Minimal Compactifications Research Articles

Minimal compactifications of the affine plane with nef canonical divisors

We shall consider minimal analytic compactifications of the affine plane with singularities. In previous work, Kojima and Takahashi proved that any minimal analytic compactification of the affine plane, which has at worse log canonical singularities, is a numerical del Pezzo surface (i.e., a normal complete algebraic surface with the numerically ample anti-canonical divisor) and has only rational singularities. In this article, we show that any minimal analytic compactification of the affine plane with the nef canonical divisor has an irrational singularity.

Lifting Chern classes by means of Ekedahl–Oort strata

The moduli space of principally polarized abelian varieties $A_g$ of genus g is defined over the integers and admits a minimal compactification $A_g^*$, also defined over the integers. The Hodge bundle over $A_g$ has its Chern classes in the Chow ring of $A_g$ with rational coefficients. We show that over the prime field $F_p$, these Chern classes naturally lift to $A_g^*$ and do so in the best possible way: despite the highly singular nature of $A_g^*$ they are represented by algebraic cycles on $A_g^*\otimes F_p$ which define elements in its bivariant Chow ring. This is in contrast to the situation in the analytic topology, where these Chern classes have canonical lifts to the complex cohomology of the minimal compactification as Goresky-Pardon classes, which are known to define nontrivial Tate extensions inside the mixed Hodge structure on this cohomology.

The plectic weight filtration on cohomology of Shimura varieties and partial Frobenius

We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovar and Scholl. This is achieved with the help of Morel's work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.

The Plectic Weight Filtration on Cohomology of Shimura Varieties and Partial Frobenius

Abstract We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovář and Scholl. This is achieved with the help of Morel’s work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.

Amenability of representations and invariant Hahn–Banach theorems

Invariant Hahn–Banach theorems are proved in the context of right amenable, weakly almost periodic representations of semigroups on locally convex spaces. Applications are made to invariant means, invariant measures, and to Banach limits on weakly almost periodic flows. Additionally, invariant linear Hahn–Banach extension operators are shown to exist on suitable invariant subspaces of Banach spaces. The key construct on which these results depend is the lifting of a representation on a semigroup S to its coefficient compactification, the minimal compactification of S for which the lift is injective.

Chern classes of automorphic vector bundles, II

We prove that the $\ell$-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over $\bar{ \mathbb{Q}}_p$, descend to classes in the $\ell$-adic cohomology of the minimal compactifications. These are invariant under the Galois group of the $p$-adic field above which the variety and the bundle are defined.

Strata Hasse invariants, Hecke algebras and Galois representations

We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks mathop {Ghbox {-}{} mathtt{Zip}}nolimits ^{mu }. Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl–Oort stratum closure of a general Hodge-type Shimura variety. A key tool is the construction of a stack of zip flags mathop {Ghbox {-}{} mathtt{ZipFlag}}nolimits ^mu , fibered in flag varieties over mathop {Ghbox {-}{} mathtt{Zip}}nolimits ^{mu }. It provides a simultaneous generalization of the “classical case” homogeneous complex manifolds studied by Griffiths–Schmid and the “flag space” for Siegel varieties studied by Ekedahl–van der Geer. Four applications are obtained: (1) Pseudo-representations are attached to the coherent cohomology of Hodge-type Shimura varieties modulo a prime power. (2) Galois representations are associated to many automorphic representations with non-degenerate limit of discrete series Archimedean component. (3) It is shown that all Ekedahl–Oort strata in the minimal compactification of a Hodge-type Shimura variety are affine, thereby proving a conjecture of Oort. (4) Part of Serre’s letter to Tate on mod p modular forms is generalized to general Hodge-type Shimura varieties.

COMPACTIFICATIONS OF SUBSCHEMES OF INTEGRAL MODELS OF SHIMURA VARIETIES

We study several kinds of subschemes of mixed characteristic models of Shimura varieties which admit good (partial) toroidal and minimal compactifications, with familiar boundary stratifications and formal local structures, as if they were Shimura varieties in characteristic zero. We also generalize Koecher’s principle and the relative vanishing of subcanonical extensions for coherent sheaves, and Pink’s and Morel’s formulas for étale sheaves, to the context of such subschemes.

Compactifications of splitting models of PEL-type Shimura varieties

We construct toroidal and minimal compactifications, with expected properties concerning stratifications and formal local structures, for all integral models of PEL-type Shimura varieties defined by taking normalizations over the splitting models considered by Pappas and Rapoport. (These include, in particular, all the normal flat splitting models they considered.)

COMPACTIFICATIONS OF PEL-TYPE SHIMURA VARIETIES IN RAMIFIED CHARACTERISTICS

We show that, by taking normalizations over certain auxiliary good reduction integral models, one obtains integral models of toroidal and minimal compactifications of PEL-type Shimura varieties which enjoy many features of the good reduction theory studied as in the earlier works of Faltings and Chai’s and the author’s. We treat all PEL-type cases uniformly, with no assumption on the level, ramifications, and residue characteristics involved.

