On the Theory of Coconvex Bodies
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Abstract
If the complement of a closed convex set in a closed convex cone is bounded, then this complement minus the apex of the cone is called a coconvex set. Coconvex sets appear in singularity theory (they are closely related to Newton diagrams) and in commutative algebra. Such invariants of coconvex sets as volumes, mixed volumes, number of integer points, etc., play an important role. This paper aims at extending various results from the theory of convex bodies to the coconvex setting. These include the Aleksandrov–Fenchel inequality and the Ehrhart duality.
Keywords
Coconvex bodies AleksandrovFenchel inequalities Volume Valuations on polytopes Virtual convex polytopes1 Introduction
The geometric study of coconvex bodies is motivated by singularity theory. The connections between coconvex geometry and singularity theory are similar to the connections between convex geometry and algebraic geometry. Many local phenomena studied by singularity theory are local manifestations of global algebraic geometry phenomena. Thus it would be natural to expect that many properties of coconvex bodies are manifestations of properties of convex bodies. In this paper, we prove a number of results of this spirit.
In the first subsection of the introduction, we briefly overview the connections of convex geometry with algebraic geometry, of algebraic geometry with singularity theory and, finally, of singularity theory with coconvex geometry. These were the main motivations of the authors, however, neither algebraic geometry, nor singularity theory appear later in the text. Thus the following subsection can be omitted.
1.1 Overview
The theory of Newton polytopes founded in 1970s revealed unexpected connections between algebraic geometry and convex geometry. These connections turned out to be useful for both fields. According to a theorem of Kouchnirenko and Bernstein [4, 15], the number of solutions of a polynomial system \(P_1=\cdots =P_d=0\) in \((\mathbb {C}{\setminus }\{0\})^d\) equals \(d!\) times the mixed volume of the corresponding Newton polytopes \(\Delta _1\), \(\ldots \) , \(\Delta _d\). Recall that, for a complex polynomial \(P=\sum _{\alpha \in \mathbb {Z}^d} c_\alpha z^\alpha \), the Newton polytope of \(P\) is the convex hull in \(\mathbb R^d\) of all points \(\alpha \in \mathbb {Z}^d\) with \(c_\alpha \ne 0\). The relationship between Algebra and Geometry contained in the Kouchnirenko–Bernstein theorem allowed to prove the Aleksandrov–Fenchel inequalities using transparent and intuitive algebraic geometry considerations [10, 20], to find previously unknown analogs of the Aleksandrov–Fenchel inequalities in algebraic geometry [8], to find convexgeometric versions of the Hodge–Riemann relations that generalize the polytopal Aleksandrov–Fenchel inequalities [18, 21]. The number of integer points in polytopes is a classical object of study in geometry and combinatorics. The relationship with algebraic geometry has enriched this area with explicit formulas of Riemann–Roch type [12], has allowed to connect the Ehrhart duality with the Serre duality (from topology of algebraic varieties). The integration with respect to the Euler characteristic, inspired by the connections with algebraic geometry, has allowed to find a better viewpoint on classical results of McMullen, simplify and considerably generalize them [11, 12]. Connections with algebraic geometry led to important results in combinatorics of simple (and nonsimple) convex polytopes, and had many more followups.
