The topology of conjugate varieties

Math. Ann. 305, 287-309 (1996)

m �9 Soanm-V~r~ 1996

The topology of.conjugate varieties

David Reed*

Mathematical Institute. 24-29 St. Giles', Oxford OX1 3LB, UK

Received: 7 November 1994 1 Revised version: 11 July 1995

Mathematics Subject Classification (1991): 11G35, 14F45

1. Introduction

If V is an algebraic variety defined over k, a finitely generated extension of Q, for each embedding ~ : k ~ C we can extend scalars to form the complex algebraic variety V~ defined by the following cartesian square (base change or extension of scalars)

V r xQC ~ V

Spec C ---* Speck.

The complex points of this variety V~(C) form a topological space and the topological types of two such spaces, V,,(C) and Vr(C) for two different embeddings k --, C can be compared. We will refer to varieties V~ and Vr obtained in this manner as conjugate varieties.

Serre [Se 64] and Abelson [Ab 74] have produced examples of conjugate varieties whose complex points constitute non-homeomorphic topological spaces. Since the appearance of these papers there has been little published work in this area. The principal result of the research reported upon here are sufficient conditions for the topological spaces of complex points of conjugate varieties to be homeomorphic. We begin by recalling definitions due to Matsusaka-Shimura- Koizumi [Ko 72].

(1.0.1) Definition. For a divisor X on a projective variety V define ~ ( X ) to be the class of all divisors X ~ on V for which there are integers m, n with mX =_ nX'

" Present address: Department of Mathematics, Duke University, Durham, NC 27708-0320, USA

288 D. Reed

(algebraic equivalence). I f ~ ( X ) contains an ample divisor then it is called a polarization (this terra is also applied to the ample divisor in the class). An isomorphism of projective varieties f : V --* W is said to be a isomorphism of polarized projective varieties if there are polarizations ~o, ~ , on V and W respectively, such that the map on divisors induced by f takes ~ ' to

(1.0.2) Definition. The field of modulifor a polarized projective variety (V , ~ ) is the fieM k such that for a e Aut(C), o" is in fact in Aut(C/k ) if and only i f V ~ ~- V as polarized varieties.

A discussion of when fields of moduli exist in general for varieties can be found in [Ko 72]. To ~tvoid these rather delicate questions we abstract a property possessed by varieties defined over their fields of moduli.

(1.0.3) Definition. Let V be a projective variety defined over k, a finitely generated extension of Q. Let k ~ be a purely transcendental subfield of k with tr.deg k' = tr.deg k. I f there is a subfield of k J over which k is generated by i = v / - 1 call this ko, otherwise we relabet k' as ko. Choose an embedding ~bo : ko ~ C. V will be said to have non-isomorphic conjugates with respect to ~o if there are no embeddings or1, or2 : k ---, C agreeing with (bo on ko and such that V~ ~- V~: (isomorphism as polarized projective varieties).

The reasons for the choice of ko in this definition become evident on consideration of an example due to Shimura of a hyperelliptic curve (discussed below in Sect. 3.2) which does not have the non-isomorphic conjugates property and, which, indeed, is not defined over its field of moduti.

It is not difficult to see that if V has non-isomorphic conjugates with respect to one choice of ~b0 then this will be the case with respect to any other q~. Just pick a, a generator of k over ke. Then we can always find a ~- 6 Aut(C) which

fixes a and satisfies 7- o ~o = 4~. If V~ ~ Vg~ for or; agreeing with ~b~ on/co, conjugating f by ~r will give V~, ~ V~ 2 for ai agreeing with ~bo on/co. From now on we will simply say that V has non-isomorphic conjugates and we will refer to a choice of 0o only when necessary. Clearly if V has a field of moduli and is defined over this field then V has non-isomorphic conjugates. We adopt the convention that if tr.deg k = 0 then all embeddings agree on k0.

The principal result proved here, Theorem (3.2.5), states that if V is a polarized projective variety defined over k, a finitely generated extension of Q, where V is assumed to have non-isomorphic conjugates, and if, in addition, a further condition on the stratification of families in which V lies is satisfied, as explained below, then the topological type of Vv(C) is independent of ~. The proof of this Theorem relies on a strengthened form of Thom's stratified Isotopy Theorem due to Verdier which is described in Sect. 2 below. The Theorem itself is proved in Sect, 3.

It was previously known that certain types of varieties whose topology is rather easily described, such as non-singular curves, abelian varieties, K - 3

Topology of conjugate varieties 289

surfaces, and simply connected non-singular surfaces of general type, have topological types which do not vary under conjugation.

In Sect. 4 we show that complete intersections in projective space, and, more generally in homogeneous varieties, have non-isomorphic conjugates, hence they will also belong on this list if the further stratification condition of the Theorem can be shown to hold as well. This is straightforward in the case of smooth varieties, and we recover the known fact that conjugate smooth complete intersections are homeomorphic and extend this to smooth complete intersections in homogeneous varieties. The result can be further extended to certain restricted classes of singular varieties as well but we are unable to verify the stratification condition in the case of arbitrary complete intersections.

In Sect. 5 we point out a few of the many questions that remain open in this area.

2. Stratified isotopy theorems and the topology of conjugate varieties

2.1. Whitney stratifications t~ la Verdier

The most natural setting for the study of stratifications of singular spaces is the category of real analytic subspaces of smooth (real analytic) manifolds and proper maps between them. The best current reference is [G-M 88].

We wish to employ this theory in the context of complex algebraic varieties and to take our stratifications to be constructible sets whose fields of definition we can control. Following a suggestion of Bernstein, Beilinson, Deligne [BBD 81] Sect. 6.1.6, we find that such a version of stratified isotopy theory has been given by Verdier.

We give a notion of Whitney stratification which is adapted to algebraic varieties. The standard Whitney conditions a and b will be replaced by a single condition w which is also local in nature.

We use the following notion of distance between sub-vector spaces in a finite dimensional Euclidean space E, 6(F, G) defined by

6(F, G) := sup dist(x, G) x E F IIx I1-1

In particular, iS(F, G) = 0 =>, F C G.

