $W$ a closed subvariety of $D$. We assume that $X \setminus R$ is of dimension $n \ge1$ and $D \setminus R$ is pure of dimension $n-1$. Assume $W \subset D$ is closed, of dimension at most $n-2$, and such that $X \setminus(R \cup W)$ and $D \setminus(R \cup W)$ are smooth. In this section we consider the mixed Hodge theory approach for the open pair $(X \setminus R,D \setminus R)$. All subvarieties, inverse images, intersections and closures are taken with their reduced structures. Homology and cohomology have rational coefficients.

Construction 3.11. Take $X$ as the compactification of $X \setminus R$. By Hironaka’s embedded resolution theorem [19], we can resolve singularities of $X$ in the following way. There is a morphism $\eta:T \to X$ from a smooth projective variety $T$ that is an isomorphism over $X \setminus(R \cup W)$, because this latter set is contained in the smooth locus of $X$. Set

where the closures are taken in $T$. Now, we use Hironaka’s principalization theorem, see also [Wło05, Theorem 1.0.1], applied to the ideal sheaf given by the product of the ideal sheaves of $B$, $D'$, and $W'$ as defined in [14 Chapitre 0, (4.1.6)]. This gives a morphism $\pi:\mathcal{X} \to T$ between smooth compact complex varieties. Moreover, setting

and $f := (\eta\circ\pi)|_{\mathcal{X} \setminus\mathcal{B}}$, principalization implies that $\mathcal{B} \cup\mathcal{D} \cup\mathcal{W}$ is a simple normal crossing divisor in $\mathcal{X}$. Also note that $\mathcal{W} \subset\mathcal{D}$. Then $f$ induces an isomorphism over $X \setminus(R \cup W)$ and a modification of the pairs

Lastly, set

This construction has the following properties.

Proof. We start by showing 1. This follows directly from the principalization theorem [19] since, $\mathcal{D}$ and $\mathcal{W}$ are the closures of the preimages of $\mathcal{D}'$ and $\mathcal{W}'$ away from $\mathcal{B}$, whose ideal sheaves we simultaneously principalized. This shows in particular that $(\mathcal{X}, \mathcal{D})$ is a good compactification of $(\mathcal{X} \setminus\mathcal{B}, \mathcal{D} \setminus\mathcal{B})$ by Remark 3.10.

We show 2 next. By construction we have

So $\mathcal{D}_{W} \cup\mathcal{W}$ is a simple normal crossing divisor because $\mathcal{D}$ is by 1. Moreover, $\mathcal{D}_{W}$ and $\mathcal{W}$ do not share any component. Next, we show that $\mathcal{D}_{W}$ is smooth. Note that each of its irreducible components is smooth by 1., so the only singular points would be the intersections of the components. For the sake of contradiction, suppose two of the components in $\mathcal{D}_{W}$ intersect in a subvariety $C$. Then, $C$ must be of codimension two in $\mathcal{X}$ and it must also be contained in a component of $\mathcal{B} \cup\mathcal{W}$, as $\mathcal{D}_{W} \setminus(\mathcal{B}_{D} \cup\mathcal{W}_{D}) \cong\mathcal{D} \setminus(W \cup R)$ was assumed smooth. Hence any intersection must lie over $W \cup R$. However, $\mathcal{B} \cup\mathcal{D}_{W} \cup\mathcal{W}$ is a simple normal crossing divisor so any such component $C$ lying in a component of $\mathcal{B} \cup\mathcal{W}$ must be of codimension 3 in $\mathcal{X}$. Thus we arrive at a contradiction and $\mathcal{D}_{W}$ must be smooth.

We next show 3. The restriction $f_{D}$ is proper, since $f$ is proper and $\mathcal{D}_{W} \setminus\mathcal{B}_{D}$ is closed in $\mathcal{X} \setminus\mathcal{B}$. Moreover,

by construction and $f_{D}$ is an isomorphism over $\mathcal{D} \setminus(R \cup W)$ as $f$ was. Thus $f_{D}$ is a modification of pairs.

Finally, we show 4. Restricting the simple normal crossing divisor $\mathcal{B} \cup\mathcal{W}$ to the smooth divisor $\mathcal{D}_{W}$ yields a simple normal crossing divisor $\mathcal{B}_{D} \cup\mathcal{W}_{D}$ in $\mathcal{D}_{W}$. Suppose $\mathcal{B}_{D} \cup\mathcal{W}_{D}$ shared a common component $C$, then this would be codimension 2 in $\mathcal{X}$. However $C$ would also be contained in $\mathcal{D}_{W}$ by construction, again a contradiction to $\mathcal{B} \cup\mathcal{D}_{W} \cup\mathcal{W}$ being a simple normal crossing divisor. Hence, $(\mathcal{D}_{W}, \mathcal{W}_{D})$ is a good compactification. ∎

In dimension zero, we use the canonical map sending $\sum_p a_p[p]$ to the function $p \mapsto a_p$. This applies to the boundary pair when $n = 1$, in which case $W = \emptyset$.

Proof. The strategy is to apply Brown and Dupont’s construction of the canonical map $\Omega$ to the good compactification $(\mathcal{X},\mathcal{D})$ of the modification $(\mathcal{X} \setminus\mathcal{B},\mathcal{D} \setminus\mathcal{B})$ of $(X \setminus R,D \setminus R)$ in Construction 3.11.

Start by recalling that the pushforward $f_*$ in (1) gives an isomorphism of mixed Hodge structures

by [4 Proposition 1.3], where we used $f^{-1}(D \setminus R) = D \setminus\mathcal{B}$, see (2). In particular, this also implies that $g(\mathcal{X} \setminus\mathcal{B},\mathcal{D} \setminus\mathcal{B}) = g(X \setminus R,D \setminus R)$. The inclusion of pairs $\iota:(\mathcal{X} \setminus\mathcal{B},\mathcal{D} \setminus\mathcal{B}) \hookrightarrow(\mathcal{X},\mathcal{D})$ induces a map

Then, combining the equality of genera induced by $f_*$ from above with [4 Proposition 3.19] for the good compactification $(\mathcal{X},\mathcal{D})$ we get

Since $\mathcal{X} \setminus\mathcal{D}$ is smooth, the canonical map $\Omega:H_n(\mathcal{X},\mathcal{D}) \to\Omega_{\log}^n(\mathcal{X} \setminus\mathcal{D})$ is well-defined. Therefore,

is well-defined.

