Grothendieck ring of varieties I. Eckart Viehweg, informal notes for a seminar talk, Essen 6.0.08 Based on [Nic] and on some of the references given there.. Basic definitions Definition.. Let S be a noetherian scheme. () The Grothendieck group K 0 (Var S ) is the abelian group generated by isomorphism classes [X/S] of separated S-schemes X of finite type, with the relation [X/S] = [(X \ Y )/S] + [Y/S] if Y X is a closed immersion. (2) A product is defined by the fibre product [X/S] [Y/S] = [X S Y/S], and the resulting ring is called the Grothendieck ring. (3) L S := [A S ]. Remarks.2. ) Here (X \ Y ) is the open subscheme with underlying space (X red \ Y red ) and with the scheme structure inherited by X. In particular, for Y = X red one gets [X/S] = [X red /S red ] = [X red /S]. 2) So for a field k and S = Spec(k) one can as well consider reduced, separated, not necessarily irreducible k-schemes (= k-varieties) and K 0 (Var S ) = K 0 (Var k ) = Z [ [X]; [X] an isomorphy class of k-varieties X ] where [ ] satisfies [X] = [U] + [Z] if Z X is closed and U = X \ Z. We write L k = L S or just L in this case. Example.3. Let C be a complex rational curve with a cusp, and let τ : P C be the normalization. Then [C] = [P ] = L + in K 0 (Var C ). Definition.4. Given a morphism π : T S of noetherian schemes, one has the base change map π : K 0 (Var S ) K 0 (Var T ). If π : T S is separated and of finite type one also has the forgetful map In particular this will allow to consider π : K 0 (Var T ) K 0 (Var S ). ι : K 0 (Var k ) K 0 (Var K ) with [X] [X k K] for field extensions k K (and ι : Spec(K) Spec(k)) for finite field extensions. ι : K 0 (Var K ) K 0 (Var k ) with [X/K] [X/k] Remarks.5. π is a ring homomorphism, whereas π is just a morphism of (additive) groups. General Principle. Several geometric data of k-varieties respect the relation. Needed:The data must be compatible with the relation. In particular this holds for cohomology theories if one takes the alternating sums over dimensions.
2 Examples.6. i. Euler Characteristic: k = C. Then χ top : K 0 (Var k ) Z with X i 0 ( ) i dim(h i c(x(c), Q)) is well defined. Moreover χ top (L) =. ii. Counting points: Let k be a finite field. Then # : K 0 (Var k ) Z with X #X(k) is well defined. iii. Hodge-Deligne: k = C. Then one defines the Hodge-Deligne polynomial HD(X; u, v) := k 0( ) k h p,q (H k c (X(C), C))u p v q, where h p,q (H k c (X(C), C)) denotes the dimension of the (p, q) part of Deligne s mixed Hodge structure. Then HD( ; u, v) defines a ring morphism Definition.7. HD : K 0 (Var C ) Z[u, v]. K (bl) 0 (Var k ) = Z [ [X] bl ; [X] bl an isomorphy class of smooth projectivek-varieties ], where [ ] bl satisfies the blow-up relations: [ ] bl = 0 If Y is a closed subvariety of X, smooth over k, if X X is the blow-up of X with center Y, and if E = X X Y is the exceptional divisor, then [X ] bl [E] bl = [X] bl [Y ] bl. Again the product is induced by fibre products over k (one has to check the compatibility with the blowup relation for [ ] bl ). Theorem.8 (Franziska Bittner/Heinloth). If char(k) = 0 the natural map K (bl) 0 (Var k ) K 0 (Var k ) with [X] bl [X] is an isomorphism. Idea of Proof. Obviously the map is well defined. Hironaka & Chow s Lemma & Stratification = surjective. For the injectivity, one uses two steps: First replace non-singular, projective by projective. Then one has to modify the equivalence relation to [X] [Y ] = [X ] [Y ] X \ Y = X \ Y. In particular the equivalence relation is generated by ones, induced by birational maps. Finally, if one adds non-singular, this makes only sense for Y non-singular. Here one has to use the: Theorem.9 (Weak Factorization (see [AKMW])). Let φ : X X 2 be a birational map between complete nonsingular algebraic varieties X and X 2 over an algebraically closed field k of characteristic zero, and let U X be an open set where φ is an isomorphism. Then φ can be factored into a sequence of blowing ups
and blowing downs with smooth irreducible centers disjoint from U, namely, there exists a sequence of birational maps between complete nonsingular algebraic varieties ϕ ϕ 2 X = V 0 V ϕ i V i ϕ i+ V i+ ϕ i+2 ϕl V l ϕ l V l = X 2 where () φ = ϕ l ϕ l ϕ 2 ϕ, (2) ϕ i are isomorphisms on U, and (3) either ϕ i : V i V i+ or ϕ i : V i+ V i is a morphism obtained by blowing up a smooth irreducible center disjoint from U. Furthermore, there is an index i 0 such that for all i i 0 the map V i X is a projective morphism, and for all i i 0 the map V i X 2 is a projective morphism. In particular, if X and X 2 are projective then all the V i are projective. Definition.0. () Two irreducible smooth projective k-varieties X and Y are stably birational, if X k P n k is birational to Y k P m k. (2) Let SB denote the set of equivalence classes for stably birational, and let Z[SB] be the free abelian group on SB. Consider the map from the free abelian group over all isomorphy classes of nonsingular projective varieties to Z[SB]. By the Theorem of Franziska it factors through K 0 (Var k ). Theorem. (Larsen-Lunts). If char(k) = 0 the map [X] bl stably birational equivalence class of X, defines an isomorphism Ψ SB : K 0 (Var k )/LK 0 (Var k ) Z[SB]. Idea of Proof. One has L = [P ], hence modulo L one finds in Theorem.8 that [E] = [Y ]. So dividing by L just leaves birational equivalence as relation. Let Z[AV k ] denote the free abelian group generated by the isomorphism classes of abelian varieties, equipped with the ring structure given by products. Corollary.2. If char(k) = 0 then the Albanese map induces Alb : K 0 (Var k ) Z[AV k ]. In particular, for abelian varieties A and B one has: [A] = [B] K 0 (Var k ) A = B. Proof. Alb is invariant under stably birational equivalence. 