(2.1.1) Definition. A Verdier-Whitney stratification of (the complex points o39 a complex algebraic variety V where V is a k-variety of finite type, k a finitely generated extension of Q, is a finite disjoint partition of V by smooth constructible sets At

n

V = U A i i=l

such that

290 D. Reed

1. the "boundary property" holds, namely A-"i fq Aj r 0 implies Aii D Aj and 2. if A-'~, D Aj with i ~ j then the pair (Ai,Aj) satisfies the foUowing property

"w" at every point y EAf: Consider Ai and Aj as real analytic manifolds and take coordinate patches around y to some Euclidean space E, then there exists a neighborhood U C E of(the image of) y and a positive real number C such that Vx E U NAi and y~ E U f3Aj (here Ai und Aj are taken to mean the images of some small open subsets around y in E) we have

t~(Ty,Aj, Txai) < C II x - y ' II

where TxAi is the tangent plane to Ai at x and ~ is as defined above.

A number of remarks arc called for here. Since we are restricting ourselves to algebraic varieties, it is sufficient to

consider stratifications with finite collections of subsets whereas consideration of analytic varieties would require infinite collections of subsets. This permits a certain amount of simplification in the definition and the subsequent arguments. It also permits us to speak of the (common) field of definition of the stratification as being the smallest field containing the fields of definition of the Ai.

On the other hand the condition w (so-called by Verdier) replaces the more familiar conditions 'a' and 'b'. Condition w implies condition a simply because it is a uniform version of it, hut w does not imply b in general (consider the logarithmic spiral at 0). The key fact however, is that this implication does hold when Al is a smooth subanalytic subspace of a real analytic space and Aj is a smooth analytic subspace of A//(Kuo, [Ku 70]).

Verification of property w does not depend on the choice of coordinates (for this it is important that the strata Ai are rex]uire~ to be smooth). The next key fact is that using resolution of singularities, we can stratify arbitrary complex algebraic varieties in much the same way as real analytic manifolds. Recall to begin with that the singular locus of a variety X over a field k of characteristic 0 is defined over k and is nowhere dense, hence its inverse image under monoidal transforms is nowhere dense. We can summarize the portion of Hironaka's resolution of singularities that we require by saying that, starting with an arbitrary variety X, defined over a field k of characteristic O, resolution of singularities produces a smooth variety X t and a proper morphism f : X' ~ X such that the inverse image of the singular locus of X becomes a divisor with normal crossings D in X' and X', D and f : X -* X are defined over k as well.

The first Theorem we will require is,

(2~1,2) Theorem. If V is a complex algebraic variety as above and V~ a finite family of constructible subsets of V then there is a Verdier-Whitney stratification of V such that each V~ is obtained as the union of stratc~ The stratification is defined over the (common) field of definition of V and the V~.

The proof of this is based on


(2.1.3) Theorem. V as above, M, M' smooth, connected, locally closed subsets such that M N M' = O, M' C "M and -M - M' is closed (all for the Zariski topology), then there is a Zariski open Y C M' contained in the locus of points y E M' such that (M, M') has the property w at y, and M' - Y is Zariski closed. Y is defined over the (common) field of definition of V, M and M'.

Proof of(2.1.3). In what follows we review the principal steps in the construction of V is to indicate that the methods involved do not require extension of the field of definiton. Details can be found in [Ve 76].

We use resolution of singularities to find a smooth complex algebraic variety W and a proper morphism lr : W ~ V with 7r(W) = M" and I r - l (M ') = D C W is a divisor whose singularities are at worst normal crossings. But resolution of singularities takes place without extension of the field of definition so that both W and D are defined over the same field as V. The next step is to produce the desired open set by removing "bad points", in this case, the normal crossings singularities.

Consider the subset Dq C D of points of D where at least q irreducible local components of D meet (this set is empty for q >> 0) let/gq be the normalization of Dq (separating the points lying on the various components) and let i : l)q ~ Dq be the normalization map./)q is a smooth algebraic variety defined over the same field as D. We consider d(Tr o iq) and ask for the locus of points where this map is not surjective so that we do not have a submersion. These form a Zariski closed subset Rq C Dq [EGA IV, Sect. 17.15.13] defined by a Jacobian condition and hence this singular locus is defined over the same field as Dq. iq(Rq) is closed since iq is finite and hence a closed map (the "going-down" Theorem) and the collection of iq(Rq) is finite so that S := 7r(Uq iq(Rq)) C "M" is Zariski closed. We thus have that

Y : = M ' N ( M - S ) c M -

is a Zariski open subset of M defined without extension of the base field such that Y is dense in M z and it is smooth by construction. The proof of (2.1.3) now proceeds by considering the local analysis of the smooth real analytic varieties underlying M, M ' and Y, see [Ve 76], Sect. 2. 13

Proof. of Theorem (2.1.2). The remainder of the proof is an induction on dimension. An important ingredient is the following Lemma which we state without proof.

(2.1.4) Lemma. Let X be an algebraic variety, Y~ a finite family of constructible subsets of X, then there is another finite family of subsets of X, Ba satisfying:

1. for all a, -Ba and -Ba - Ba are Zariski closed and the Ba are smooth and k-irreducible;

2. the Ba partition X and each Y~ is the union of a collection of Ba;

292 D. Reed

and the Ba are defined and irreducible over the (common) field of definition of the Ya.

Note that it is not the case that the Ba can be taken to be connected nor is it the case that the (geometric) connected components of the B~ are defined over k.

2.2. Stratified morphisms

The next ingredient is the demonstration that quite general algebraic-geometric morphisms behave well with respect to stratifications.

Recall that a morphism of stratified spaces f : X -~ Y is called a stratified morphism if it is proper and if the inverse image of a stratum of Y under f is a union of strata of X and each component of these strata is mapped submersively (as a real analytic manifold) to Y.

We now have a version of the 2 na isotopy theorem for complex algebraic varieties,

(2.2.5) Theorem. (Verdier) l f f : X -+ Y is a morphism of complex algebraic varieties and f is proper then there are Verdier-Whimey stratifications S and T of X and Y, defined over the (common)field of definition of X, Y and f , such that f is transverse to S and T.