Lastly, note that any good compactification can be chosen to give an isomorphism $\mathcal{X} \setminus(\mathcal{D} \cup\mathcal{B}) \cong X \setminus(D \cup R)$. Identify the common open subsets of the two models with $X \setminus(R \cup D)$ via their respective modifications. By naturality of $\Omega$ and of Poincaré duality, the restrictions of the two constructed forms represent the same cohomology class on $X \setminus(R \cup D)$, since both come from the same relative homology class $\sigma$, see [4 §1.3]. □

Let $\dot{\omega}_X$ be the dualizing sheaf on $X$. There is a natural injection

Proof. Write $\bar{f}:=\eta\circ\pi:\mathcal{X}\to X$ for the composition of the maps in Construction 3.11. Let $V$ be the complement of the open subset in $X$ over which $\bar{f}$ is an isomorphism. Since $X$ is normal, $\operatorname{codim}_X V\geq2$. The restriction $r$ of $\bar{f}$ to $\bar{f}^{-1}(X\setminus V)$ induces the map $r_*:H^0(\mathcal{X},\omega_{\mathcal{X}}(D))\hookrightarrow H^0(X\setminus V,\dot{\omega}_X(D))$. It is well-defined since $D$ is reduced and contained in $\bar{f}^{-1}(D)$, and it is injective since $\bar{f}^{-1}(X\setminus(D\cup R))$ is dense in $\mathcal{X}$. Therefore, the injection induced by the restriction $r=\bar{f}|_{\bar{f}^{-1}(X\setminus V)}$ and the isomorphism furnished by $\bar{f}$ give

where the last isomorphism follows from [17 Proposition 1.6], since $\dot{\omega}_X(D)$ is reflexive by construction.

Remark 3.15. By Lemmas 3.13 and 3.14, the forms in $\Omega_{\log}^{n}(\mathcal{X}\setminus D)$ determine rational differential top forms on $X$ which are regular as sections of $\dot{\omega}_X$ on $X\setminus D$ and have at most simple poles along $D$. In particular, they have no iterated poles over $R$. Their pullbacks to $\mathcal{X}$ have logarithmic poles only along $D$. So they are candidates for canonical forms.

Let $W\subset D$ be a closed subset. Then, recall that (partial) boundary maps

as in [4 §1.1.3] are also well-defined for quasiprojective varieties. Recall the notation from Lemma 3.12. Then, there is also a well-defined residue map

Proof. We show this by using the commutative property for the canonical map $\Omega$ for the good compactifications from Construction 3.11. Let

be the restriction of the inclusion $\iota$, and let $\partial_{\mathcal{D}_W}$ be the partial boundary map associated with $\mathcal{D}=\mathcal{D}_W\cup\mathcal{W}$. By modification invariance and the invariance of genera for good compactifications, see [4 Proposition 3.19], both compact pairs below have genus zero. Moreover, by Lemma 3.12 both $\mathcal{X}$ and $\mathcal{D}_W$ are smooth, thus the canonical maps $\Omega$ are well-defined for these pairs. Consider

The left square commutes by naturality of boundary maps and excision [4 §1.1.3], and the right square by the proof of [4 Proposition 2.14]. Compositions of the horizontal maps are the maps $\overset{\circ}{\Omega}$ for the two pairs, so the outer rectangle yields the claim. □

The main theorem

In what follows we write $\mathcal{X}(\mathbb{R})$ for the real locus of a real variety $\mathcal{X}$, and $[S]$ for the class of a semi-algebraic set $S$ in an appropriate relative homology group.

Proof. Let $p\in\partial S\cap(W\setminus D_W)$. There is an analytic neighborhood $U\subset(\mathcal{X}\setminus D')(\mathbb{R})$ of $p$ such that $W\cap U$ is a smooth hypersurface in $U$. By assumption $\mathcal{X}$ is smooth in a neighborhood of $p\in\partial S\cap(W\setminus D_W)$, hence we can choose local coordinates $x_1,\ldots,x_n$ such that $U\cap W$ is a coordinate hyperplane in $U$ given by the vanishing of $x_1$. The hypersurface $W$ divides $U$ into two connected components, which a priori are each either contained in $S$ or disjoint from $S$. The set $S$ must contain exactly one of these components because $p$ lies on the boundary of $S$. Thus $U \cap S$ is homeomorphic to the half-ball $\{x \in\mathbb{R}^n \mid x_1 \ge0\}$. This implies that the boundary of $S$ is locally at $p$ given by $W$, i.e.,

Since $W$ is locally given by the vanishing of $x_1$, it follows that $p \in\operatorname{int} F$. By varying $p$, this shows $F \setminus D_W \subset\operatorname{int} F$. As we assumed the reverse inclusion, we get

Combining this with $\operatorname{int}(F \cap U) \cong\mathbb{R}^{n-1}$ shows that $\operatorname{int} F$ is an orientable manifold, since it is contained in the smooth open variety $W \setminus D_W$, with orientation induced by $S$. Moreover, we get $\partial F \subset D_W$ and $\partial_a F \subset D_W$.

