Geometry of generalized virtual polyhedra Askold Khovanskii October 4 2022

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Geometry of generalized virtual polyhedra

Askold Khovanskii ∗

October 4, 2022

Abstract

Partial generalizations of virtual polyhedra theory (sometimes un-

der diﬀerent names) appeared recently in the theory of torus manifolds.

These generalizations look very diﬀerent from the original virtual poly-

hedra theory. They are based on simple arguments from homotopy

theory while the original theory is based on integration over Euler

characteristic. In the paper we explain how these generalizations are

related to the classical theory of convex bodies and to the original

virtual polyhedra theory. The paper basically contains no proofs: all

proofs and all details can be found in the cited literature. The paper

is based on my talk dedicated to V. I. Arnold’s 85-th anniversary at

the International Conference on Diﬀerential Equations and Dynamical

Systems 2022 (Suzdal).

1 Introduction. Virtual convex polyhedra and their

polynomial measures

Convex polyhedra in the linear space Rnform a convex cone in the following

way. One can multiply a convex polyhedron ∆ by any nonnegative real

number λ(i.e. take its dilatation λ∆ centred at the origin with the factor

λ) and add two convex polyhedra ∆1,∆2in Minkowski sense. Recall that

the Minkowski sum of ∆1,∆2⊂Rnis the set ∆ of the points zrepresentable

in the form z=x+y, where x∈∆1, y ∈∆2.

A convex chain is a function on Rnrepresentable as a ﬁnite linear com-

bination with real coeﬃcients of characteristic functions of closed convex

polyhedra (of diﬀerent dimensions).

Convex chains form a real vector space in a natural way. One can further

deﬁne a product f∗gof two chains f, g as follows. If fand gare characteristic

∗The work was partially supported by the Canadian Grant No. 156833-17.

arXiv:2210.01070v1 [math.AG] 3 Oct 2022

functions of closed convex polyhedra ∆1,∆2⊂Rnthen, by deﬁnition, the

chain f∗gis the characteristic function of ∆ = ∆1+∆2(where the addition

is understood in the Minkowski sense). This product can be extended by

linearity to the space of convex chains.

Note that it is not obvious at all that the above product is well deﬁned.

Indeed, convex chain can be represented as a linear combination of charac-

teristic functions in many diﬀerent ways, and independents of product f∗g

of such representations of fand gis not obvious. One can prove [1] that the

product is well deﬁned using as the tool integration over Euler characteristic

[2].

Convex chains in Rnwith the multiplication ∗form a real algebra with

the identity element 1, which is the characteristic function of the origin in

Rn. The characteristic function χ∆of a closed convex polyhedron ∆ ⊂Rn

are invertible in the algebra of convex chains. More precisely, the following

theorem holds.

Theorem 1.1. Let ∆⊂Rnbe a convex polyhedron and let −∆0be the

set of interior points (in the intrinsic topology of ∆) of the polyhedron −∆

symmetric to ∆with respect to the origin. Then

(−1)dim ∆χ−∆0∗χ∆=1.

In other words, the convex chain (−1)dim ∆χ−∆0is inverse to ∆with respect

to the addition in Minkowski sense (extended to the space of convex chains).

Algebra of convex chains contains the multiplicative subgroup generated

by characteristic functions of closed convex polyhedra. Elements of that

group are called virtual polyhedra in Rn.

Let us ﬁx closed convex polyhedra ∆1,...,∆k⊂Rn. For any k-tuple of

nonnegative integral numbers n= (n1, . . . , nk) one can deﬁned the polyhe-

dron ∆(n) = Pni∆i.

The following sentence can be considered as a slogan of virtual polyhedra

theory: “A natural continuation of the function ∆(n) (whose values are

convex polyhedra) to k-tuples n= (n1, . . . , nk) of integral numbers (some

of which could be negative) is a convex chain ˜

∆(n) deﬁned by the following

formula

∆(n) = χn1

∆1∗ · · · ∗ χnk

∆k.

This slogan has a following justiﬁcation: value of a polynomial measures

(see an example of such measure below) on a chain ˜

∆(n) is a polynomial

of n. Generalizations of virtual polyhedra theory suggest other families of

cycles depending on parameters, such that integrals against such cycles of a

diﬀerential form with polynomial coeﬃcients are polynomial in parameters.

Let us present an example of a polynomial measure on convex polyhedra

with integral vertices and a justiﬁcation of the slogan of virtual polyhedra

theory. Let P:Rn→Rbe a polynomial of degree m. With Pone can asso-

ciate the following measure µon convex polyhedra ∆ with integral vertices:

µ(∆) = Px∈Zn∩∆P(x). One can prove that the function µ(∆(n) is a degree

≤(n+m) polynomial on k-tuples nof nonnegative integral numbers.