Proof. We first show that if X ~ Y is proper and S a stratification of X such as given in Theorem (2.1.2), then there is a Zariski open U c f ( X ) C Y which is smooth in Y and (Zariski) dense i n f ( x ) such tha t f [ f - l ( U ) ~ U takes the connected components of S n f - l ( u ) submersively to U.

Set

Xq= U so dim S~ <_ q

this is a closed, smooth, constructible subset of X and we write fq for f ] Xq : Xq -~ Y.

Once again, let Rq be the set of points in Xq where dfq is not onto and set

= r - Yr g a ( U q'<_q

where Yreg is the set of regular points of Y (as noted above this set is defined without extension of the field of definition in characteristic 0). The proof now proceeds by induction on q to show that

fq I Uq "fql(Vq)---+ Uq

maps the connected components o fS nfql(Uq) tofq(Uq) where we always have Uq Zafiski dense in Y. Note that Y - Uq is automatically a Zariski closed subset since the fq, (Rq,) are.


So let S~ C Xq be a stratum. If dim So, < q then there is a qr < q such that fq, maps the connected components of S nf~;l(Uq,) to Uq,. Thus we can assume dimSa = q so S,~ C Xq is Zariski open and smooth and is a connected component of W = Xq - Xq-l which is also open and smooth.

Now R e N f - I ( U q _ l ) C W since Vq' < q

coker (dfq,) ~ coker (dfq)

is surjective over points of Xq,, as fq agrees with fq, on Xq,. fq(Rq)n Uq-i is a Zariski closed subset and Uq-i - (fq(Rq) N Uq-l) is dense in Uq-i and fq I Uq takes S M f~- 1 (Uq) submersively to fq (Uq).

fq(Rq) is empty for q >> 0 and we can form U := Nq Uq. This is dense in Y with Y - U an algebraic Zariski closed set and, by construction, f takes the connected components of S N f - l ( U ) submersively to U.

It is now relatively straightforward to construct the full stratification using Theorem (2.1.2). []

2.3. Stratified isotopy

Lastly we state a version of the 1 st isotopy Theorem:

(2.3.6) Theorem. (Thorn) X, Y are real analytic spaces, S and T stratifications, f : X ~ Y proper and submersive on the connected components of the strata of X. Set Yo E Y. Write Xo = f- l (yo) , So = Xo fq S. Then there is an open neighborhood (in the complex topology) Yo E V C Y and a homeomorphism ~b : ( f - l ( V ) , S M f - I ( V ) ) --* (X0 x V, So x V) preserving the stratifications and compatible with projections to V.

Proof. Classically this is proved using techniques from differential topology and is quite difficult. Using the condition w in place of the more standard a and b conditions Verdict is able to give a fairly self-contained proof in a few pages [Ve 76], Sect. 4. Nonetheless we will pass this over in silence since our objective is not to see how the result is proved but rather to show how it can be applied to give useful transversality properties for the algebraically defined stratifications described above. See [Ve 76] for the missing details. D

When combined with Verdier's results this gives:

(2.3.7) Corol lary. Let X f-~ Y be a proper morphism of algebraic varieties, then there are Verdier-Whitney stratifications S and T of X and Y respectively such that the topological type of the fibers o f f over a connected component of a stratum of T is constant.

Proof. By Theorem (2.2.5) there are stratifications S and T of X and Y respectively, defined over the common fields of definition of X, Y and f , such that

2 ~ D . R ~ d

the inverse image of any stratum of T is a union of strata of S, f - l ( T ~ ) = US a, and each connected component of a stratum of S is mapped submersively onto a stratum of T. Thus only the last statement requires discussion. Consider a connected component of a stratum of T, and call it W. Partition W into subsets such that the topological type of the fibers o f f are constant on each member of the partition. The sets partitioning W are then open by Theorem (2.3.6) and they are disjoint by construction. Since W is connected only one of the sets in the partidon is non-empty. []

We note again that although the stratifications S and T are defined without extending the field of definition this is not necessarily the case for the connected components of the strata.

3. Principal results

3.1. A sufficient condition for topological stability under conjugation

The results described above can be applied to give a criterion for an algebraic variety and its conjugates to have the same topological type. We consider fields k which have the following property: given a choice of k0 and ~0 as in Definition (1.0.3), then the set of distinct embeddings k --* C has at least 3 members.

(3.1.1) Theorem. Let V be a k-variety, k a finitely generated extension of Q which has the above property and suppose there exists a family f : Y ~ B, that is, a proper morphism of complex algebraic varieties such that:

1. all of the conjugate complex algebraic varieties V~. are isomorphic as C- varieties to fibers o f f (in other words for each a : k ~ C there is a point b~ E B and C-isomorphism Vo ~_f-l(b~)); and

2. f : Y ~ B arises by base extension from Q, that is, there are Q-varieties Y/~ and B/Q and a morphism f/Q : Y/Q ~ B/Q such that

Y -_ v/Q • C , v:Q

B "" B/Q x C ~ B/Q ~B

commutes,

3. the b~ form a Q-irreducible algebraic set in B, and 4. the Verdict-Whitney stratum containing the bu is (geometrically) connected;


then the topological type of V~(C) is independent of a.

Proof. By Corollary (2.3.7) we can stratify Y and B with stratifications S and T defined over (~ so that the map f is topologically locally trivial over each connected component of the strata. We need only show therefore that the points b,~, be 2 corresponding to conjugate varieties V~ 1, V~ 2 must lie in a single connected component of a stratum of S.

We first comment on the scheme structure associated to the b~,. If we fix a set of o'i agreeing with a given 4~0 on ko (recall Definition (1.0.3)) then the tri are a closed subset of points in B and we give this closed set the induced reduced subscheme structure. This maps under ~ to a subscheme of B/r - in fact a single closed point.