Now we choose a simplicial triangulation of the compact set $S$ that is compatible with the semi-algebraic sets $\partial S$, $F$, and $S \cap D'$, which exists by [20]. We denote the simplicial chains in this compatible triangulation representing the fundamental classes of $S$ and $F$ by $c_S$ and $c_F$, respectively. Since our chosen orientation of $\operatorname{int} F$ is the one induced by $S$, and since $\partial S \setminus D' = \operatorname{int} F$, we get

where $d_n : C_n(S) \to C_{n-1}(S)$ is the simplicial differential and $c'$ is a chain supported on $D' \cap\partial S$. Next, we use the partial boundary map in [4 Eq. (10)]

where the last isomorphism follows from excision. By the above equation (3) we get

where the class $[c']$ is 0 in $H_{n-1}(F, \partial F)$ because $c'$ was supported on $D' \cap\partial S$. Composing with the pushforward induced by the inclusion $\iota: X(\mathbb{R}) \hookrightarrow X$, we get the desired identity of cycles under the map $H_n(X, W \cup D') \to H_{n-1}(W, D_W)$. □

Let us briefly illustrate which topological situations are avoided by the assumptions made in Lemma 3.17.

Example 3.18. Consider the union $\Delta\subset\mathbb{R}^2$ of the two regions $\Delta_1$ and $\Delta_2$ shown in Figure 2. Its algebraic boundary is given by the union of lines defined by $x$, $y$ and $x+y \pm1$, where $x,y$ are the coordinates of $\mathbb{R}^2$. Consider the face $F$ of $\Delta$ given by restricting it to $y=0$, which is the line segment $[-1,1] \cong[-1,1] \times\{0\}$. However, $\operatorname{int} F = (-1,1)$ is not contained in $V(y) \setminus D_y = [-1,1] \setminus\{-1,0,1\}$, contradicting the assumptions in Lemma 3.17. Indeed, the partial boundary map $\partial_y : H_2(\mathbb{P}^2, \partial_a\Delta) \to H_1(V(y), D_y)$ sends $[\Delta]$ to $[0,1] - [-1,0]$, which is not the class $[0,1] + [-1,0]$ of $[F]$. ◇

We are now ready to formulate the main result of this section, namely Theorem A from the introduction. Here we write $\mathring{\Omega}$ for the canonical map from Lemma 3.13, $[S]$ for the homology cycle in $H_n((X \setminus R)(\mathbb{R}), (D \setminus R)(\mathbb{R}))$ of the semi-algebraic set $S$, and $\iota: ((X \setminus R)(\mathbb{R}), (D \setminus R)(\mathbb{R})) \hookrightarrow(X \setminus R, D \setminus R)$ for the inclusion inducing the pushforward $\iota_*$ in relative homology. Finally, $r_*$ is the map induced by restriction from Lemma 3.14.

Proof. For simplicity we will write $(\mathring{X},\mathring{D})$ for the pair $(X \setminus R, D \setminus R)$ and $\sigma$ for the homology class $[S]$. Since $(\mathring{X},\mathring{D})$ is a recursive genus zero pair, and $X$ is smooth, the map

exists by Lemma 3.13. Here $(\mathcal{X},\mathcal{D})$ is a good compactification of a modification of $(\mathring{X},\mathring{D})$ as in Construction 3.11. By Lemma 3.14 we get a rational differential $n$-form $r_*\mathring{\Omega}(\sigma)$ on $X$, which is regular on $X \setminus D$, has logarithmic poles along $D$ and has no iterated poles along $R$.

We first show that $r_*\mathring{\Omega}(\sigma)$ satisfies the recursive property. We prove the statement by recursion on the dimension of the faces. Since $S$ is a face of $S$, the interior of $S$ is orientable. Let $V$ be a facet stratum, that is, an irreducible component of $D = \partial_a S$, so that the corresponding facet $F$ is the closure of the interior of $V \cap\partial S$. So $F$ is non-empty, and it is regular by construction. Moreover, since $\mathcal{F}(S)$ is graded of rank $n$, the algebraic boundary $\partial_a F \subset D_V$ is pure of codimension one in $V$.

Every pair $(V \setminus R,D_V \setminus R)$ for $V \in\mathcal{L}(D) \setminus\mathcal{R}(S)$ has genus zero and the complement $V \setminus W$ is smooth by assumption. Moreover, for every facet $F$ the boundary $\partial F \cap V \setminus D_V$ must be smooth, which can be seen as follows. Any codimension two stratum in $W$ is contained in the intersection of two distinct irreducible components of $D_V$ by construction. Hence, $W \cap V \subseteq D_V$ and thus $V \setminus D_V \subseteq V \setminus W$ where the latter is smooth by assumption. So $V$ is smooth at every point of $\partial S \cap(V \setminus D_V)$. Therefore, Lemma 3.17 applies, and we can use the recursiveness built into the Brown-Dupont framework as follows, by combining this Lemma with Proposition 3.16, yielding

where $\operatorname{Res}_{V}$ is the iterated residue map along any chain in $\mathcal{F}(S)$ leading to $V$.

Now we can continue recursively by the assumption that the pair $(X \setminus R,D \setminus R)$ is a recursive genus zero pair, and because the face poset of any face $F$ is a graded sub-poset of $\mathcal{F}(S)$ of rank $\dim F$, since it is a principal lower order ideal. This shows that the iterated residue of $r_*\mathring{\Omega}(\iota_*[S])$ has the correct logarithmic poles along the faces of $S$. By construction of the map $\mathring{\Omega}$ (see Remark 3.15), the form $r_*\mathring{\Omega}(\iota_*[S])$ has vanishing iterated residues along the components of $\mathcal{R}(S)$, i.e. it has poles exactly along the faces of $S$.

We proceed by showing that the canonical forms appearing in the recursion have residues that evaluate to $\pm1$ at the vertices. Since we assumed that every edge of $S$ contains exactly two vertices, the residues at these vertices are non-zero and can be normalized to $\pm1$ by the global residue theorem. In particular, this shows that the form $\mathring{\Omega}(\iota_*[S])$ and the iterative residues along faces are non-zero. Since the 1-skeleton is connected, the residue of $\mathring{\Omega}(\iota_*[S])$ at any vertex evaluates to $\pm1$. This shows that $\mathring{\Omega}(\iota_*[S])$ satisfies conditions (a), (b) and (c) from Definition 2.5.