The theory of Newton polytopes has a local version, which studies singularities of sufficiently generic polynomials with given Newton diagrams at the origin. This theory connects singularity theory with somewhat unusual geometric objects, namely, Newton diagrams. A Newton diagram is the union of all compact faces of an unbounded convex polyhedron lying in a convex cone (which in this case coincides with the positive coordinate orthant) and coinciding with the cone sufficiently far from the origin. The complement in the cone of the given unbounded convex polyhedron is, in our terminology, a coconvex body (except that it is also convenient to remove the apex of the cone from the coconvex body for reasons that will become clear later). Computations of local invariants in algebraic geometry and singularity theory have persistently led to volumes and mixed volumes of coconvex bodies, the number of integer points in coconvex bodies, etc. Computation of local invariants is often reduced to computation of global invariants. Let us illustrate this effect on the following toy problem: compute the multiplicity of the zero root of a polynomial \(P(z)=a_kz^k+\cdots +a_nz^n\) with \(a_k\ne 0\) and \(a_n\ne 0\). The multiplicity of the zero root of the polynomial \(P\) equals the number of nonzero roots of a polynomial \(P_\varepsilon =P+\varepsilon \) that vanish (i.e., tend to \(0\)) as \(\varepsilon \rightarrow 0\). The Newton polytopes of the polynomials \(P_\varepsilon \) (\(\varepsilon \ne 0\)) and \(P\) are, respectively, the intervals \([0,n]\) and \([k,n]\). The lengths \(n\) and \(nk\) of these intervals are equal to the number of nonzero roots of the polynomials \(P_\varepsilon \) and \(P\) (the answers in global problems). It follows that exactly \(k=n(nk)\) solutions vanish as \(\varepsilon \rightarrow 0\). Thus, in this simplest case, the multiplicity \(\mu =k\) of the root 0 (the local invariant) equals the difference of two global invariants, namely, the lengths \(n\) and \(nk\) of the intervals \([0,n]\) and \([k,n]\).
Similarly to this simple example, many questions of singularity theory (local questions) reduce to questions of algebraic geometry (global questions). Computing various local invariants for generic collections of functions with given Newton diagrams reduces to computing global algebrogeometric invariants for collections of generic polynomials with given Newton polytopes.
A systematic development of the coconvex bodies theory became a pressing need when, several years ago, relationships between convex and coconvex geometry on one side, algebraic geometry and singularity theory on the other side, were found that are far more general than the relationships based on Newton diagrams and Newton polytopes. For example, these relationships have allowed Kaveh and Khovanskii [9] to deduce nontrivial commutative algebra inequalities from a version of the Brunn–Minkowski inequality for coconvex bodies (this version follows from Theorem A). Computing Hilbert polynomials of algebraic varieties and their local versions for algebraic singularities leads to problems of counting integer points in lattice convex and coconvex polytopes.
1.2 Terminology and Notation
We start by recalling some terminology from convex geometry, see e.g. [19] for a detailed exposition. The Minkowski sum of two convex sets \(A\), \(B\subset \mathbb R^d\) is defined as \(A+B=\{a+b\,\, a\in A,\ b\in B\}\). For a positive real number \(\lambda \), we let \(\lambda A\) denote the set \(\{\lambda a\,\, a\in A\}\). By definition, a convex body is a compact convex set, whose interior is nonempty.
Let \(C\subset \mathbb R^d\) be a convex cone with the apex at \(0\) and a nonempty interior. Consider a closed convex subset \(\Delta \subset C\) such that \(C{\setminus }\Delta \) is bounded and nonempty. Note that sets \(\Delta \) with specified properties exist only if \(C\) is a salient cone, i.e., if \(C\) contains no affine subspace of dimension 1. Then the set \(A=C{\setminus }(\Delta \cup \{0\})\) is called a coconvex body. When we talk about volumes, we may replace \(A\) with its closure \(\overline{A}\). However, for the discussion of integer points in coconvex bodies, the distinction between \(A\) and \(\overline{A}\) becomes important. If \(A\) and \(B\) are coconvex sets with respect to the same cone \(C\), then we can define \(A\oplus B\) as \(C{\setminus }((\Delta _A+\Delta _B)\cup \{0\})\), where \(\Delta _A\) and \(\Delta _B\) are the unbounded components of \(C{\setminus }A\) and \(C{\setminus }B\), respectively. It is clear that any \(C\)coconvex set can be represented as a settheoretic difference of two convex bounded sets. This representation allows to carry over a number of results concerning the convex bodies with the operation \(+\) to the coconvex bodies with the operation \(\oplus \).
In this paper, we describe several results of this type. Although the reduction from the “convex world” to the “coconvex world” is always simple and sometimes straightforward, the results obtained with the help of it are interesting because, firstly, they are related (through Newton diagrams) with singularity theory [2, 14] and commutative algebra [9], and, secondly, they are intrinsic, i.e., do not depend on a particular representation of a coconvex set as a difference of two convex sets. Coconvex Aleksandrov–Fenchel inequalities also appeared in [7] in the context of Fuchsian groups. This paper extends the earlier very short preprint [13] of the authors, in which just the coconvex Aleksandrov–Fenchel inequality has been discussed.