To obtain the other embeddings of k we note that k is generated over k0 by an algebraic number a and hence we can compose the o" i with those 71" E Aut(C) which move k0 but fix ,~. Thus from any si we get an open dense subset of B. This set maps to a genetic point of B/Q under ~ so we give it the structure of an extension of scalars of a generic point of B/Q via an embedding of ko in C.

In general the full set of b~ is a product of these two schemes. We will frequently focus attention on a finite set of embeddings tri agreeing with a given embedding of ko. Note that if we fix a ~ri and consider the set of embeddings obtained by varying the embedding of ko, then the image of this in B/O always satisfies Condition 3) above.

So fix an embedding ~o of ko and assume that ,7t and or2 lie in the finite set o f 0r i which agree with ~50. The b~r, are mapped to a Q-irreducible subscheme b/o of B/Q. We let b/Q be defined by the ideal ~ .

Take the inverse image of b/Q under ~B in B ~- B/Q x C; call it b c. It is defined by the ideal ~ | C (which in general is not prime). The points bat and b,, 2 lie in bc by commutativity of the diagram in (3.1.1).

By hypothesis, the stratification T of B is defined over Q, and, since the strata containing the b,, are assumed to be (geometrically) connected, the connected components of these strata are thus also defined over Q. We claim that the points b~ and b,~ 2 cannot be separated by the connected components of a stratum of T.

Assume otherwise. By Verdier's results b/c is a finite union of strata t.JTi which are algebraic sets defined over Q. By induction on the number of strata we may assume TI U/'2 is all of b/c. By the definition of a stratification, if ~lfq T2 ~ 0 then ~ D T2. But both/'1 n {ba} and T2 n {bo} map to the genetic point of b/o so this is impossible. Hence we must have T-~lN T2 = 0 and similarly ~2 f3/'1 = 0. But then the set of the b,r would not be Q-irreducible, contrary to assumption.

To conclude, note that since the collection of bo obtained from a given b~ under the action of those elements in Aut(C) which fix a given generator, a , of k over ko, is an open dense set in B which is defined over Q, and no proper subset of these b~ is Q-rational. By an argument as above this cannot be separated by connected components of strata. Since the full collection of bo is the product of the bo, by such an open dense set, we are done. []

296 D. Reed

(3.1.2) Remarks

- Note that condition 4) can be verified by checking whether the connected components of the strata each contain a rational point. Since the morphism Y ---, S arises from base change over Q we have a stratification T of S defined over Q as well. Let Tl be a stratum containing b~ and let tl be the connected component of Tl containing b~,. Assume that h has a rational point, call it x. The other connected components t i of T1 are then Q-conjugate to tl (Tl is Q-irreducible), that is, ti = r for some r E Aut(C). But r = x hence r r ~ so r = h since they are the connected components of Tl. As this holds for any r E Aut(C) we have that the field of definition of h (and hence of all the ti) is Q. But then the stratum must be geometrically connected since it is defined over Q, is Q-irreducible, and has its (geometric) connected components defined over Q as well.

- If the field k has only 2 distinct embeddings which agree with a choice of r on /co then one of these embeddings can be chosen to be complex conjugation. Since complex conjugation is the only continuous automorphism of C it induces a homeomorphism of topological spaces. Hence the restriction of ths field k in the Theorem does not affect the generality of the result.

3.2. Corollaries

Following [BBD 81] Sect. 6.1.6, it is easy to see that any complex projective variety V (say defined over a field k) can be embedded in a family defined over Q. One merely considers the coefficients caa of the homogeneous ideal defining V as indeterminates. This produces a "family", that is, a projective morphism of C-schemes f : Y ---r S over an affine base S, with the original variety V isomorphic to the fiber o f f over the point of S corresponding to the ca~. This family clearly contains all of the conjugates of V, since these are obtained by conjugating the coefficients in its homogeneous ideal. Furthermore, since S is affine this family is actually "arises via base extension from Q" in the sense of Theorem (3.1.1). We now explore geometric properties relating to Condition 3) in the theorem.

(3.2.3) Lemma. Let V be a projective k variety and fix a r : ko --* C with ko as above. Suppose V has non-isomorphic conjugates and let si be the set of points in S parametrizing the conjugates Vai for ai : k --* C which agree with a given r on ko. Suppose further that "r E Aut(C/ko). Then i f7 acts non-trivially on k, there is no linear isomorphism "y : S -* S, acting solely by permutation of coordinates, such that 7(si) = "r(si) for all i.

Proof. Assume for a contradiction that there is such a 7, then it induces a map on the whole Y --* S construction


Y ~ Y

s ~ s

and, when restricted to the si we get A/(Si) = 7 - ( S i ) = Sj for some j , j r i. ,~ Write the finite collection of points si as B C S. It is a closed subset of S and

h-~(B) C Y, its inverse image under h is closed in Y. Take the induced, reduced subscheme structures on these two closed subsets. The projections/3 restrict to these subschemes which thus arise by base extension from schemes over Q, call them B/Q and h - 1 (B/~).

If there exists a 7 as above then it restricts to a map B ~ B since 7- fixes the collection of si. Then we have two diagrams, the first

h- I (B) ~ h - l ( B )

B ~ B

which is Cartesian by construction, and

Z := h- l (B) • B .E} h - l ( B )

B "v B

Since 7 is an isomorphism, the top arrow in this second diagram is an isomorphism as well. We wish to compare these two diagrams. Note that since we require that/~ and 7- agree on a complete set of conjugates, we would like to consider them as acting on the image of B under/3. A priori however, r is not a C-morphism and hence is not invariant under conjugation by elements of Aut(C). Note however, requiring that 7- agree with 7 acting by permutation of coordinates when restricted to B, implies that, in this specific case, the action of 7- on B is invariant under conjugation by elements of Aut(C) and thus is actually defined over Q (in the statement of the Proposition we could have replaced the condition on q' by the condition that it be defined over Q). When we extend scalars back to C we recover 7- and thus exhibit 7- as a C-morphism when restricted to B.