It remains to prove that the canonical forms above are unique in each iterative step. Assume for the sake of contradiction that for the $d$-dimensional pair $(V,F)$ of a face $F \subset S$ with face stratum $V$, there exist two forms $\omega_1$ and $\omega_2$ satisfying conditions (b) and (c). Since $V$ is normal, we have that $\omega_1-\omega_2$ is a holomorphic $d$-form on $V$ and thus is equal to zero since $H^d(V,\mathcal{O}_V)=0$. This can be shown as follows. Then note that the dualizing sheaf $\dot{\omega}_V$ is isomorphic to the extension of the canonical line bundle $\Omega_V^d$ on the smooth locus of $V$ by [24 Proposition 5.75]. Therefore, by construction of the dualizing sheaf we have $0=H^d(V,\mathcal{O}_V)^\vee\cong H^0(V,\dot{\omega}_V) \cong H^0(V,\Omega_V^{[d]})$, where $\Omega_V^{[d]}:=\det(\Omega_V^1)^{\vee\vee}$. Hence there are no non-zero holomorphic $d$-forms on $V$.

Example 3.20. Theorem 3.19 allows one to recover the known results that polytopes in $\mathbb{P}^n$ and the totally non-negative Grassmannian $(\operatorname{Gr}(k,n),\operatorname{Gr}(k,n)_{\geq0})$ are positive geometries. For polytopes all the assumptions of the theorem are trivially satisfied. For the totally non-negative Grassmannian, the necessary properties follow from [37] and [23]. ◇

of the other 3 points, one in each of the three intersections $H_i \cap H_j \cap Q$ with $i \ne j$, $i,j < 4$, and three lines, namely $H_1 \cap H_4$, $H_2 \cap H_4$, and $H_3 \cap H_4$. The adjoint is therefore the product of the hyperplane $V(\ell_4)$ containing the three lines and the hyperplane spanned by the three points, which is defined by the linear form $\ell= 2360x - 2145y - 1036741z + 2451652w$. So the canonical form of $S$ is

This is not a canonical form of a positive geometry because it does not have a pole along the boundary component $H_4$. This is because the boundary curve $V(\ell_4,q) = H_4 \cap Q$ does not contain a vertex of the cycle. In fact, from the point of view of the mixed Hodge theory framework, the reason is that the canonical form of the homology cycle in $H_3(X,D)$ defined by the semi-algebraic set $z \ge0$, $9 \ge x^2 + y^2 + z^2$ is mapped to the zero-form: the canonical form is a global section of $\Omega^3_{\log}(\mathbb{P}^3 \setminus(H_4 \cup Q)) \cong\Omega^3_{\mathbb{P}^3}(H_4 \cup Q)$, where we used that $H_4 \cup Q$ is a simple normal crossing divisor. However, $\Omega^3_{\mathbb{P}^3}(H_4 \cup Q) \cong\mathcal{O}_{\mathbb{P}^3}(-1)$, which does not have any global sections. ⋄

Our goal for the rest of the article is to apply Theorem 3.19 to the $m = 2$ amplituhedron $A_{n,k,2}(Z)$. This is not at all straightforward, as by [29 Theorem 4.1] the pair $(\operatorname{Gr}(k,k+2), \partial_a A_{n,k,2})$ has strictly positive genus for $k \ge3$ and large enough $n$. We will therefore produce a different pair which will be a recursive genus zero pair and in which the amplituhedron will give rise to a relative homology cycle.

Faces of the $m = 2$ amplituhedron

In this section we concentrate on the faces of the $m = 2$ amplituhedron and for the sake of brevity write $A_{n,k} := A_{n,k,2}(Z)$. The main goal of this section is to show that for generic choices of $Z$ the faces of the amplituhedron $A_{n,k}$ are supported on varieties of genus zero, and that any Zariski open subset of such varieties also has genus zero. Moreover, as we show in [29 Proposition 4.11], the obvious obstructions for the pair $(\operatorname{Gr}(k,k+2), \partial_a A_{n,k})$ to have vanishing genus come from residual strata $W \in\mathcal{L}(\partial_a A_{n,k})$. Therefore, we conclude that any obstruction for the pair $(\operatorname{Gr}(k,k+2), \partial_a A_{n,k})$ to have genus zero must come from the residual arrangement $\mathcal{R}(A_{n,k})$. We will use this in the proof of our main Theorem B to bound the genus of the open pair $(\operatorname{Gr}(k,k+2) \setminus R, \partial_a A_{n,k,2}(Z) \setminus R)$, where $R$ is the union of all residual components in the residual arrangement.

This section is organized as follows. We start in Subsection 4.1 by recording a number of technical results that are useful for computing genera of varieties with rational singularities. Then, in Subsection 4.2 we show that face strata of the amplituhedron are unirational varieties with precisely this type of singularities. Moreover, we show that residual strata are disjoint from the amplituhedron, which is one of the conditions of Theorem 3.19. Lastly, in Subsection 4.3 we show that all face strata of the amplituhedron are genus zero varieties.

Rational singularities, sheaf cohomology, and the genus

In this subsection we present a number of technical results governing the behavior of the genus of a variety $X$ under resolutions of singularities and, more generally, surjective morphisms. We will use these results in Subsection 4.3, by applying them to the face strata of $\mathcal{A}_{n,k}$.

In the following let $X$ be a complex projective variety, $\mathcal{O}_X$ its structure sheaf, and, if $X$ is normal, $\omega_X$ the canonical sheaf. For a birational morphism $f:Y \to X$ of projective varieties also denote by $R^k f_* \mathcal{F}$ the higher direct image sheaves on $X$ given by the sheafification of $U \mapsto H^k(f^{-1}(U),\mathcal{F})$, where $\mathcal{F}$ is a coherent sheaf on $Y$. We start with the definition of the type of singularities we will encounter when dealing with amplituhedra.

