Deformations of Abelian Varieties and Serre Tate Theory

36 Pages·2016·0.33 MB·English

by  Drew Moore

Deformations
ofAbelian
Varietiesand
SerreTate
Theory
DrewMoore
Deformations of Abelian Varieties and
Deformations
ofSchemes
Serre Tate Theory
Deformations
ofAbelian
Varieties
Deformations
ofp-Divisible
Groups Drew Moore
SerreTate
Coordinates
November 10
Context of this Talk within the Seminar
Deformations
ofAbelian The main players we’ve been dealing with in this seminar have been
Varietiesand
SerreTate algebraic varieties over Fp, Zp, Qp, formal schemes over Zp, and rigid
Theory
analytic varieties over Q , and their analogues for extensions of F
p p
DrewMoore and Q . Just as a reminder, these categories sit in a commutative
p
Deformations diagram
ofSchemes
DofeAfobrmeliaatnions Sch/Zp Sch/Qp
Varieties
Deformations Sch/Fp
ofp-Divisible
Groups FSch/Zp Rig/Qp
SerreTate
Coordinates For any scheme (resp. formal scheme) X over SpecZ (resp. SpfZ ),
p p
we have its specialization map X→X to its special fiber X ∈Sch/Fp.
This talk: take X to be a moduli space of abelian varieties. Pick a
closed point x in X (corresponding to an ab var A ). Study the
0 0
preimage of x under the specialization map, i.e. its deformations to
0
characteristic 0.
Deformations
ofAbelian
Varietiesand In Matt’s second talk this quarter, he described the rigid generic fiber
SerreTate
Theory functor FSch/Zp →Rig/Qp only for p-adic formal schemes, which was
DrewMoore Raynaud’s original construction. I.e., those formal schemes for which
the topology on the open affines is given by the ideals (pn).
Deformations
ofSchemes
Deformations Example: SpfZ (cid:104)(cid:104)T(cid:105)(cid:105)(cid:32) the closed unit polydisk.
ofAbelian p i
Varieties
The preimage of a point in the special fiber of this example is an
Deformations
ofp-Divisible open unit polydisk. Would like a formal model for this. Berthelot
Groups
SCeororredTinaattees extended this functor to all of FSch/Zp, which will solve this.
Example: SpfZ [[T]], with ideal of definition (p,T). Then its rigid
p i i
generic fiber is the open unit polydisk.
Conclusion: we expect that the deformations of x to characteristic 0
0
will be parametrized by something like SpfZ [[T]].
p i
The Setup: Coefficients and Categories
Deformations
ofAbelian Let k be a field and W a complete local ring with residue field k. For
Varietiesand
SerreTate example:
Theory
k =W =C
DrewMoore
k =F and W =Z
Deformations p p
ofSchemes k =F , W =W(F )
Deformations p p
ofAbelian Let C be the category
Varieties
Deformations
ofp-Divisible Artin local rings R with maxop
Groups  
C = ideal m equipped with an
SerreTate R
Coordinates  isomorphism R/m →k 
R
I will call C the category of Artin schemes. Similarly C(cid:98)is the
category defined similarly, but replace “Artin” with “complete
Noetherian.” Objects of C(cid:98)are inductive limits of objects of C, and
are affine formal schemes.
Why C and C(cid:98)?
Deformations
ofAbelian Suppose we are in the situation k =W =C. Let E be an elliptic
Varietiesand 0
SerreTate curve over k =C.
Theory
DrewMoore
E :y2 =x3+a x +b a ,b ∈C
0 0 0 0 0
Deformations
ofSchemes
We want to study deformations of E . A deformation over C[[t]]
Deformations 0
ofAbelian could be something like
Varieties
Deformations
ofp-Divisible E :y2 =x3+a(t)x +b(t) a(0)=a ,b(0)=b
Groups 0 0
SerreTate
Coordinates We could imagine that a and b have a positive radius of convergence,
so then plugging in sufficiently small values of t would give
“deformations” E of E .
t 0
SpfC[[t]] is an object of C(cid:98). To study deformations of E0 to C[[t]], it
is easier to first study deformations to the intermediate rings
C[t]/(tn), which correspond to schemes in C.
Why C and C(cid:98)?
Deformations k will be “where we start” - i.e., we will have an object over k, and
ofAbelian
Varietiesand we will aim to “deform” it over W. For example, if k =W =F ,
SerreTate p
Theory then we will be restricting to deformations that are characteristic p.
DrewMoore But if W =Z , the deformations could be mixed characteristic.
p
Deformations