1.3 Aleksandrov–Fenchel Inequalities
 Brunn–Minkowski inequality: the function \(\mathrm {Vol}_\alpha ^{\frac{1}{d}}\) is concave, i.e.$$\begin{aligned} (\mathrm {Vol}_\alpha (tu+(1t)v))^\frac{1}{d}\geqslant t\mathrm {Vol}_\alpha (u)^\frac{1}{d}+(1t)\mathrm {Vol}_\alpha (v)^\frac{1}{d},\quad t\in [0,1]. \end{aligned}$$

Generalized Brunn–Minkowski inequality: the function \(\big (L_{v_1}\ldots L_{v_k}\mathrm {Vol}_\alpha \big )^\frac{1}{dk}\) is concave.
 First Minkowski inequality:$$\begin{aligned} \big (\frac{1}{d!}L_{u}L_{v}^{d1}(\mathrm {Vol}_\alpha )\big )^d\geqslant \mathrm {Vol}_\alpha (u)\mathrm {Vol}_\alpha (v)^{d1}. \end{aligned}$$
 Second Minkowski inequality: if all marked points coincide with \(u\), then$$\begin{aligned} B_\alpha (u,v)^2\geqslant \mathrm {Vol}_\alpha (u)\, B_\alpha (v,v) \end{aligned}$$
1.4 Coconvex Aleksandrov–Fenchel Inequalities
Theorem A
The inequality stated in Theorem A is called the coconvex Aleksandrov–Fenchel inequality. In recent paper [7], Theorem A is proved under the assumption that \(C\) is a fundamental cone of some discrete group \(\Gamma \) acting by linear isometries of a pseudoEuclidean metric, and \(C{\setminus }(g(v)\cup \{0\})\) is the intersection of some convex \(\Gamma \)invariant set with \(C\), for every \(v\in \Omega \). Theorem A is motivated by an Aleksandrov–Fenchel type inequality for (mixed) intersection multiplicities of ideals [9].
 Reversed Brunn–Minkowski inequality: the function \(\mathrm {Vol}_\beta ^{\frac{1}{d}}\) is convex, i.e.$$\begin{aligned} (\mathrm {Vol}_\beta (tu+(1t)v))^\frac{1}{d}\leqslant t\,\mathrm {Vol}_\beta (u)^\frac{1}{d}+(1t)\mathrm {Vol}_\beta (v)^\frac{1}{d},\quad t\in [0,1]. \end{aligned}$$

Generalized reversed Brunn–Minkowski inequality: the function \(\displaystyle \big (L_{v_1}\cdots L_{v_k}\mathrm {Vol}_\beta \big )^\frac{1}{dk}\) is convex.
 First reversed Minkowski inequality:$$\begin{aligned} \big (\frac{1}{d!}L_{u}L_{v}^{d1}(\mathrm {Vol}_\beta )\big )^d\leqslant \mathrm {Vol}_\beta (u)\mathrm {Vol}_\beta (v)^{d1}. \end{aligned}$$
 Second reversed Minkowski inequality: if all marked points coincide with \(u\), then$$\begin{aligned} B_\beta ^C(u,v)^2\leqslant \mathrm {Vol}_\beta (u)\, B_\beta ^C(v,v) \end{aligned}$$
1.5 Coconvex Polytopes as Virtual Convex Polytopes
The set of convex polytopes is closed under Minkowski addition but not closed under “Minkowski subtraction”. Virtual convex polytopes are geometric objects introduced in [11] that can be identified with formal Minkowski differences of convex polytopes. We will now briefly recall the notion of a virtual convex polytope.
Lemma 1.1
Suppose that \(A\) and \(B\) are convex polytopes in \(\mathbb R^d\). Then \(\mathbb {I}_{A+B}=\mathbb {I}_A*\mathbb {I}_B\).