Remark 4.2. Note that varieties with rational singularities are normal by [42 Remark 2.7]. This essentially follows from the condition $\pi_* \mathcal{O}_{\widetilde{X}} \cong\mathcal{O}_X$.

Proof. By construction of the Hodge filtration we have

see, e.g., [6 §1.2.1]. Then,

Now for complex projective varieties there is a surjection

(see, e.g., [26 Eq. (1.5)]). Thus, by symmetry in the mixed Hodge numbers we have $g(X) + \operatorname{cr}(X) = \dim\operatorname{gr}^{0}_{F}H^d(X)_{\mathbb{C}} \le\dim H^d(X,\mathcal{O}_X)$ proving the first part of the statement. Now if $X$ has rational singularities, it follows from [28 Theorem S] that $X$ has Du Bois singularities. In this case by [26 Eq. (1.3)] the surjection above is an isomorphism. $\square$

We will need the following technical result, which we believe to be standard but have not found in the literature.

Proof. Since $X$ and $Y$ are varieties with the same function field, the natural map $\mathcal{O}_X \to f_*\mathcal{O}_Y$ between coherent sheaves is injective. Denote by $\mathcal{F}$ its cokernel so that we have the short exact sequence of coherent sheaves on $X$

As $f$ is an isomorphism over a dense open $U \subseteq X$, the support $\operatorname{supp}\mathcal{F}$, i.e. the set of points $x \in X$ such that $\mathcal{F}_x \ne0$, is contained in $X \setminus U$, which implies $\dim\operatorname{supp}\mathcal{F} \le d-1$. By Grothendieck’s vanishing theorem [16 Thm. III.2.7], the top sheaf-cohomology group $H^d(X,\mathcal{F})$ vanishes. Therefore, the long exact sequence in cohomology, induced by the short exact sequence of sheaves above, gives a surjection

Next, for $i \ge1$, the theorem on formal functions (see, e.g., [15 Cor. 4.2.2]) implies

Since $f$ is birational and $\dim X=\dim Y=d$, the fiber dimension theorem [16 Ex. II.3.22 (d)] gives $\dim X_{\ge i} \le d-i-1$ for $i \ge1$. This can be seen as follows. Take a dense open $U$ of $X$ over which $f$ is an isomorphism. Then, for $i \ge1$ we have $X_{\ge1} \subset X \setminus U$ and $\dim f^{-1}(X_{\ge i}) \le d-1$ as it must be a proper closed subset of $Y$ with $\dim Y=d$. Then, the fiber dimension theorem applied to $f^{-1}(X_{\ge i}) \to X_{\ge i}$ yields $\dim X_{\ge i}+i \le\dim f^{-1}(X_{\ge i}) \le d-1$. Combined with Grothendieck vanishing this yields

Consider the Leray spectral sequence $E_2^{p,q}:=H^p(X,R^qf_*\mathcal{O}_Y)\Rightarrow H^{p+q}(Y,\mathcal{O}_Y)$, with differentials $d_r^{p,q}:E_r^{p,q}\to E_r^{p+r,q+1-r}$ for any $r\ge2$. Then, (5) shows that $d_2^{d-i,i}=0$ for any $i\ge1$, as their domains $E_2^{d-i,i}$ must vanish (also note that this implies $E_r^{d-i,i}=0$ for any $r\ge2$ as these are sub-quotients of $E_2^{d-i,i}$). Therefore, the only outgoing differential for $E_2^{d,0}$ in weight $d$ is $d_2^{d,0}$. Hence, the filtration on $H^d(Y,\mathcal{O}_Y)$ collapses i.e. $E_\infty^{d,0}\cong H^d(Y,\mathcal{O}_Y)$, and we get a surjective morphism

Lastly, we compose the surjective maps $H^d(X,\mathcal{O}_X)\to H^d(X,f_*\mathcal{O}_Y)$ from (4) and $H^d(X,f_*\mathcal{O}_Y)\to H^d(Y,\mathcal{O}_Y)$ from (6), which yields the natural surjective map $f^*:H^d(X,\mathcal{O}_X)\to H^d(Y,\mathcal{O}_Y)$. This shows the first claim. It remains to prove that if $f$ is a resolution of singularities and $X$ has rational singularities, then $f^{*}$ is an isomorphism. To that end it suffices to note that $R^{q}f_{*}\mathcal{O}_{Y}=0$ for all $q>0$ by Definition 4.1. Therefore, the Leray spectral sequence degenerates on the second page

as all higher differentials $d_{r}^{p,q}:E_{r}^{p,q}\to E_{r}^{p+r,q+1-r}$ vanish, since only the row of the second page $E_{2}^{p,q}$ with $q=0$ is non-zero. This provides an isomorphism $H^{d}(Y,\mathcal{O}_{Y})\cong H^{d}(X,f_{*}\mathcal{O}_{Y})$. Since the singularities of $X$ are rational, we also have $f_{*}\mathcal{O}_{Y}\cong\mathcal{O}_{X}$, see Definition 4.1, which yields the claim.

Faces and the residual arrangement in the twistor embedding

In this subsection we will show that face strata of the amplituhedron are varieties with rational singularities. This will allow us to apply the results of the preceding subsection in the proof of Proposition 4.9, the main result of this section.

The algebraic boundary stratification of $\mathcal{A}_{n,k}$ was completely described in [33], and we now recall the necessary results and notation. This description uses the notion of the twistor embedding

where $X\in\operatorname{Gr}(k,k+2)$ is given as the rowspan of a $k\times(k+2)$ matrix, and $\langle X_{i_{1}i_{2}}\rangle$ are twistor coordinates, obtained by stacking $X$ together with the rows $i_{1}$ and $i_{2}$ of the matrix $Z$ defining the amplituhedron map and taking the determinant of the resulting matrix. By [33 Lemma 1], the map $T_{Z}$ is indeed an embedding.