In the above Cartesian diagrams we then have that Z and h- l (B ) are (canonically) isomorphic and hence there is an algebraic geometric isomorphism h- l (B ) "~ h- l (B) . We thus get

h - l ( s ~ ) = V~ ~_ (V~,)" = h - l ( ( s , , ) ~')

contradicting the assumption that V is of weak moduli type. 13

Failure to have non-isomorphic conjugates and the image of the points parametrizing the V~ under/3 are related as follows.

(3.2.4) Proposition. Let V be a projective k-variety and let ~ : k --~ C. Then if V has non-isomorphic conjugates, the image under/3 in S/~ of the points s~

298 D. Reed

parametrizing the conjugate varieties Va, for all embeddings cr : k --+ C which agree with 0 on ko, satisfies condition 3) of Theorem (3.1.1).

Proof. We assume for a contradiction that there is a non-trivial partition Si of the si into disjoint algebraic subsets defined over Q. Write sij for the i th coordinate associated to the jth embedding. Let �9 be an elementary symmetric polynomial in n = d i m s variables. �9 is defined over Q and hence ~((si) ~') = (~(si)) ~. The partition Si of the si induces a partition of the values of ~(si), call it ~i and since the collection of Si is stable under all Aut(C), the same must hold for the ~/i i. That is, we have a set of at least three conjugate elements of C which is partitioned into algebraic subsets and the partition is stable under all ~- E Aut(C). Since the values of the coordinates of the si generate k this can only happen if the partition is trivial. This in turn says that the values of �9 on each Si are a complete set of conjugates and hence �9 takes the same values on all of the Si, thus for some -r E Aut(C) we have

�9 ( s l j , . . . , s , j ) ~- = ~(s l ,k , . . . , s~,k)

while (sij) r ~ si,k so this can only happen if r permutes the sit ] . That is si, k =

sp(i)j for some permutation of n letters, p. On the other hand, such a permutation commutes with the action of all elements of Aut(C) and hence the same p gives the action of ~- on all the Sy. Thus we have a linear map 7 : S ~ S, acting solely by permutation of coordinates, which is such that 7 ( s l j , . . . , s , j ) = 1-(sl j , . . . s~j) for all i and a contradiction. D

We can draw the topological consequences as follows.

(3.2.5) Theorem. Let V be a projective variety defined k, a finitely generated extension of Q, which has non-isomorphic conjugates. Let cr : k --~ C be an embedding and let V~ be the associated extension of scalars. Define the morphism f : Y ---, S as above, by letting the coefficients {caB} of the homogeneous ideal of V~ vary in C. Assume that we have stratified f : F ---* S ~ la Verdier and assume that the stratum containing the point s~ E S parametrizing V~ is (geometrically) connected. Then all of the conjugates of V are isomorphic to fibers f - l ( s o ) where the s~ lie in a single connected component of a Verdier-Whitney stratification of S and hence have the same topological type.

Proof. The morphism Y -~ S is proper and that each of the conjugates of V can be identified with a f iberf- l(s~,) for some s~ E S. We consider embeddings ai which agree with a on k0. Then the image of the b~,~ satisfies condition 3) of Theorem (3,1.1) by virtue of Proposition (3.2.4).

Now if we choose a a t which does not agree with a on ko then, as we have noted previously, we can find an element of Aut(C) which takes tT(ko) to at(ko) but leaves a generator a , k over ko, fixed. Thus the collection of b~ for all embeddings also satisfies the conditions of the Theorem. D


The examples of varieties with non-homeomorphic conjugates produced by Serre and Abelson do not have non-isomorphic conjugates. We consider Serre's examples which are constructed from the product of an abelian variety - itself a product of elliptic curves with real j-invariant and a ~ diagonal hypersurface quotiented by the action of a cyclic group (this action varies with the chosen embedding of the field of definition of the abelian variety). It is then not difficult to see that the variety and its complex conjugate are isomorphic while the field of definition of the variety is not a real field.

4. Complete intersection type varieties

4.1. Generalities

We can use Theorem (3.1.1) to exhibit further classes of varieties whose topological type is invariant under conjugation. These classes of varieties may be parametrized by vector spaces of sections of vector bundles and this linear structure provides a natural method of descending from C to Q.

The Deformation Theory of these varieties has been studied by a number of authors including [K-S 58], [S 75], [B 83] and [W 84] in a series of papers with results extending from smooth hypersurfaces through to the more general cases. The most general result along these lines is:

(4.1.1) Theorem. (Wehler) Let Z = G / H be a non-singular homogeneous complex variety, quotient of a simple, simply connected Lie group G by a parabolic subgroup H, E = @}=l~2(dj) a vector bundle, s E H~ a section and X a complex variety described by the zero locus of s such that codimX = r and X is not a K-3 surface. Then the vector space H ~ parametrizes a complete set of small deformations of X and these deformations are given by the family

Y := {(z,s) ]z E Z , s E H~ = 0} ~ H~

If we are to apply the Theorem to this case we must show that the family Y ~ H~ E) satisfies the conditions of Theorem (3.1.1). To do this we reprove the theorem using algebraic geometric techniques to obtain a family over Q which will have the required properties. In particular, we show that the deformation theory of these varieties can be 'descended' to their field of definition. The proofs given here therefore are modelled on Wehler's but adapted to algebraic rather than analytic geometry. The key in both the analytic and algebraic approach is to establish a close connection between the deformation theory of the objects and their Hilbert schemes as will be explained shortly. For another approach to the algebraic deformation theory of complete intersections see [Ma 68].

300 D. Reed

4.2. Comparison of Hilbert scheme and deformation functors

Let Z be an arbitrary non-singular projective variety over a field k and E ---* Z an algebraic k-vector bundle over Z. Consider the scheme X C Z defined by a global section so E H~ E). We construct this scheme as follows: the section s defines a map of sheaves Cz --* E" sending the section 1 of ~z to the section s (we are abusing notation here by not distinguishing between the vector bundle and its associated locally free sheaf). There is a dual map g : ~; --) ~z and X is said to be defined by s if ~ fits into an exact sequence of sheaves

X is sometimes referred to as the zero scheme of s.