ofSchemes Geometrically, the isomorphism R/mR →k allows us to concretely
Deformations pick out the “special fiber” over S =SpecR.
ofAbelian
Varieties
Closed embeddings S →S(cid:48) in C are called thickenings. If the kernel
Deformations
ofp-Divisible of the corresponding ring map R(cid:48) →R has square zero, then S →S(cid:48)
Groups
SerreTate is said to be first order. Examples:
Coordinates
SpecC[t]/(tn)→SpecC[t]/(tn+1)
SpecZ/pnZ→SpecZ/pn+1Z
Every thickening is a composition of first order thickenings.
Fact: C(cid:98)is equivalent to the category of formal schemes equipped
with an isomorphism of its special fiber with the point Speck.
Deformation Definitions, Definitely
Deformations
ofAbelian Suppose that X0 is a smooth scheme over k. Let S =SpecR be an
Varietiesand
SerreTate element of C. An infinitesimal deformation of X0 to S is a smooth
Theory
S-scheme X →S along with an isomorphism of the special fiber of X
DrewMoore
with X :
0
Deformations X −∼→= X × k
ofSchemes 0 S
DofeAfobrmeliaatnions Suppose S →S(cid:48) =SpecR(cid:48) is a thickening in C, and X is smooth
Varieties over S. Then a deformation along S →S(cid:48) is a smooth scheme X(cid:48)
Dofepfo-rDmivaitsiiobnles over S(cid:48) along with an isomorphism X ∼=X(cid:48)×S(cid:48) S.
Groups
SerreTate Current goal: understand when we can deform X to S, and if so, in
Coordinates 0
how many different ways? We will do this by studying deformations
along first order thickenings.
Namely, we are interested in the following contravariant functor:
Def : C →Sets S (cid:55)→{deformations of X to S}/∼=
X0 0
Cohomology and Deformations
Deformations Theorem
ofAbelian
Varietiesand Suppose S →S(cid:48) is a first order thickening, and X is a deformation of
SerreTate
Theory X to S. Let I be the kernel of R(cid:48) →R. Then:
0
DrewMoore
There exists an obstruction
Deformations
ofSchemes
ω(X/S)∈H2(X,T ⊗I)
Deformations X/S
ofAbelian
Varieties
which is zero if and only if there exists X(cid:48)/S(cid:48) deforming X to S(cid:48).
Deformations
ofp-Divisible Moreover, ω is natural with respect to isomorphisms in X/S.
Groups
SerreTate If ω(X/S)=0, then the set of deformations of X to S(cid:48) (up to
Coordinates
isomorphism) can be put in bijection with
H1(X,T ⊗I)
X/S
If X(cid:48) is a deformation of X along S →S(cid:48), then the infinitesimal
automorphisms of X(cid:48) over X are naturally identified with
H0(X,T ⊗I)
X/S
Proof in special case, idea of proof
Deformations
ofAbelian For simplicity, suppose we are in the case S =Speck and
Varietiesand
SerreTate S(cid:48) =Speck[(cid:15)]. Let X/k[(cid:15)] be a deformation of X0/k. Then the ideal
Theory
sheaf of X (cid:44)→X is just ((cid:15)) considered as an O -module. Hence, the
0 X
DrewMoore conormal bundle of X in X is ((cid:15))∼=O
0 X0
Deformations
ofSchemes Hence the conormal sequence gives (using ((cid:15))∼=O ),
Deformations X0
ofAbelian
Varieties X i X
Deformations 0
ofp-Divisible (cid:32) 0→O →i∗Ω →Ω →0
Groups X0 X/k[(cid:15)] X0/k
SerreTate
Coordinates Speck Speck[(cid:15)]
On the right, we have an extension of Ω by O , and thus an
X0/k X
element of Ext1 (Ω ,O )∼=H1(X ,T ).
OX0 X0/k X0 0 X0/k
Idea of proof: first prove it when X is affine (i.e. there exists a
unique deformation), and then use Cech cocycles in the general case.
Abelian Schemes
Deformations
ofAbelian Let S be a scheme (think S ∈C). An abelian scheme over S is an
Varietiesand
SerreTate S-group scheme A→S which is proper, flat, finitely presented, and
Theory
has smooth geometrically connected fibers.
DrewMoore
Deformations When S =Speck is the spectrum of a field, an abelian scheme A /S
ofSchemes 0
is just an abelian variety.
Deformations
ofAbelian
Varieties An infinitesimal deformation of A to S ∈C is an abelian scheme
0
Deformations A/S over S along with an isomorphism of the special fiber
ofp-Divisible
Groups A −∼→= A× S. Important: we are deforming not just the scheme A ,
SerreTate 0 k 0
Coordinates but the group structure too.
Consider the following contravariant functor:
(cid:26) (cid:27)
infinitesimal deformations
Def : C →Sets S (cid:55)→ /∼=
A0 of A to S
0