The stratification $\mathcal{L}(\mathcal{A}_{n,k})$ can be studied in the twistor embedding and by [33] it is a subposet of the positroid stratification of $\operatorname{Gr}(2,n)_{\geq0}$. That is, any stratum in $\mathcal{L}(\mathcal{A}_{n,k})$ is isomorphic to the intersection of a positroid variety $\Pi_{\mathcal{N}}$ and the image $T_{Z}(\operatorname{Gr}(k,k+2))$. Denote the set of all rank 2 positroids on $n$ elements yielding face (not residual) strata of $\mathcal{A}_{n,k}$ by $\mathcal{P}_{n,k}$. We start by recording the following consequence of these definitions.

Proof. By definition the face stratification $\mathcal{F}(\mathcal{A}_{n,k})$ is the sub-poset of the algebraic boundary stratification $\mathcal{L}(\mathcal{A}_{n,k})$ consisting of all strata which are not residual. These strata are precisely intersections of the positroid varieties $\Pi_{\mathcal{N}}$ for $\mathcal{N}\in\mathcal{P}_{n,k}$ as defined in [33] with $T_{Z}(\operatorname{Gr}(k,k+2))$. Hence, the claim follows from Theorem 25 in op. cit.

We now use this description to record the following topological property of the amplituhedron $\mathcal{A}_{n,k}$, which we will need in § 5.

Proof. By [33 Proposition 38], any residual stratum $V \in\mathcal{R}(\mathcal{A}_{n,k,2})$ is isomorphic, under the twistor embedding $T_Z$, to the intersection of $T_Z(\operatorname{Gr}(k,k+2))$ with a positroid variety $\Pi_{\mathcal{N}}$ with $\mathcal{N} \notin\mathcal{P}_{n,k}$. By [33 Theorem 23] the only positroid varieties intersecting $\mathcal{A}_{n,k}$ are of the form $\Pi_{\mathcal{N}}$ for $\mathcal{N} \in\mathcal{P}_{n,k}$. Thus, $V \cap\mathcal{A}_{n,k,2} = \varnothing$.

The description of $\partial_a\mathcal{A}_{n,k}$ from [33] allows one to study the algebro-geometric properties of the elements of $\mathcal{L}(\mathcal{A}_{n,k})$. For instance, the proof of [33 Theorem 25] shows that for generic choices of the matrix $Z$ these varieties are normal. We now use this description to show that the open face strata are smooth and that the closed face strata are varieties with rational singularities. The smoothness of open strata is one of the conditions of Theorem 3.19, and rational singularities allow one to keep control of the genus under birational modifications, as explained in the previous subsection.

Proof. By [33 Theorem 25 and Proposition 38] each $W \in\mathcal{L}(\mathcal{A}_{n,k})$ is isomorphic to the intersection of a positroid variety $\Pi$ in $\operatorname{Gr}(2,n)$ with the Grassmannian $\operatorname{Gr}(2,\operatorname{span}(Z))$, which is the image of $\operatorname{Gr}(k,k+2)$ under $T_Z$ (here $\operatorname{span}(Z)$ denotes the column span of $Z$). Moreover, we have $\mathring{W} \cong\mathring{\Pi} \cap\operatorname{Gr}(2,\operatorname{span}(Z))$. Positroid varieties are projective and have rational singularities [23 Theorem 5.13] and their open counterparts are smooth. Moreover, $\operatorname{Gr}(2,\operatorname{span}(Z))$ is a smooth linear section of the ambient $\operatorname{Gr}(2,n)$ (see, e.g., [18 Lecture 6]).

Since positroid varieties have rational singularities, for any such variety $\Pi$ in $\operatorname{Gr}(2,n)$ of dimension at least $2k$ the set of $2k$-dimensional linear sections of $\Pi$ that also have rational singularities and whose intersection with $\mathring{\Pi}$ is smooth is locally open and constructible, see, e.g., [10 Remark 3.4.11 (3)]. Our goal is to show that the special linear sections of the form $\Pi\cap\operatorname{Gr}(2,\operatorname{span}(Z))$ are in this set for a general $Z$. Since the set is locally open and constructible, it is sufficient to give one example of $Z$ for which $\Pi\cap\operatorname{Gr}(2,\operatorname{span}(Z))$ has rational singularities and $\mathring{\Pi} \cap\operatorname{Gr}(2,\operatorname{span}(Z))$ is smooth. For an $n \times(k+2)$ matrix $Z$ containing a $(k+2) \times(k+2)$ identity submatrix in some subset of $k+2$ columns and having all other columns zero such an intersection is again a positroid variety, and thus has the desired properties by [23 Theorem 5.13]. Note that $Z$ has to be chosen so that $\Pi\cap\operatorname{Gr}(2,\operatorname{span}(Z))$ has the expected dimension.

The number of positroid varieties in $\operatorname{Gr}(2,n)$ is finite and for each of them the set of matrices $Z$ such that $\Pi\cap\operatorname{Gr}(2,\operatorname{span}(Z))$ has rational singularities and $\mathring{\Pi} \cap\operatorname{Gr}(2,\operatorname{span}(Z))$ is smooth is locally open and constructible. A finite intersection of locally open constructible sets is again a non-empty locally open constructible set. Thus, the set of matrices $Z$ for which every intersection $\Pi\cap\operatorname{Gr}(2,\operatorname{span}(Z))$ has rational singularities and $\mathring{\Pi} \cap\operatorname{Gr}(2,\operatorname{span}(Z))$ is smooth is also locally open and constructible. This shows the claim of the lemma.

Faces have genus zero

In this subsection we will show that every face stratum of $\mathcal{A}_{n,k}$ has genus zero. Our strategy is as follows. First, we will show that every face stratum is a unirational variety, which follows from the results of [33]. Moreover, by Lemma 4.7 each face stratum has rational singularities. We then exploit the results from Subsection 4.1 to show that complex projective unirational varieties with rational singularities have genus zero. Finally, with a view towards Theorem 3.19, we show that any Zariski open subset of such a variety has genus zero.

