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**Voronoï’s geometric theory. Finiteness properties for families of lattices and similar objects.
(Théorie de Voronoï géométrique. Propriétés de finitude pour les familles de réseaux et analogues.)**
*(French.
English summary)*
Zbl 1085.11033

In this article, the author extends and generalizes the approach to develop a completely geometric Voronoï theory which he began in [J. Reine Angew. Math. 482, 93–120 (1997; Zbl 1011.53035)] where he introduced the concept of generalized systoles. In the present article, the author introduces the notion of nondegenerate point and systematically studies points satisfying particular properties such as perfection and eutaxy whose definitions are motivated by those from the classical Voronoï theory of lattices. Particular emphasis is laid upon the study of onfigurations and of finiteness results. As applications, the author obtains new results on the invariants of Bergé-Martinet and of Hermite-Humbert that are attached to number fields, and on Riemann surfaces.

The theoretical framework of this theory is laid out in §1 of the present paper. Let \(E\) be a finite-dimensional \({\mathbb R}\)-vector space, \({\mathcal F}\) a finite set of vectors in \(E\) and \(K\) their convex hull. \({\mathcal F}\) is said to be perfect if it generates \(E\) affinely, and eutactic if the origin is in the affine interior of \(K\). Now let \(V\) be a smooth and connected variety with \({\mathcal C}^1\)-functions \(f_s:V\to{\mathbb R}\) indexed by some set \(C\). \((f_s)_{s\in C}\) is called a system of length functions if for every \(p\in V\), every neighborhood \(U\) of \(p\) and every \(L\in{\mathbb R}\) one has \(f_s >L\) on \(U\) for almost all \(s\in C\). The generalized systole \(\mu (p)\) in a point \(p\in V\) is now defined to be \(\mu (p)=\min_{s\in C}f_s(p)\). Then \(p\in V\) is called perfect resp. eutactic if the family of differentials \((df_s(p))_{s\in S_p}\) is perfect resp. eutactic in the cotangent space \(T^*_pV\), where \(S_p=\{ s\in C; f_s(p)=\mu (p)\}\), and (strictly) extreme if \(\mu\) has a (strict) local maximum in \(p\). \(V\) can be partitioned into so-called minimal classes, where two points \(p,q\in V\) belong to the same class iff \(S_p=S_q\).

Let now \(V\) be equipped with a connection (assumed to be geodesic). A \({\mathcal C}^1\)-function \(f:V\to{\mathbb R}\) is said to be (strictly) convexoïdal if any critical point of \(f\) restricted to a geodesic is a (strict) local minimum. Now suppose in addition that \((f_s)_{s\in C}\) is a family of convexoïdal length functions on \(V\). Then a point \(p\in V\) is said to be nondegenerate if for every geodesic emanating from \(p\) there is at least one \(f_s\) that is strictly convexoïdal on that geodesic in some neighborhood of \(p\). With these definitions, it follows readily that if \(V\) is equipped with a family of convexoïdal length functions, then perfect points are always nondegenerate (Proposition 1.4), and that Voronoï’s theorem (i.e. a point is extreme iff it is perfect and eutactic) holds iff every extreme point is nondegenerate iff every extreme point is perfect. This holds in particular whenever the length functions are strictly convexoïdal (Proposition 1.5). Furthermore, the set of points that are eutactic and nondegenerate is discrete, so in particular also the set of points that are perfect and eutactic (Corollary 1.7).

The author then obtains certain somewhat technical results on isolation of points that are eutactic and nondegenerate by which he can prove that if \(V\) is a smooth subvariety of \(P_n\), the space of unimodular positive definite symmetric \(n\times n\) matrices, defined by polynomials with algebraic coefficients in \({\mathbb R}\), then there are only finitely many perfect points for \(\mu^D\) (relative to \(V\)) and they are all algebraic over \({\mathbb Q}\) as are all nondegenerate eutactic points under some additional assumption on \(V\) (Corollary 1.12). Here, \(\emptyset\neq D\subset{\mathbb Z}^n \setminus\{ 0\}\) and \(\mu^D(A)=\min_{s\in D}A[s]\) for \(A\in P_n\). The author points out that the interest in this result lies in the fact that it applies readily to all natural and interesting types of lattices, for example autodual lattices.

In §2, the author applies these foundational results in certain situations to obtain finiteness results, notably to \(P_n\) as defined above and connected complete totally geodesic subvarieties thereof, and in particular their \(\rho\)-invariant forms for some representation \(\rho :\Pi\to \text{GL}_n({\mathbb Z})\) for a finite group \(\Pi\) (such \(\rho\)-invariant forms correspond in a natural way to \(\Pi\)-lattices). The main results concern finiteness of minimal classes containing weakly eutactic points and finiteness of perfect or eutatic nondegenerate points (Theorem 1), criteria for nondegeneracy (Theorem 2), the rank of minimal vectors for perfect points (Theorem 3), finiteness of the number of minimal classes modulo a certain action in the hermitian, symmetric or antisymmetric bilinear case (Theorem 4).

Using the methods developed in this paper, the author shows in Theorem 5 that the invariant of Bergé-Martinet satisfies Voronoï’s theorem (also generalized to the hermitian case). He then interprets the invariant of Hermite-Humbert for Humbert forms over the ring of integers \({\mathcal O}_L\) in some algebraic number field \(L\). In this new setting, he obtains Voronoï’s theorem for a certain \(\mu_L\) which is defined in a way so that the notions of perfection and eutaxy coincide with the classical ones for Humbert forms.