C-scattered type and minimal compactifications with countable remainder

R(X) is the closed subset of a space X consisting of the points that do not have a compact neighbourhood. For each ordinal α, Rα(X) is defined inductively as follows: R0(X)=X, Rα+1(X)=R(Rα(X)) and, if α is a non-zero limit ordinal, Rα(X)=⋂β<αRβ(X). According to R. Telgársky, X is called C-scattered if Rα(X)=∅ for some α. The first α for which Rα(X) is locally compact is called here the C-scattered level of X. A space is called a minimal space if it is homeomorphic to its perfect images, and a compactification Y of X is called a minimal compactification of X if for every compactification Z of X smaller than Y, Y∖X and Z∖X are homeomorphic.For each α<ω1, we define a C-scattered countable metric space Nα of level α which is contained as a closed subset in every metric C-scattered space of level α. We characterize the spaces Nα topologically and prove that they are minimal spaces. We use the results to refine results of Zippin and Hoshina by showing that a rim-compact separable metrizable space X with Rω0(X)=∅ has a minimal metrizable compactification with remainder homeomorphic to Nα or the direct sum of countably many copies of Nα, for a certain α≤ω0.

Sur une conjecture de Kottwitz au bord

We are interested in the intersection cohomology of the minimal compactification of Siegel modular varieties at some places of bad reduction. We compute the semi-simple trace of the Frobenius morphism on the fibers of the nearby cycles of the intersection complex. We obtain a common generalization of results of Morel and Haines and Ngo.

Notes on minimal compactifications of the affine plane

In this paper, we consider the minimal compactifications X of the complex affine plane with at most log canonical singular points. We classify the surfaces X in the case X has at least one non-log terminal singular point.

On the moduli space of the Schwarzenberger bundles

For any odd n, we prove that the coherent sheaf F A on P n C , defined as the cokernel of an injective map f : O ○+2 Pn → O Pn (1) ○+(n+2) , is Mumford-Takemoto stable if and only if the map f is stable, when considered as a point of the projective space P(Hom(O Pn (-1) ⊗2 ,O ⊗(n+2) Pn )*) under the action of the reductive group SL(2) × SL(n + 2). This proves a particular case of a conjecture of J.-M.Drezet and it implies that a component of the Maruyama scheme of the semi-stable sheaves on P n of rank n and Chern polynomial (1 + t) n+2 is isomorphic to the Kronecher moduli N(n + 1, 2, n + 2), for any odd n. In particular, such scheme defines a smooth minimal compactification of the moduli space of the rational normal curves in P n , that generalizes the construction defined by G. Ellinsgrud, R. Piene and S. Stromme in the case n = 3.

Dynamics of polynomial mappings of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /

We study the dynamics of polynomial self mappings f of C 2 . We construct, for a large class of mappings, an invariant measure μ which is mixing and of maximal entropy h μ (f) = max (log d t (f), log λ 1 (f)), where d t (f) is the topological degree of f and λ 1 (f) its first dynamical degree. To achieve this, we look at the meromorphic extensions of f to smooth minimal compactifications of C 2 . When a good compactification is found, we construct an f * -invariant Green current T which contains many dynamical informations. When δ:= d t (f)/λ 1 (f) > 1, the measure μ is obtained as p = dd c (vT), where v is a partial Green function defined on the support of T. When δ < 1, μ = T ∧ T - where T - is a globally defined f * -invariant current.

An internal characterization of sets of functions determining minimal compactifications

We look for internal necessary and sufficient conditions for a subset F of the algebra of all continuous real-valued bounded functions on a Tychonoff space X to have the property that there exists a minimal compactification of X over which every function from F is continuously extendable. Further, we apply some of our ideas to a nonlocally compact case of Magill's theorem, i.e., to the problem of when the continuous image of a remainder of a not necessarily locally compact Tychonoff space X is again a remainder of X.

Errata to the Paper “On the Minimal Compactifications of C2”

Errata to the Paper “On the Minimal Compactifications of C2”

Minimal martin compactifications of riemann surfaces with densities

Given an arbitrary nonnegative locally Holder continuous 2-form P on an arbitrary open Riemann surface R. We will show the existence of an nonnegative C ∞ 2-form Q on R such that Q ≥ P on R and the Martin compactification R ∗ Q of R which is the smallest possible Martin compactification of R.

Appendix to "The Iwasawa Conjecture for Totally Real Fields": Arithmetic Minimal Compactification of the Hilbert-Blumenthal Moduli Spaces

The arithmetic theory of toroidal compactification of the Hilbert-Blumenthal moduli spaces was described in Rapoport's thesis [Rap], while the arithmetic theory of minimal compactification has been lacking a long time. After Faltings' work on the arithmetic compactification of Siegel moduli spaces [Fa], it was obvious to the experts that the same things could be done in the HilbertBlumenthal case in similar but simpler ways. (In fact, what emerged was a machinery to compactify moduli spaces coming from abelian varieties.) It turns out that the arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces could have been done more than ten years ago. [Rap] contains everything that is needed, so in some sense I am writing a summary and an appendix to [Rap]. The purpose of this. appendix is to explain this remark, and show that this theory of minimal compactification can be useful in number theory. For instance, one can replace for almost all by for all primes outside the level and the discriminant in the main result of [Br-La]. Two more applications are given here, one about p-adic monodromy, the other about positivity of the Hodge line bundle (the sections of which are Hilbert modular forms). For a systematic exposition of the theory of semi-abelian degeneration and compactification, the reader is referred to the book [Fa-Ch], where the (more

Minimal Compactifications and their Associated Function Spaces

Minimal Compactifications and their Associated Function Spaces