Remark 4.10. Note that $V$ having rational singularities is essential for Proposition 4.9 to hold. In general, the genus of a variety can be higher than that of the resolution of singularities, see, e.g., [4 Example 3.6].

since $D \setminus R$ has codimension at least two in $U$, by [4 proof of Proposition 3.9]. Moreover, $(\mathcal{X},\mathcal{D})$ is a good compactification of $(\mathcal{X} \setminus\mathcal{B},\mathcal{D} \setminus\mathcal{B})$, which is a modification of $(U,\mathcal{D} \setminus R)$. In particular, by [4 Propositions 1.3 and 3.18], we get

Next, consider the long exact sequence in relative cohomology

After applying the exact functor $\operatorname{gr}^{n}_{F}$, the outermost terms vanish by the Hodge bounds and the inequality $\dim(\mathcal{B} \setminus\mathcal{D}) < n$; see, for example, [7 Théorème 8.2.4 (i), (ii)] or [4 §1.2.2]. Therefore, we obtain an isomorphism

The latter is naturally identified with $H^{0}(\mathcal{X},\omega_{\mathcal{X}}(\mathcal{D}))$ by [4 Eq. (20)], since $\mathcal{D}$ is a simple normal crossing divisor.

We now show that $H^{0}(\mathcal{X},\omega_{\mathcal{X}}(\mathcal{D}))$ vanishes as follows. Since $\mathcal{D}$ is effective, we have an inclusion of sheaves $\omega_{\mathcal{X}} \hookrightarrow\omega_{\mathcal{X}}(\mathcal{D})$ on $\mathcal{X}$. By left-exactness of the pushforward, this gives an inclusion $\bar{f}_{*}\omega_{\mathcal{X}} \hookrightarrow\bar{f}_{*}\omega_{\mathcal{X}}(\mathcal{D})$ of sheaves on $X$. Write $j:X_{\mathrm{reg}} \hookrightarrow X$ for the open embedding of the regular locus. The restriction of $\bar{f}_{*}\omega_{\mathcal{X}}(\mathcal{D})$ to $X_{\mathrm{reg}}$ is $\omega_{X_{\mathrm{reg}}}$, because $\mathcal{D}$ is disjoint from $\bar{f}^{-1}(X_{\mathrm{reg}})$ and $\bar{f}_{*}\omega_{\mathcal{X}}=\omega_{X}$ since $X$ has rational singularities. Therefore, we have

This gives the inclusion

because $\bar{f}_{*}\omega_{\mathcal{X}}(\mathcal{D})$ is torsion-free. Thus we have established the following relations:

Finally, the composition of both maps is the natural map, i.e., the unique map restricting to the identity on $X_{\mathrm{reg}}$. Both ends of this chain are equal to the dualizing sheaf of $X$. First, on the left, the assumption that $X$ has rational singularities implies $\bar{f}_{*}\omega_{\mathcal{X}}=\omega_{X}$. Moreover, since $X$ is normal (see Remark 4.2), the dualizing sheaf of $X$ is equal to the pushforward $\omega_{X}=j_{*}\omega_{X_{\mathrm{reg}}}$, under the inclusion of the canonical sheaf on the regular locus; see [17 Proposition 1.6], see also [27 Eq. (5.3)]. So the composition of the two injections in the above equation gives an isomorphism, which implies that all sheaves on $X$ in the above equation are equal. Thus the global sections of $\omega_{\mathcal{X}}(D)$ and $\omega_{\mathcal{X}}$ must be the same, i.e.

Since $\bar{f}:\mathcal{X}\to X$ is birational, $\mathcal{X}$ is also unirational. Therefore,

by [30 Corollary 1.56]. Thus we have

which after taking duals shows the first claim $\operatorname{gr}^{0}_{F}H^n(U)=0$. The remaining claim follows from the definition of the genus.

The $m=2$ amplituhedron is a positive geometry

In this section we achieve our main goal and show that for generic choices of $Z$ the pair $(\operatorname{Gr}(k,k+2),\mathcal{A}_{n,k,2}(Z))$ is a positive geometry. We start with the following auxiliary statement. It is a generalization of [4 Corollary 3.14], which covers the case of simple normal crossing divisors. It first appeared in the PhD thesis of the first author [25]. Recall that $\mathcal{L}(D)$ denotes the algebraic stratification of a subvariety $D$ of a complex variety $X$.

Proof. We use the bound provided by [4 Corollary 3.13] recursively by dimension. We start in dimension $n=\dim(X)$ with $g(X,D)$, where this result gives the inequality $g(X,D)\leq g(X)+\sum_i g(D_i,D_i\cap\bigcup_{j\neq i}D_j)$. For an irreducible component $V\subset D_i\cap\bigcup_{j\neq i}D_j$ that has codimension at least two in $D_i$, [4 Proposition 3.9] tells us that $g(D_i,D_i\cap\bigcup_{j\neq i}D_j)$ equals $g(D_i,(D_i\cap\bigcup_{j\neq i}D_j)\setminus(V\setminus\bigcup_j D_i\cap D_j))$. So we can replace the subset $D_i\cap\bigcup_{j\neq i}D_j$ of $D_i$ by the union of all its irreducible components $D_i^{(j)}$ of codimension one in $D_i$. Then, [4 Corollary 3.13] implies the inequality

Now, since the dimension of $D_i$ is $n-1$, we can apply the bound of [4 Corollary 3.13] to all new pairs $(D_i^{j},D_i^{(j)}\cap\bigcup_{k\neq j}D_i^{(k)})$, which eventually gives an inequality

where $m_V$ are non-negative integers counting how often the same variety $V$ appears in the recursion. Since the algebraic stratification $\mathcal{L}(D)$ is closed under taking intersections by construction, all $V$ that appear in the recursion are contained in $\mathcal{L}(D)$. Hence, $g(V)=0$, and all terms in the sum above vanish, showing the claim.

