Deformations and elements of deformation theory (original) (raw)
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Level mmm stratifications of versal deformations of ppp-divisible groups
Journal of Algebraic Geometry, 2008
Let k be an algebraically closed field of characteristic p > 0. Let c, d, m be positive integers. Let D be a p-divisible group of codimension c and dimension d over k. Let D be a versal deformation of D over a smooth k-scheme A which is equidimensional of dimension cd. We show that there exists a reduced, locally closed subscheme s D (m) of A that has the following property: a point y ∈ A(k) belongs to s D (m)(k) if and only if y * (D)[p m ] is isomorphic to D[p m ]. We prove that s D (m) is regular and equidimensional of dimension cd−dim(Aut Aut Aut(D[p m ])). We give a proof of Traverso's formula which for m >> 0 computes the codimension of s D (m) in A (i.e., dim(Aut Aut Aut(D[p m ]))) in terms of the Newton polygon of D. We also provide a criterion of when s D (m) satisfies the purity property (i.e., it is an affine A-scheme). Similar results are proved for quasi Shimura p-varieties of Hodge type that generalize the special fibres of good integral models of Shimura varieties of Hodge type in unramified mixed characteristic (0, p).
Journal of Algebraic Geometry, 2011
Let k be an algebraically closed field of characteristic p > 0. Let D be a p-divisible group over k which is not isoclinic. Let D (resp. D k) be the formal deformation space of D over Spf(W (k)) (resp. over Spf(k)). We use axioms to construct formal subschemes G k of D k that: (i) have canonical structures of formal Lie groups over Spf(k) associated to p-divisible groups over k, and (ii) give birth, via all geometric points Spf(K) → G k , to p-divisible groups over K that are isomorphic to D K. We also identify when there exist formal subschemes G of D which lift G k and which have natural structures of formal Lie groups over Spf(W (k)) associated to p-divisible groups over W (k). Applications to Traverso (ultimate) stratifications are included as well.
Stratifications of Newton polygon strata and Traverso's conjectures for p-divisible groups
Annals of Mathematics, 2013
The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field is the least positive integer m such that D[p m ] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latter disproves a conjecture of Traverso. As an application, we answer a question of Zink on the liftability of an endomorphism of D[p m ] to D.
Purity of the stratification by Newton polygons
Journal of the American Mathematical Society, 1999
Let S S be a variety in characteristic p > 0 p>0 . Suppose we are given a nondegenerate F F -crystal over S S , for example the i i th relative crystalline cohomology sheaf of a family of smooth projective varieties over S S . At each point s s of S S we have the Newton polygon associated to the action of F F on the fibre of the crystal at s s . According to a theorem of Grothendieck the Newton polygon jumps up under specialization. The main theorem of this paper is that the jumps occur in codimension 1 1 on S S (the Purity Theorem). As an application we prove some results on deformations of iso-simple p p -divisible groups.
Newton polygons and p-divisible groups: a conjecture by Grothendieck
2002
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A Superficial Working Guide to Deformations and Moduli
This is the first part of a guide to deformations and moduli, especially viewed from the perspective of algebraic surfaces (the simplest higher dimensional varieties). It contains also new results, regarding the question of local homeomorphism between Kuranishi and Teichmueller space, and a survey of new results with Ingrid Bauer, concerning the discrepancy between the deformation of the action of a group G on a minimal models S, respectively the deformation of the action of G on the canonical model X. Here Def(S) maps properly onto Def(X), but the same does not hold for pairs: Def(S,G) does not map properly onto Def(X,G). Indeed the connected components of Def(S), in the case of tertiary Burniat surfaces, only map to locally closed sets. The last section contains anew result on some surfaces whise Albanese map has generic degree equal to 2.
Deformations of Local Systems and Eisenstein Series
Geometric and Functional Analysis, 2008
Introduction 0.1. The goal of this paper is to realize a suggestion made by V. Drinfeld. To explain it let us recall the general framework of the geometric Langlands correspondence.
2010
Using first order deformation theory of pointed curves we show that the semigroup of a generic Weierstrass point whose semigroup has first nonzero element k consists only of multiples of k until after its greatest gap value, and that on the moduli space of curves two components of the divisor of points corresponding to curves possessing exceptional Weierstrass points intersect nontransversely. As usual let J( be the moduli space of curves of genus g. Many subloci of Jtg defined in terms of Weierstrass points have received much study. (1) Definition. For an integer k with 2 < k < g, Dk k = {[C] e Jfg. C possesses a Weierstrass point p with h°(C, kp) > 2). It is known that Dk k is irreducible of dimension 2g 3 + k and a general point in Dk k corresponds to a curve C with a Weierstrass point p with h°(C,kp) = 2. See Rauch [1], Lax [1, 2], Arbarello [1, 2]. Here we will prove: (2) Theorem. A generic point in Dk k corresponds to a curve with a Weierstrass point whose semigroup c...
2001
This work is the first part in a series of three dedicated to the foundations of integral aspects of Shimura varieties and of Fontaine's categories. It deals mostly with the unramified context of (arbitrary) mixed characteristic (0,p). Among the topics covered we mention: the generalization of the classical Serre-Tate theory of ordinary p-divisible groups and of their canonical lifts, the generalization of the classical Serre-Tate-Dwork-Katz theory of (crystalline) coordinates for ordinary abelian varieties, the strong form of the generalized Manin problem, global deformations in the generalized Shimura context, Dieudonn\'e's theories (reobtained, simplified and extended), the main list of stratifications of special fibres of integral canonical models of Shimura varieties of preabelian type, the uniqueness of such models in mixed characteristic (0,2), the existence (in many situations) of such models in mixed characteristic (0,2), steps towards the classification of S...