2. Zero-divisors in K 0 (Var k ) Theorem 2. (Poonen). If char(k) = 0, then K 0 (Var k ) has zero-divisors. To be more precise: Theorem 2.2 (Poonen). There exist abelian varieties A and B, defined over Q such that ([A] [B]) ([A] + [B]) = 0 but with ι [A] ι [B] for all field extensions Q k. Proof. One has to find A and B with A 2 = B 2 but with A k = Bk for A k = A Q k and B k = B Q k. In fact, by Corollary.2, the latter implies that ι [A] = [A k ] [B k ] = ι [B]. 3
4 To find such A and B Poonen starts with two isogenous but non-isomorphic abelian varieties A and B. In particular the kernel K of A B is part of the n-division points A(n), for some n but not the whole group. If A A has enough automorphisms, one can hope that there, say τ : A A A A such that the image of K K maps isomorphically to {e} A(n). Then B B would be isomorphic to A A. This is perhaps a bit naive, and Poonen uses newforms, Eichler-Shimura and computer algebra to find A and B as abelian varieties over Q. Example 2.3 (Kollár). Let k be a field, char(k) = 0. Assume that there is a nontrivial form of P, i.e. a non-singular projective curve C 0 of genus 0, which is not isomorphic to P k, but becomes isomorphic to P K over some field K, algebraic over k. As well known, a non-trivial form C 0 of P has no k-rational point (in particular k is not C ) and one can choose the extension K/k to be of degree 2. For example, over Q one can take the zero-set C 0 of X 2 + y 2 + z 2 in P 2 Q = P(Q[x, y, z]). In general, if S is a base scheme, and if C S is a form of P, e.g. a conic bundle, then C is S-isomorphic to P S if and only if C S has a section. Choosing the diagonal as a section one finds that C 0 k C 0 = P C0 = P k k C 0. So [C 0 ] ([C 0 ] [P k ]) = [C 0 k C 0 ] [C 0 k P k ] = 0. If [C 0] = [P k ], apply Theorem. (Here we need that char(k) = 0). So C 0 P n k has to be birational to Pn+ k. In particular C 0 P n k (k), which implies that C 0(k), contradicting the choice of C 0. Then [C 0 ] [P k ] and [C 0] is a zero-divisor in K 0 (Var k ). Theorem 2.4 (Nicaise, Rökaeus). If k is a non-separably closed field, then there are zero-divisors in K 0 (Var k ). Construction. Choose a non-trivial finite Galois extension K of k of degree d. Then Spec(K) k Spec(K) = d Spec(K) and hence [Spec(K)] (Spec(K) d) = 0. It remains to verify: Claim 2.5. [Spec(K)] 0 and [Spec(K)] d in K 0 (Var k ). The first inequality is obvious. For the second one, assume first that k is a finite field. Then the number of k-rational points of [Spec(K)] is zero, whereas #(d) = d. In general, one replaces k by a field, finitely generated over Q or F p, and then one specializes to a finite field. By using the Chebotarev density theorem, one can do so in such a way, that the corresponding field extension does not split completely. 3. Questions and comments Question 3.. Is L k a zero-divisor in K 0 (Var k )? Of course L k = [P ] [(0, )]. So the Question 3. is related to: Question 3.2. Let X be a projective smooth k-variety and let x X(k) be a rational point. Can [X \ x] be a zero-divisor in K 0 (Var k )?
One can ask the question, whether there are non-trivial forms of A n, i.e. varieties Y over k with A n K = Y k K over some finite separable extension K of k. For A k the answer is no, since G m Aut(A ) G a 0 is exact, and hence H ét (k, Aut(A )) = 0. T. Kambayashi has shown that there is no form of A 2. Nevertheless the cut and paste relation allows to construct forms of A 2 in K 0 (Var k ). Example 3.3. Let C k P 2 k be a non-trivial form of P k over some field k, and C K = P K. Then [P 2 k \ C k] = [P 2 k ] [C k] is a form of [A 2 k ]. In fact, [P 2 K \ C K ] = [P 2 K] [C K ] = [P 2 K] [P K] = [A 2 K]. A more complicated example, due to [Esnault, Viehweg], give singular forms of [A 2 k ]. They also show that [Lo, Lemma 5.] does not extend to the equivariant version of K 0 (Var k ). Example 3.4. For k = Q K = Q( ) there exists an element [Z] K 0 (Var k ) with [Z] 2 L k = [A 2 k ] but with ι [Z] = 2 L K = [A 2 K ] in K 0(Var k ). Moreover [Z] is a singular variety, and a quotient of [A 2 K ] by a group G, with 0 Z/2 G Z/2 = Gal(K/k) 0. References [AKMW] Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; W lodarczyk, Jaros law: Torification and factorization of birational maps. J. Amer. Math. Soc. 5 (2002), 53 572 [Lo] Looijenga, E.: Motivic measures, Séminaire Bourbaki 999/2000, exp. 865-879. volume 276 Astérisque, 267-297. [Nic] Nicaise, Johannes : A trace formula for varieties over a discretely valued field. preprint 5