Therefore, we have an immediate consequence for recursive genus zero pairs.

Proof. Since $\mathcal{L}(D)$ is closed under intersections, Proposition 5.1 applies to any pair $(W,D_W)$, where $D_W$ is the union of all elements of $\mathcal{L}(D)$ that are codimension one subsets of $W$.

We are now ready to prove our main result.

Proof. Let $R$ be the residual arrangement of $\mathcal{A}_{n,k,2}(Z)$ and consider the pair $(\mathring{X},\mathring{D}):=(\operatorname{Gr}(k,k+2)\setminus R,\partial_a\mathcal{A}_{n,k,2}(Z)\setminus R)$. By Proposition 4.6 we have $\mathcal{A}_{n,k,2}(Z)\subseteq\mathring{X}$ and $\partial\mathcal{A}_{n,k,2}(Z)\subset\mathring{D}$, so $\mathcal{A}_{n,k,2}(Z)$ gives rise to a well-defined relative cycle in $H_n(\mathring{X},\mathring{D})$. We will now show that the assumptions of Theorem 3.19 are satisfied for this pair and $S=\mathcal{A}_{n,k,2}(Z)$.

Our first step is to show that $(\mathring{X},\mathring{D})$ is a recursive genus zero pair. Since we have removed the residual arrangement, every element in the algebraic boundary stratification of $\mathcal{A}_{n,k,2}(Z)$ in $\mathring{X}$ is a Zariski open subset of a face stratum $V$ of $\mathcal{A}_{n,k,2}(Z)$. By Lemma 4.8 $V$ is a unirational variety, and by Lemma 4.7 it has rational singularities. Thus by Lemma 4.11 each element in the boundary stratification of $\mathcal{A}_{n,k,2}(Z)$ in $\mathring{X}$ has vanishing genus. Then, $(\mathring{X},\mathring{D})$ is a recursive genus zero pair by Corollary 5.2.

Our second step is to show that faces of $\mathcal{A}_{n,k}$ satisfy the three conditions outlined in Theorem 3.19. Consider a face $F$ with Zariski closure $V$ in a closed subvariety of $\operatorname{Gr}(k,k+2)$. By the proof of [33 Theorem 25] $V$ is normal and moreover its singularities are contained in the Zariski closures the strata of $\mathcal{L}(D)$ that have codimension two in $V$. Both follow from Kleiman transversality together with [23] and [3 Theorem 2.7], where these properties where shown for positroid varieties. Thus, Condition 1 of Theorem 3.19 is satisfied. Now, by Proposition 4.9 we have $H^d(V,\mathcal{O}_V)=0$ if $d:=\dim V>0$, implying Condition 2. Finally, Condition 3 requires that the interior of $F$ be orientable and contained in $V\setminus D_V$. This again follows from [33 Theorem 25]: the faces are connected components of linear sections of an affine chart of $\operatorname{Gr}(2,n)$ which is isomorphic to $\mathbb{C}^{2n-4}$ and thus orientable.

Note that $\mathcal{A}_{n,k,2}(Z)$ has a graded face stratification of rank $2k$ by [33 Theorem 25 and Theorem 33]. It remains to show that every face of $\mathcal{A}_{n,k}$ has a vertex, the $1$-skeleton of this set is connected, and each $1$-dimensional face contains two vertices. This follows from the fact that $\mathcal{F}(\mathcal{A}_{n,k})$ is an Eulerian poset by [33 Theorem 33]. Therefore, the semi-algebraic set $\mathcal{A}_{n,k}$ together with the pair $(\operatorname{Gr}(k,k+2)\setminus R,\partial_a\mathcal{A}_{n,k}\setminus R)$ satisfies the conditions of Theorem 3.19. Thus, $(\operatorname{Gr}(k,k+2),\mathcal{A}_{n,k})$ is a positive geometry with canonical form $r_*\overset{\circ}{\Omega}(\iota_*[\mathcal{A}_{n,k}])$, where $r_*$ is the map induced by restriction in Lemma 3.14, $\overset{\circ}{\Omega}$ is the canonical map from Lemma 3.13, and $\iota_*$ is the pushforward $H_{2k}(\widetilde{X}(\mathbb{R}),\widetilde{D}(\mathbb{R})) \to H_{2k}(\widetilde{X},\widetilde{D})$ induced by inclusion of pairs.

Remark 5.4. Our proof does not give an explicit canonical form of $\mathcal{A}_{n,k,2}(\mathbb{Z})$. A natural way to construct it would be to sum the canonical forms of the amplituhedron tiles from [39]. However, proving that such a sum is indeed the canonical form of the amplituhedron is a separate subtle technical task.

AI Use. We used AI (Claude Opus 5.5 and GPT-6 Astra Pro) as a search engine, for proofreading purposes of drafts of this work, and to generate the initial code for Figure 1 and Figure 3. Parts of the proofs of Lemmas 3.17 and 4.4 were suggested by AI and subsequently rewritten and streamlined by the authors. The text of the article is entirely written by the authors, and they take full responsibility for its content, including all mathematical results.

Acknowledgements. The authors are grateful to Daniele Agostini, Clément Dupont, Thomas Lam, and Christian Lehn for helpful discussions. This research is funded by the European Union (ERC, UNIVERSE PLUS, 101118787). Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. The authors would like to thank the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), University of Vienna (Austria), for the opportunity to participate in the Thematic Program “Amplitudes and Algebraic Geometry” where an important part of this work has been accomplished. RS was supported by the SPP 2458 “Combinatorial Synergies”, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project number 539677510.

Authors’ addresses:

Joris Koefler, Institute for Advanced Study, Princeton E-Mail: [email protected]

Dmitrii Pavlov, Max Planck Institute for Physics, Garching bei München E-Mail: [email protected]

Rainer Sinn, Georg-August-Universität Göttingen E-Mail: [email protected]