(4.2.2) Definition. The Hilbert functor of X (in Z), is the functor Hilb from ~ , the category of local artin rings with residue field k to the category Sets which assigns to each object A in ~ the set of schemes Y which fit into the following diagram

Z D X ~ _ Y x k A

Spec k

�9 Y C Z xSpecA

1 , SpecA

with Y flat over Spec A.

(4.2.3) Definition. The (affine) projective cone (hereinafter simply "the cone") Cx of (or on) a projective variety X C IP n is given by

Cx := Spec(Gx �9 Gx(1) �9 ~x(2) e . . . )

The vertex p of Cx is the (closed) subscheme defined by the augmentation ideal, ker e, where

: (~x �9 ~ ( 1 ) r ~ ( 2 ) e . . ) --,

Spec(e) thus defines a map X --* Cx whose image is p.

For future use we recall that the cone with vertex removed (6point6)

Cx -p ~ ~7(~2(-I)) "' Spec(@~x(n))

where V denotes the operation of taking the vector bundle associated to a locally free sheaf. There is an action of G,n on Cx (and on Cx - p ) with integral weights coming from the action on each tensor power ~'(n). For details see [EGA II Sect. 8.4 - 8.6].


Next we define a deformation functor Defcx as the functor which assigns to each object A in ~ the set of deformations ~'Cx,p(A) of the k-algebra C'Cx,p which is the local ring of Cx at p.

There is a natural morphism h from the Hilbert functor to this deformation functor obtained by assigning to a scheme Y as above the local ring of the vertex of the projective cone on Y. This is naturally a deformation of the local ring of the vertex of the projective cone on X. The morphism thus consists of "forgetting the embedding in ~ ' or, in terms of the underlying rings, forgetting the gradings.

Comparison Theorems between Hilbert functors and Deformation functors go back (at least) to Schlessinger [Sch 71] and can be found in [P 74], [K 79] and [Wa 92]. "We will use a version due to Kleppe which employs Andr6-Quillen cohomology to avoid unnecessary smoothness conditions.

Since X is given as the zero-scheme of a section of a vector bundle we want to describe its Hilbert scheme in the same way. So define functors Fso from the category i~ to Sets which, for any given section So E H~ assign to an object A of ~ the set of zero schemes in Z x SpecA of sections sA E H~ x SpecA, E x SpecA) which reduce to So over the closed point k under the map Spec k ~ Spec A.

If we further assume that codimX=rankE we have that X is (at least) a local complete intersection. Moreover, such zero schemes sA are fiat over SpecA. By [EGA IV Sect. 19.3.8] the zero scheme defined by any of the SA is a local complete intersection as well. Thus we obtain elements of Hilbx(A) as sections

of vector bundles E x SpecA and a morphism of functors Fso / Hilbx. We now study this morphism of functors.

(4.2.4) Proposition. Let Z, E and A be as above and let X be the zero set of a section so with codim X=rankE, write ~ C ~ for the ideal sheafofX and suppose that

HI(Z,E | = O

then the above morphism of functors f : F --~ Hilb is surjective on tangent spaces.

Proof The tangent space to H at ho, is Hilbso(kH) and it is standard that

T ( H , ho ) = H o m ~ (,.~: , CX ) = Horn ~ (,~ /.~x 2 , r )

(for the first isomorphism see [G 61], for the second see [H 77] Ch. II, Sect. 8). We also have

Hom~ (~/: / ~ 2, :--:x) ~- H~ ,Nx /z) ~- H~ ,E ]x)

where N is isomorphic to the tangent space to Hilb. But the tangent space to F := F(k[e]) is the vector space of sections of E x Spec(k[e]) ~ Z x Spec(k[e]) which reduce to so over k and this is just H~ E). Consider the standard short exact sequence

302 D. Reed

tensor it with E and take cohomology to get

0 --~ H ~ | --* H ~ --~ H ~

---} H l ( Z , E | o~7x ) ---~ . . .

Hence if H I(Z, E | ~x) = 0 we have the desired surjectivity. []

So far we have developed portions of a triangle of functors

Fs0

Hilbso h

To fill in the morphism marked with "?" note that the local ring of the vertex of the cone of the zero scheme of s,a is naturally a deformation of the local ring of the vertex of the cone of the zero scheme of so. It is clear that the triangle commutes because y is just the composition of f and h. We now invoke the comparison Theorem relating the Hilbert functor and the deformation functor

(4.2.5) Theorem. Suppose that V is projectively normal and that Tt (V ),the tangent space to Defer, is negatively graded (in a sense to be made precise below), then the natural morphism o f functors

h : Hilbv ~ Defer

is smooth.

Proof. See [K 79] []

As we will see in the case of the zero schemes of sections of certain vector bundles, this result together with Proposition (4.2.4) will enable us to establish the existence of families of the desired type. The codimX=rankE condition that we have been placing on our zero schemes of sections of E ensures projective normality (assuming that the varieties are normal in the first place) by a straightforward adaptation of the proof for complete intersections in projective space. It is the grading condition in the comparison theorem which is the more difficult of the two to verify and we will reduce it to a condition on vanishing of cohomology.

Recall that Defcx is the deformation functor of the vertex on the projective cone Cx over X and that there is an action by Gin(k) on Cx with weights ranging over the integers. TI(X) := Defcx(k[e]) is a vector space and the action of G,n on on Cx becomes an action on TI(X) , so that we get a decomposition TI (X) = m ~ T1r and TI(X) becomes a graded vector space (as p is an


isolated singularity in Cx, in fact T1(~,) = 0 for v >> 0 and for ~, << 0 but we will not need this). We say that T1(X) is negatively graded if Tt(X)(t,) = 0 for

> 0. To demonstrate that this condition obtains we show that Hilb(k[e]) too has a grading, which is negative in this sense and that the map Th is surjective and respects both gradings.

(4.2.6) Proposition. Let X be the zero scheme of the section So of E --* Z and assume that X projectively normal (e.g. codim X = rank E) and dimX > 2. Suppose the conclusion of Proposition (4.2.4) holds, that is, the morphism of functors Fso --~ Hilbx is surjective on tangent spaces. Let 9 : Fso --* Defcx be the natural morphism taking an element of Fso (A) to an element of Defc~ (A). Denote

the map on tangent spaces by H~ E) ~ TI(x) . If this map is surjective then T 1 (X) is negatively graded and the morphism offunctors h is smooth.