A final application concerns systoles of Riemann surfaces (Theorem 6).

The theoretical framework of this theory is laid out in §1 of the present paper. Let \(E\) be a finite-dimensional \({\mathbb R}\)-vector space, \({\mathcal F}\) a finite set of vectors in \(E\) and \(K\) their convex hull. \({\mathcal F}\) is said to be perfect if it generates \(E\) affinely, and eutactic if the origin is in the affine interior of \(K\). Now let \(V\) be a smooth and connected variety with \({\mathcal C}^1\)-functions \(f_s:V\to{\mathbb R}\) indexed by some set \(C\). \((f_s)_{s\in C}\) is called a system of length functions if for every \(p\in V\), every neighborhood \(U\) of \(p\) and every \(L\in{\mathbb R}\) one has \(f_s >L\) on \(U\) for almost all \(s\in C\). The generalized systole \(\mu (p)\) in a point \(p\in V\) is now defined to be \(\mu (p)=\min_{s\in C}f_s(p)\). Then \(p\in V\) is called perfect resp. eutactic if the family of differentials \((df_s(p))_{s\in S_p}\) is perfect resp. eutactic in the cotangent space \(T^*_pV\), where \(S_p=\{ s\in C; f_s(p)=\mu (p)\}\), and (strictly) extreme if \(\mu\) has a (strict) local maximum in \(p\). \(V\) can be partitioned into so-called minimal classes, where two points \(p,q\in V\) belong to the same class iff \(S_p=S_q\).

Let now \(V\) be equipped with a connection (assumed to be geodesic). A \({\mathcal C}^1\)-function \(f:V\to{\mathbb R}\) is said to be (strictly) convexoïdal if any critical point of \(f\) restricted to a geodesic is a (strict) local minimum. Now suppose in addition that \((f_s)_{s\in C}\) is a family of convexoïdal length functions on \(V\). Then a point \(p\in V\) is said to be nondegenerate if for every geodesic emanating from \(p\) there is at least one \(f_s\) that is strictly convexoïdal on that geodesic in some neighborhood of \(p\). With these definitions, it follows readily that if \(V\) is equipped with a family of convexoïdal length functions, then perfect points are always nondegenerate (Proposition 1.4), and that Voronoï’s theorem (i.e. a point is extreme iff it is perfect and eutactic) holds iff every extreme point is nondegenerate iff every extreme point is perfect. This holds in particular whenever the length functions are strictly convexoïdal (Proposition 1.5). Furthermore, the set of points that are eutactic and nondegenerate is discrete, so in particular also the set of points that are perfect and eutactic (Corollary 1.7).

The author then obtains certain somewhat technical results on isolation of points that are eutactic and nondegenerate by which he can prove that if \(V\) is a smooth subvariety of \(P_n\), the space of unimodular positive definite symmetric \(n\times n\) matrices, defined by polynomials with algebraic coefficients in \({\mathbb R}\), then there are only finitely many perfect points for \(\mu^D\) (relative to \(V\)) and they are all algebraic over \({\mathbb Q}\) as are all nondegenerate eutactic points under some additional assumption on \(V\) (Corollary 1.12). Here, \(\emptyset\neq D\subset{\mathbb Z}^n \setminus\{ 0\}\) and \(\mu^D(A)=\min_{s\in D}A[s]\) for \(A\in P_n\). The author points out that the interest in this result lies in the fact that it applies readily to all natural and interesting types of lattices, for example autodual lattices.

In §2, the author applies these foundational results in certain situations to obtain finiteness results, notably to \(P_n\) as defined above and connected complete totally geodesic subvarieties thereof, and in particular their \(\rho\)-invariant forms for some representation \(\rho :\Pi\to \text{GL}_n({\mathbb Z})\) for a finite group \(\Pi\) (such \(\rho\)-invariant forms correspond in a natural way to \(\Pi\)-lattices). The main results concern finiteness of minimal classes containing weakly eutactic points and finiteness of perfect or eutatic nondegenerate points (Theorem 1), criteria for nondegeneracy (Theorem 2), the rank of minimal vectors for perfect points (Theorem 3), finiteness of the number of minimal classes modulo a certain action in the hermitian, symmetric or antisymmetric bilinear case (Theorem 4).

Using the methods developed in this paper, the author shows in Theorem 5 that the invariant of Bergé-Martinet satisfies Voronoï’s theorem (also generalized to the hermitian case). He then interprets the invariant of Hermite-Humbert for Humbert forms over the ring of integers \({\mathcal O}_L\) in some algebraic number field \(L\). In this new setting, he obtains Voronoï’s theorem for a certain \(\mu_L\) which is defined in a way so that the notions of perfection and eutaxy coincide with the classical ones for Humbert forms.

A final application concerns systoles of Riemann surfaces (Theorem 6).

Reviewer: Detlev Hoffmann (Nottingham)

### MSC:

11H55 | Quadratic forms (reduction theory, extreme forms, etc.) |

11E12 | Quadratic forms over global rings and fields |

11H06 | Lattices and convex bodies (number-theoretic aspects) |

11H50 | Minima of forms |

14K99 | Abelian varieties and schemes |

30F45 | Conformal metrics (hyperbolic, Poincaré, distance functions) |

30F99 | Riemann surfaces |

53C05 | Connections (general theory) |

53C99 | Global differential geometry |