Thus the Minkowski sum of convex polytopes corresponds to the Minkowski product in \(Z_c(\mathbb R^d)\).
Proof
It follows from Lemma 1.1 that \(\mathbb {I}_{\{0\}}\) is the identity element of the ring \(Z_c(\mathbb R^d)\). It is proved in [11] that, for every convex polytope \(A\) in \(\mathbb R^d\), the indicator function \(\mathbb {I}_A\) is an invertible element of the ring \(Z_c(\mathbb R^d)\), i.e., there exists an element \(\varphi \in Z_c(\mathbb R^d)\) with the property \(\varphi *\mathbb {I}_A=\mathbb {I}_{\{0\}}\) (we write \(\varphi =\mathbb {I}_A^{1}\)). The function \(\varphi :\mathbb R^d\rightarrow \mathbb {Z}\) admits a simple explicit description: it is equal to \((1)^{\dim (A)}\) on the relative interior of the set \(\{x\,\, x\in A\}\) and to 0 elsewhere. Virtual (convex) polytopes are defined as elements of \(Z_c(\mathbb R^d)\) of the form \(\mathbb {I}_A*\mathbb {I}_B^{1}\), where \(A\) and \(B\) are convex polytopes. If we identify convex polytopes with their indicator functions, then virtual polytopes are identified with formal Minkowski differences of convex polytopes. Virtual polytopes form a commutative group under Minkowski multiplication. Note that we do not deal with more general “virtual convex bodies” than virtual polytopes; the corresponding theory is more involved and is still developing, see e.g. [16].
Fix a linear function \(\xi :\mathbb R^d\rightarrow \mathbb R\) such that \(\xi \geqslant 0\) on \(C\) and \(\xi ^{1}(0)\cap C=\{0\}\); for every subset \(X\subset C\), we set \(X_t=X\cap \{\xi \leqslant t\}\). The following theorem is a general principle that allows to reduce various facts about coconvex polytopes to the corresponding facts about convex polytopes.
Theorem B
 (1)The function \(\mathbb {I}_A\) is a virtual polytope. Moreover, we havefor all sufficiently large \(t\in \mathbb R\).$$\begin{aligned} \mathbb {I}_A=\mathbb {I}_{\Delta _t}*\mathbb {I}_{C_t}^{1} \end{aligned}$$
 (2)If \(A\) and \(B\) are \(C\)coconvex polytopes, then$$\begin{aligned} \mathbb {I}_{A\oplus B}=(\mathbb {I}_A)*(\mathbb {I}_B). \end{aligned}$$
Theorem B explains our definition of a coconvex body, in particular, the choice of the boundary points that need to be included into it. Theorem A can be deduced from Theorem B and the convex Aleksandrov–Fenchel inequalities. There are many other consequences of Theorem B that deal with \(C\)coconvex integer polytopes. Some of them are stated below.
Define a convex integer polytope as a convex polytope with integer vertices. Let \(\mathcal {R}_c(\mathbb {Z}^d)\) be the minimal subring of the ring of sets \(\mathcal {R}(\mathbb R^d)\) containing all convex integer polytopes. Similarly, let \(Z_c(\mathbb {Z}^d)\) be the minimal subalgebra of \(Z_c(\mathbb R^d)\) containing the indicator functions of all convex integer polytopes. Clearly, any function in \(Z_c(\mathbb {Z}^d)\) is measurable with respect to \(\mathcal {R}_c(\mathbb {Z}^d)\). A valuation on integer polytopes is by definition a finitely additive measure on \(\mathcal {R}_c(\mathbb {Z}^d)\). For a valuation \(\mu \) on integer polytopes and an element \(\varphi \) of the ring \(Z_c(\mathbb R^d)\), we define \(\mu (\varphi )\) as the integral of the function \(\varphi \) with respect to the measure \(\mu \). Note that \(\mu (A)\), the measure \(\mu \) evaluated at a measurable set \(A\), is the same as \(\mu (\mathbb {I}_A)\). Similarly to valuations on integer polytopes, we define more general valuations on polytopes as finitely additive measures on \(\mathcal {R}_c(\mathbb R^d)\).