The following Theorem justiﬁes the slogan of virtual polyhedra theory.

Theorem 1.2. Let Pbe a polynomial of degree mand let ˜

F(n)be the

function on k-tuples n= (n1, . . . , nk)of integral numbers (which could be

negative) deﬁned by the formula

F(n) = X

x∈Zn

χn1

∆1(x)∗ · · · ∗ χnk

∆k(x)P(x).

Then ˜

F(n)is a degree ≤(n+m)polynomial on k-tuple nwhich coincides

with F(n)on k-tuples with nonnegative components.

Virtual polyhedra theory allows to develop a general theory of polyno-

mial ﬁnite additive measures on convex polyhedra (see [1]), which contains

wide generalizations of the above theorem.

The virtual polyhedra theory was motivated by cohomology theory of

complete toric varieties, with coeﬃcients in sheafs invariant under the torus

action. In particular it provides a combinatorial version of Riemann–Roch

theorem for such varieties [3], which also could be considered as a multidi-

mensional version of the classical Euler–MacLuren formula (see [3]).

The general theory is applicable to singular polynomial measures on

polyhedra (such as the measure, which associates to a polyhedron the num-

ber of integral points in it) which could take nonzero value on polyhedra

∆ with dim ∆ < n. However, if one is interested in nonsingular polynomial

measures, which vanish on polyhedra whose dimension is smaller than n, one

can totally neglect all polyhedra of dimension < n in convex chains. This

leads to a signiﬁcant simpliﬁcation of virtual polyhedra theory, which cap-

tures smooth polynomial measures (an which is not appropriate for studying

singular measures).

Simpliﬁed theory is still useful. In particular, it allows to provide a

topological proof of Bernstein–Koushnirenko–Khovanskii (BKK) theorem.

More generally, using a description of algebras with Poincare duality (see

for example [4, Section 6] or [5]) it allows to describe the cohomology ring

H∗(M, Z) of a smooth complete toric variety Min terms of volume func-

tion on virtual integral convex polyhedra (so-called Khovanskii–Pukhlikov

description of the ring H∗(M, Z)).

In this paper we only deal with simpliﬁed versions of the virtual polyhe-

dra theory which deal only with nonsingular measures as well as its gener-

alizations. We also mention some topological applications of these general-

izations. We start with geometric meaning of a virtual convex body and its

volume for the diﬀerence of two strictly convex bodies with smooth bound-

aries. We also will present some applications of mixed volume and virtual

polyhedra in algebra.

2 Virtual strictly convex bodies and their volumes

Formal virtual convex body is a formal diﬀerence of compact convex bodies

(which in general are not polyhedra).

Similar to polyhedra, compact convex bodies in Rnform a convex cone

with respect to Minkowski addition and dilation with positive factors cen-

tered at the origin. Moreover, the addition of convex bodies satisﬁes the can-

celation property, i.e. if for a convex body ∆ the identity ∆1+ ∆ = ∆2+ ∆

implies that ∆1= ∆2. Hence one can generate a group by formal diﬀerences

of convex bodies with ∆1−∆2= ∆3−∆4whenever ∆1+ ∆4= ∆3+ ∆2.

By Minkovsky’s Theorem, the volume is a homogeneous degree npoly-

nomial on the cone of convex bodies. More concretely, if ∆1,∆2are convex

bodies, λ, µ ≥0 then the volume Vol(λ∆1+µ∆2) is a homogeneous degree

npolynomial in (λ, µ). Therefore, the volume can be extended to the lin-

ear space of formal diﬀerences of convex bodies as a homogeneous degree

npolynomial. In Section 4 we give a geometric interpretation of virtual

convex bodies as well as their volumes.

Since the volume is a homogeneous polynomial of degree non the cone

of convex bodies in Rn, it admits a polarization Vol(∆1,...,∆n). That is

Vol(∆1,...,∆n) is a unique function of n-tuple of convex bodies ∆1,...,∆n

with the following properties:

1. Vol(∆1,...,∆n) is linear in each argument, with respect to Minkowski

addition;

2. Vol(∆1,...,∆n) is symmetric;

3. on a diagonal it is equal to the volume, i.e. Vol(∆,...,∆) = Vol(∆).

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GeometryofgeneralizedvirtualpolyhedraAskoldKhovanskii*October4,2022AbstractPartialgeneralizationsofvirtualpolyhedratheory(sometimesun-derdierentnames)appearedrecentlyinthetheoryoftorusmanifolds.Thesegeneralizationslookverydierentfromtheoriginalvirtualpoly-hedratheory.Theyarebasedonsimpleargumentsf...

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