Proof. We have a triangle of tangent maps

H~ E)

Hilb(k [el) Th , DefCx(k[e])

which commutes because the triangle of underlying maps does and we are assuming that Tf is onto. If we now further assume that H~ --~ T I ( x ) is onto then Hilb(k[e]) ~ Tt (x ) must also be onto as well.

Hilb(k[e]) - H~ as noted above and H~ ~_ H~ where Cx is the projective cone over X. By projective normality the singularity at the vertex p of Cx has depth=2 so global sections of Cx - p extend to global sections of Cx. Since X is a local complete intersection, this is true of all coherent sheaves on Cx by local duality, and so, in particular

H~ Ncx ) ~- HO(Cx - p, Ncx ) ~- H~ - p, Ncx-p)

But Cx - p ~- V ( ~ ( - 1)) - zerosection

so there is a natural affine map 7r : Cx - p -* X and Ncc-p ~- 7r*Nx and we can compute

H~ - p ,Ncx-e) ~- H~ - p, Tr*Nx) ~- H~ lr.rc*Nx)

~_ @~176

where Nx(v) := Nx | O(u) as usual. The second isomorphism comes from the fact that 7r is affine and the conclusion that H~ = 0 for v > 0 comes from the fact that the sections must extend over all of Cx and hence cannot have any poles at p.

304 D. Reed

The grading thus produced on Hilb(k[r arises from the action of Gm on Cx and the morphism Hilb ---, Def is contructed precisely by passing to the projective cone to go from the varieties in projective space to abstract varieties. Hence the action of Gm on T1(X) and Hilb(k[r is compatible with this morphism. The result is that TI(X) must also be negatively graded and hence we can apply the comparison Theorem to get that this morphism is smooth. []

Finally it is not difficult to reduce the requirement "H~ ~ TI(X) surjective" to a statement of vanishing of cohomology.

The tangent sheaf to a variety V is defined by Ov := Hom~, (Y2v, ~ , ) , where I2v is the sheaf of differentials of V. Since Cx C A n+l and Cx - p is smooth we have that

o --, Oc,, Ic,,-p~ OA-, Ic,,-r,-' ./t6,, I cx - :~ 0

is exact. Note that C - p is not affine so there is a longer exact cohomology sequence

0 ~ H~ -p,Ocx lCx-p) '-+ H~ -P, OA,+~lcx_~,) -">

H~ Cx - P , ./l/'ccx tcx-e ) "-' H1( Cx - p, Ocx Icx-p) -'~ H l ( Cx - P , OA,"lcx_ , ) ---'

The theory of the "cotangent complex" [Li-Sch 67] gives us the following exact sequence

0 ~ H~ .--* n~ [cx) ---* n~

--, TI(X) --~ Hl(Cx, 0r x) --, . . .

Once again since X is projectively normal, sections over Cx - p extend to Cx so that

H~ Cx , Ocx ) "~ H~ Cx - P, Ocx [cx-p)

H~ ( Cx , eA,*, Ic,) --~ H~ Cx - P , O A,** ]c~-v )

and

H~ "" H~ - P,./~Cx-p) "" H~ ~- H~

Finally Hl(X, Oz/x) ~- Hl(Cx, Ox)

Putting all of this together we see that if

HI(X, Oz lx)=O

then

H~ Ix--' TI(X) ~ 0

and since


is onto, we are done. To summarize,

H~ E) ~ H~ E Ix)

(4.2.7) Proposition. For X the zero scheme of a section s o f a bundle E over a

non-singular projective variety Z satisfying

1. codim X = rank E; 2. dimX > 2; 3. H1(Z,E | and 4. X is projectively normal

then we have the commutative triangle

where all of the arrows are smooth.

Proof. We only need to show that Fso f Hilbx is smooth since then the third side of the triangle will be smooth by [Sch 67]. We know that this arrow is surjective on tangent spaces hence we only need to show that Fso is less obstructed than

Hilbx. In other words, if B ~ A is a surjection in ~ with ker(qS) 2 = 0 and (mB)ker((k) = 0 (so ker($) is a k-vector space) and if for ~o E Hilbx(A) which is in the image o f f so, ~o =f(~o), fro E Fso(A), there is a ~ E Hilbx(B) such that Hilbx(~b)(~) = ~o then there is a ~ E Fso(B) such that Fso(~b)(~) = ~o. But Fs o is clearly unobstructed. []

Since we know that the formal scheme prorepresenting Hilbx is algebraizable [G 61], we now have the same conclusion for Defcx. All infinitesimal deformations of X come from small deformations and all small deformations lie in Z. Indeed, by versality, H~ E) is a complete deformation space and the family

Y := {(z ,s) l z E Z , s E H~ =0} c Z x H ~ 1 7 6

is a universal family both in the sense of the deformation functor and the Hilbert functor. These results are valid for X defined over any fieM.

306 D. Reed

4.3. Application of the theory

We now apply this theory to our problem. I f a normal k-variety V (k a finitely generated extension of Q) is given as the zero scheme of a section s E H~ E) (everything defined over k) and if we have

- codimV=rankE, dimV > 2; - HI(Z ,E| and - V projectively normal

then we know that the k-vector space H~ E) parametrizes a complete family of deformations of V. There is a map

H~ E) --~ H~ E)/~ | k

giving a Q structure to H~ E). This induces a morphism of functors F~ ~ F~. We observe that this morphism is smooth since first, i rA' ~ A is a surjection

in ~ then

Fk(A ') --~ Fk(A) XF~A) FQ(A ')

must also be onto. To see this note that the arrow/5 in

~t Fk(A) • FQ(A ') --+ F@(A ')

Fk(A) FQ(A)

is onto and hence/~' is also and that Fk(A ') --o F~(A ') is onto as well. Secondly the morphism also induces a bijection on tangent spaces since H~ and H~ are vector spaces of the same dimension over fields of the same cardinality.