Recall that a function \(P\) on a commutative multiplicative group \(G\) is said to be polynomial of degree \(\leqslant d\) if, for every fixed \(g\in G\), the function \(P(gx)P(x)\) is polynomial of degree \(\leqslant d1\). Polynomial functions of degree 0 are by definition constant functions. Define the group of virtual integer polytopes as the subgroup of the group of virtual polytopes generated by the indicator functions of all convex integer polytopes. Recall the following theorem of [11]: If a valuation \(\mu \) on integer polytopes is polynomial of degree \(\leqslant k\), i.e., for every convex integer polytope \(A\), the function \(x\mapsto \mu (A+x)\) is a polynomial on \(\mathbb {Z}^d\) of degree at most \(k\), then the function \(\varphi \mapsto \mu (\varphi )\) is a polynomial function of degree \(\leqslant d+k\) on the group of virtual integer polytopes with \(*\) as the group operation. An important example of a valuation on integer polytopes is the valuation \(\mu \) that assigns the number of integer points in \(X\) to every \(X\in \mathcal {R}_c(\mathbb {Z}^d)\). This valuation can be evaluated on all virtual integer polytopes. In particular, for every integer convex polytope \(A\), the number \(\mu (\mathbb {I}_A^{*n})\) depends polynomially on \(n\) (cf. [11, 17]). This polynomial function is called the Ehrhart polynomial of \(A\).
Corollary 1.2
We will use the following notation. For a \(C\)coconvex body \(A\), we write \(\mathrm {Int}(A)\) for the interior of \(A\). We set \(\overline{A}\) to be the closure of \(A\), and define \(A^\bullet \) as \(\overline{A}\cap \mathrm {Int}(C)\).
Corollary 1.3
1.6 Generating Functions for Integer Points
Note that \(G(C_a)\) can be computed explicitly, by subdividing \(C_a\) into cones, each of which is spanned by a basis of \(\mathbb {Z}^d\). If a cone \(C\) is spanned by a basis of \(\mathbb {Z}^d\), then the computation of \(G(C)\) reduces to the summation of a geometric series. In this paper, we will prove the following theorem that generalizes Brion’s theorem to coconvex polytopes.
Theorem C
Observe that usual exponential sums over the integer points of \(A\) are obtained from the rational function \(G(A)\) by substituting the exponentials \(e^{p_1}\), \(\ldots \) , \(e^{p_d}\) for the variables \(x_1\), \(\ldots \) , \(x_d\). Sums of quasipolynomials (in particular, sums of polynomials) can be obtained from exponential sums by differentiation with respect to parameters \(p_1\), \(\ldots \) , \(p_d\). A similar theory exists for integrals of exponentials, quasipolynomials, etc., over convex or coconvex polytopes.
Remark
In order to compute the number of integer points in \(A\), one is tempted to substitute \(x_i=1\) into the expression for the rational function \(G(A)\) through the generating functions of cones. However, this is problematic as the denominator of this expression vanishes at the point \((1,1,\ldots ,1)\). To obtain a numeric value, one can, e.g., choose a generic line passing through the point \((1,1,\ldots ,1)\), consider the Laurent series expansion of \(G(A)\) along this line, and then take the free coefficient of this Laurent series. The same procedure is applicable to computing a quasipolynomial sum over \(A\) for exceptional values of \(p=(p_1,\ldots ,p_d)\), for which the rational function of \(e^{p_1}\), \(\ldots \) , \(e^{p_d}\), equal to this quasipolynomial sum at generic points, has a pole.
2 Proof of Theorem A
 (1)
there exists a vector \(v_0\in \mathcal {V}\) with \(Q(v_0)>0\);
 (2)
the corresponding symmetric bilinear form \(B\) (such that \(B(u,u)=Q(u)\)) satisfies the reversed Cauchy–Schwarz inequality: \(B(u,v)^2\geqslant Q(u)Q(v)\) for all \(u\in \mathcal {V}\) and \(v\in \mathcal {V}\) such that \(Q(v)>0\).
Thus, the Aleksandrov–Fenchel inequality is equivalent to the fact that \(Q_\alpha \) has signature \((1,\ell (\alpha ))\) for every finitedimensional linear family \(\alpha \) of convex \(d\)dimensional bodies with \(d2\) marked points.
2.1 An Illustration in the Case \(d=2\)
To illustrate Theorem A in a simple case, we set \(d=2\).
2.2 Reduction of Theorem A to the Aleksandrov–Fenchel Inequality
In this subsection, we prove Theorem A. The proof is a reduction to the convex Aleksandrov–Fenchel inequality. Fix a salient convex closed cone \(C\) with the apex at \(0\) and a linear family \(\beta =(\mathcal {V},\Omega ,g)\) of \(C\)coconvex bodies. We may assume that \(g(\Omega )\) is bounded in the sense that there is a large ball in \(\mathbb R^d\) that contains all coconvex bodies \(g(v)\), \(v\in \Omega \). Since \(g(\Omega )\) is bounded, there exists a real number \(t_0>0\) such that \(g(v)=g(v)_{t_0}\) for all \(v\in \Omega \).
Choose any \(t_1>t_0\). We will now define a linear family \(\alpha =(\mathcal {V}\times \mathbb R,\Omega \times (t_0,t_1),f)\) of convex bodies as follows. For \(v\in \Omega \) and \(t\in (t_0,t_1)\), we set \(f(v,t)\) to be the convex body \(C_t{\setminus }(g(v)\cup \{0\})\). The proof of the coconvex Aleksandrov–Fenchel inequality is based on the comparison between the linear families \(\alpha \) and \(\beta \).
If \(q_1\), \(q_2\) are quadratic forms depending on disjoint sets of variables, and \((k_1,\ell _1)\), \((k_2,\ell _2)\), respectively, are signatures of these forms, then \(q_1+q_2\) is a quadratic form of signature \((k_1+k_2,\ell _1+\ell _2)\). We now apply this observation to identity \((Q)\). The first term of the righthand side, \(c't^2\), has signature \((1,0)\). The signature of the lefthand side is equal to \((1,\ell )\) for some \(\ell \geqslant 0\), by the classical Alexandrov–Fenchel inequality. It follows that the signature of \(Q^C_{\beta }\) is \((\ell ,0)\), i.e. the form \(Q^C_{\beta }\) is nonnegative.
3 Proof of Theorem B and its Corollaries
In this section, we prove Theorem B and derive a number of corollaries from it.
3.1 Proof of Theorem B
Consider a salient closed convex cone \(C\subset \mathbb R^d\) with the apex at \(0\) and a nonempty interior. A closed convex subset \(\Delta \subset C\) is said to be \(C\)convex if \(\Delta +C=\Delta \). The following lemma gives the most important example of \(C\)convex sets.
Lemma 3.1
If \(\Delta \subset C\) is a convex subset such that \(C{\setminus }\Delta \) is bounded, then \(\Delta \) is \(C\)convex.
Proof
Since \(0\in C\), we have \(\Delta +C\supset \Delta \). It remains to prove that \(\Delta +C\subset \Delta \). Assume the contrary: there are points \(x\in \Delta \) and \(y\in C\) such that \(x+y\not \in \Delta \). Consider the line \(L\) passing through the points \(x\) and \(x+y\). Since \(C{\setminus }\Delta \) is bounded, there are points of \(\Delta \) in \(L\) far enough in the direction from \(x\) to \(x+y\). Thus \(x+y\) separates two points of \(L\cap \Delta \) in \(L\). A contradiction with the convexity of \(\Delta \). \(\square \)
Lemma 3.2
Let \(\Delta \subset C\) be a convex subset such that \(C{\setminus }\Delta \) is bounded. For all sufficiently large \(t>0\) and all \(s>0\), we have \(\Delta _t+C_s=\Delta _{t+s}\).
Proof
If \(x\in \Delta _t\) and \(y\in C_s\), then \(x+y\in \Delta \) by \(C\)convexity (Lemma 3.1) and \(\xi (x+y)\leqslant t+s\) since \(\xi (x)\leqslant t\) and \(\xi (y)\leqslant s\). On the other hand, take \(z\in \Delta _{t+s}\) and consider two cases.