Thus the projection H~ ~ H~ induces a map of deformation

spaces between the functors Def k and Def ~ and Hilb~ and Hilbx ~. We have a universal family Y/@ --~ H~ and the above ensures that

we recover the corresponding universal family over k by extension of scalars

Y~ --, Y/Q

H~ ~ H~

We can also extend scalars to C to get a morphism of C-schemes in the obvious way. Thus we can now consider V, E, Z, and s as objects defined over the complex numbers it is clear that all of the conjugate varieties V~ can be obtained by conjugating the section s, that is, V~ is the zero scheme of so := a(s). Hence all of the conjugates V are in the family Y if V is. Thus the first two conditions of Theorem (3.1.I) are satisfied.

Condition 3) may also be relatively easily disposed of given the above de-

formation theoretic interpretation of the Y --~ S construction. It is clear that the


image of the points in H~ E) parametrizing a collection of conjugate varieties Vo i is irreducible in H~ E)/Q (it is a single point).

Since the above construction is entirely algebraic and does not depend on the choice of an embedding k --* C, the varieties V have non-isomorphic conjugates and hence, by Proposition (3.2.4), satisfy condition 3). '~ Condition 4) appears more difficult. If V is smooth then it is diffeomorphic to

its small deformations so that the stratum containing the conjugates will contain a small ball (in the complex topology) and hence a rational point. If V is singular however, this argument no longer applies in general and we cannot at present prove that the connected components of the stratification contain rational points. In certain specific cases the desired connectivity can be shown using known facts about complete intersection singularities, see [Di 92], Ch. 1 for a sampling, but it seems unlikely that a general result can be achieved this way.

It only remains to spell out some types of varieties which are defined as the zero sections of vector bundles satisfying our conditions.

If we assume that Z is homogeneous, that is

Z "~ G/H

where G is a simple, simply connected, split algebraic group over Q, and H a parabolic subgroup, then by algebraic versions of Bott's vanishing Theorems [Dm 76] we get that H I(Z, Oz) = 0 for all i > 0. If we further suppose that E ~ ~= t~2(d j ) with dj > 0 and X defined by a section s such that

- codim X=r - X is not a K-3 surface - d i m Z > 3 ; a n d - b z ( Z ) = 1

then calculations of Borcea [B 83] (or an alternative using using an algebraic version of the Kodaira-Nakano-Akizuki vanishing theorem) show that H 1 (Z, E | ~ ) = 0 .

To use these results we need only further specify that the section of E defining X be itself defined over k and that X be projectively normal.

We note finally that while the case of K-3 surfaces must be treated separately, the result is the same since all K-3 's are homeomorphic (indeed diffeomorphic) and conjugates of K-3 varieties are K-3.

5 . C o m m e n t s a n d o p e n q u e s t i o n s

The above discussion can be used to shed a bit of light on the nature of the Serre- Abelson examples. Both authors construct their varieties as quotients using finite group actions. In both cases the action is varied under conjugation. The difference lies in the part of the homotopy type which is affected by conjugation.

In Serre, the variety acted upon is a product of a diagonal hypersurface by an abelian variety and the group action on the abelian variety makes the 7rl of

308 D. Reed

the variety into a module in demonstrably different ways under conjugation. In Abelson's example the group acts on complete intersection which is constructed via a representation of the group and all of this varies under conjugation. Special choices of the group allow one to demonstrate variation in the Postnikov tower.

We have noted that neither the Serre nor the Abelson varieties have the non- isomorphic conjugates property and the author does not know of examples of non-homeomorphic conjugates which do have this property.

One area where such examples might be sought is among non-singular canonically embedded varieties with highly obstructed deformation theories. In these cases it is not difficult to see that these varieties have non-isomorphic conjugates, but we cannot argue that 'nearby' fibers of the Y --~ S morphism are diffeomorphic since we do not have a deformation theoretic interpretation of this morphism. This ties in with the following vague question:

- Is it possible to produce examples of non-homeomorphic conjugate varieties without using (finite) group actions?

This question can be made more specific in a number of ways. Because of the use of group actions the examples of Serre and Abelson are

rather rigid. A small deformation of one of their examples no longer maintains the structure required to compare it with its conjugates. One approach might therefore be to ask,

A1) Is it possible to construct examples of non-homeomorphic conjugate varieties which are stable under small deformations? Another way to cut out finite group actions may be to ask for simply-connected examples,

A2) Are there examples of simply connected non-homemorphic conjugate varieties? If it proved to be the case that arbitrary complete intersection varieties do not necessarily have homeomorphic conjugates we would automatically get (positive) answers to these questions.

Finally, along these lines one has the question, A4) Are the simply connected covering spaces of conjugate algebraic varieties

analytically isomorphic? The criterion developed in this paper does not provide an indication of the

minimum "necessary" conditions under which the topology of varieties remains stable under conjugation. Neither does it give any indication of the "part" (if any) of the topological type which are conjugation invariant (over and above the ~tale homotopy type which is clearly invariant). A Theorem of Deligne [De 87] shows that the nilpotent completion of the fundamental group of an algebraic variety is algebraically determined. One is led to ask,

B) Is the entire rational homotopy type a conjugation invariant? One might also pose the following question which seems to lie somewhere between the above

C) Is simple connectivity a conjugation invariant?


Acknowledgements. The results reported on here are contained in my Oxford Thesis directed by Professor A. J. Macintyre. I must thank him for having pointed out this question in the first place and for help and support every step of the way. Thanks are due also to Professors P. Deligne, B. Totaro and A.J. Scholl for having pointed out errors in earlier versions.

References

lab 741

[B 83]

[BBD 81] [B-C-R 87] [De 87]

[Dm 76] [Di 92] [G 611

[G-M 88] [H 771 [Hi 64] [i 71] [K 79] [K-S 58]

[Ko 72] [Ku 70]

[Li-Sch 67]

[Ma 68]

[P 74] [Sch 67] [Sch 71] [S 75]

[Se 64]

[Sh 68]

[Sh 72] [re 76]

[Wa 92]

[W 841

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