Case 1: we have \(\xi (z)\leqslant t\). Then \(z\in \Delta _t\), and, setting \(x=z\), \(y=0\), we obtain that \(z=x+y\), \(x\in \Delta _t\) and \(y\in C_s\).
Case 2: we have \(\xi (z)>t\). If \(\lambda =t/\xi (z)\), then \(\xi (\lambda z)=t\). We now set \(x=\lambda z\), \(y=(1\lambda )z\). The number \(t\) is sufficiently large; thus we may assume that all values of \(\xi \) on \(C{\setminus }\Delta \) are less than \(t\). Then \(x\in \Delta _t\) and \(y\in C_s\). \(\square \)
Proposition 3.3
Proof
3.2 Proof of Corollaries 1.2 and 1.3
Corollary 1.2 follows directly from Theorem B and the theorem of [11] that a polynomial valuation on integer convex polytopes defines a polynomial function on the group of virtual integer polytopes. Corollary 1.3 follows from Theorem B and from Proposition 3.4 stated below.
Proposition 3.4
Proof
3.3 Proof of Theorem C
Let \(C\), \(A\), \(\Delta \) and \(\xi \) be as above. We assume that \(C\) is an integer polyhedral cone and that \(\Delta \) has integer vertices. Let \(t>0\) be sufficiently large, so that \(A\) is contained in \(C_{t'}\) for some \(t'<t\).
3.4 A Viewpoint on Theorem A Through Virtual Polytopes
Thus we obtained another proof of Theorem A. Although this proof is no simpler than the one given in Sect. 2.2, it reveals the role of virtual convex polytopes and the fact that the coconvex Aleksandrov–Fenchel form is no different from the convex Aleksandrov–Fenchel form evaluated at certain virtual polytopes.
Remark
We now sketch an analogy, which can be easily formalized and which may shed some light to the argument presented above. Consider complex algebraic varieties \(X\), \(Y\) and a regular map \(f:X\rightarrow Y\). Fix a point \(y_0\in Y\), and assume that \(f:X{\setminus }f^{1}(y_0)\rightarrow Y{\setminus }\{y_0\}\) is an isomorphism. A Cartier divisor \(D\) in \(X\) is said to be subexceptional if the support of \(D\) maps to \(y_0\) under \(f\). Subexceptional divisors in \(X\) correspond to coconvex polytopes (under the analogy, which we are discussing). A Cartier divisor \(D\) in \(X\) is said to be offexceptional if \(D=f^*(\widetilde{D})\) for some Cartier divisor \(\widetilde{D}\) in \(Y\), whose support does not contain \(y_0\). Offexceptional divisors correspond to the cone \(C\) (we are looking at the local geometry of \(Y\) near \(y_0\), thus we do not distinguish between different offexceptional divisors). Let \(S\) and \(O\) be a subexceptional and an offexceptional divisors, respectively. If \([S]\) and \([O]\) stand for the classes of these divisors in the Chow ring, then obviously \([S]\cdot [O]=0\), since the supports of \(S\) and \(O\) are disjoint. This fact is analogous to Eq. (3.3). As is shown in [10], the Hodge index theorem implies the Aleksandrov–Fenchel inequalities for the intersection of divisors in \(X\) (under some natural assumptions on \(X\), e.g., when \(X\) is projective and smooth). These inequalities can be used to provide an analog of Theorem A for the intersection form on subexceptional divisors. To make the described analogy into a precise correspondence, one takes \(X\) and \(Y\) to be toric varieties associated with integer polytopes \(\Delta _{t_0}\) and \(C_{t_0}\).
Notes
Acknowledgments
The first named author was partially supported by Canadian Grant N 15683302. The second named author was partially supported by the Dynasty Foundation Grant, RFBR Grants 130112449, 130100969, and AG Laboratory NRU HSE, MESRF Grant ag. 11 11.G34.31.0023. The research comprised in Theorem A was funded by RScF, project 142100053.
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