Topology of Toric Varieties, Part II

Transcription

1 Topology of Toric Varieties, Part II Daniel Chupin April 2, 2018 Abstract Notes for a talk leading up to a discussion of the Hirzebruch-Riemann-Roch (HRR) theorem for toric varieties, and some consequences for Minkowski sums of polytopes. 1 Introduction This talk will be a patchwork of ideas from the topology and intersection theory of toric varieties. We will start by discussing some nice computations related to the Picard group and the rational Chow ring of a toric variety, state HRR, and venture into mixed volumes. Material is mostly from sections and of [Ful93], and the analogous sections of Chapters 9, 12, and 13 of [DS11]. 2 Some more topology of toric varieties Let Σ N R be a fan, where N is an n-dimensional lattice. We first survey some basic results: Proposition 1. The following are from 3.2 of Fulton: 1. If σ is a k-dimensional cone, then π 1 (U σ ) = Z n k. 2. If Σ contains an n-dimensional cone, then X Σ is simply-connected. More generally, if N N is the subgroup generated by σ N for all σ Σ, then π 1 (X Σ ) = N/N. 3. If σ is an n-dimensional cone, then U σ is contractible. 4. The torus O σ U σ is a deformation retract. 5. There is a canonical isomorphism H i (U σ, Z) = i M(σ). Proof. (Sketch) Recall that we have 1-parameter subgroups λ v : G m T N for v N; by taking the images of S 1 G m, we get representatives for all loops in T N, up to homotopy: N = π 1 (T N ). If v σ N, then λ v extends to a map C U σ ; whence we can contract all such loops inside U σ. If σ is n-dimensional, then σ N generates N as a group, so all loops of U σ come from N; thus π 1 (U σ ) = 0. More generally, if σ is k-dimensional, then π 1 (U σ ) = π 1 (U σ (C ) n k ) = Z n k = N(σ). Applying Van Kampen s theorem to these affine pieces (which intersect in affine pieces), we see that π 1 (X Σ ) = N/( N σ ) = N/N Page 1 / 8

2 For (3), we ll recall that U σ has a unique T N -fixed point x σ. Let v be a lattice point in the interior of σ; thinking of closed points x of U σ as semigroup homomorphisms S σ C, build H : U σ I U σ by H(x t)(u) := t (u,v) x(u), and H(x 0) = x σ. Part (4) now follows from (3) by the fact that U σ = U σ O σ. Part (5) is clear. 2.1 The Picard group and H 2 We now use the above to understand the obvious map P ic(x Σ ) H 2 (X Σ, Z). It need not be injective or surjective: for example, P ic(t N ) = 0 but H 2 (T N, Z) = M. However, it is both if, for instance, X Σ is complete: Proposition 2. If all maximal cones of X Σ are n-dimensional, then P ic(x Σ ) = H 2 (X Σ ; Z). Proof. This is a nice argument that uses the same spectral sequence that one interrogates to prove that Cech cohomology agrees with sheaf cohomology: it is built from the double complex which along the bottom row is the Cech complex for a sheaf F on X (for us, the constant sheaf Z X ), and in the vertical direction is given by using an injective resolution of F in this case, we just take the sheafified Cech complex! So we get a double complex almost tautologically. The important point is that we can take as our covering {U σ } σ Σ(n) ; these are affine open sets that are contractible (so have no cohomology in non-zero degrees) and all higher-degree intersections are still affines which are each at least connected. We have E p,q 1 = i 0 < <i p H q (U i0 U ip ) H p+q (X). First we show that E 1,1 2 = H 2 (X Σ ; Z), and from there use the structure of the spectral sequence to realize H 2 (X Σ, Z) as a kernel of a nice lattice morphism. By connectivity, E p,0 1 = I Z, which, with the horizontal differentials, becomes the Koszul resolution for Z in the q = 0 row; thus, E p>0,0 2 = 0. By contractibility of U σ, E 0,q>0 1 = 0; thus E 1,1 2 = E 1,1 = H 2 (X Σ ). Using this new description of H 2, we see that H 2 (X Σ, Z) = Ker(d 1 : E 1,1 1 E 2,1 1 ) Im(d 1 : E 0,1 1 E 1,1 1 ) = Ker(d 1 : E 1,1 1 E 2,1 1 ) (1) ( = Ker M(σ i σ j ) ) M(σ i σ j σ k ) (2) where we have used (5) of the previous proposition. i<j i<j<k This is a bit to write out. But the key point in the end is, we write out a commutative diagram that presents H 2 (X Σ, Z) as a quotient by M of Ker ( i M i<j M/M(σ i σ j ) ). The latter is exactly the collection of T -Cartier divisors by virtue of the fact that M(σ i ) = 0 since our cones are maximal. Thus we ve shown P ic(x Σ ) = H 2 (X Σ, Z). Page 2 / 8

3 3 Chow groups and intersection theory 3.1 Reminder: stars and the orbit-cone correspondence Given a cone σ Σ N R, let N σ be the lattice spanned by σ, and N(σ) the quotient: 0 N σ N N(σ) 0. The star of σ is the fan obtained by projecting Σ onto N(σ) R. If σ is k-dimensional, we get an (n k)- dimensional toric variety O σ V (σ), which is a closed toric subvariety of X Σ. Proposition 3. (Orbit-cone correspondence) There exist the following relations among orbits O τ, orbit closures V (τ), and the affine opens U σ : 1. U σ = τ<σ O τ ; 2. V (τ) = γ>τ O γ; 3. O τ = V (τ) \ γ τ V (γ) 3.2 Intersection theory for toric varieties We first mention a result which generalizes the fact that the Weil divisor class group is generated by the T -invariant divisors: Proposition 4. The Chow group A k (X) is generated by orbit closures [V (τ)] for τ a (n k)-dimensional cone. Proof. Work with the filtration = X 1 X 0 X 1 X n = X given by X i = V (τ), dim τ n i taking the reduced structure on V (τ) s. Then X i \X i 1 is a disjoint union of tori O σ where dim σ = n i. Now use the exact sequence in Chow groups for a closed embedding, and induct. Second, some other interesting intersection theory, which follows the maxim that rational coefficients enter intersection theory for simplicial things, i.e. things that are locally quotients of finite group actions: Proposition 5. If X Σ is a simplicial complete toric variety and ρ 1,..., ρ d Σ(1) are distinct, then in A d (X Σ ) Q, 1 mult(σ) [D ρ1 ] [D ρd ] = [V (σ)] if σ = ρ ρ d Σ, 0 otherwise Proof. Intersect stuff. This is all great news at the cost of ignoring torsion, we can do intersection theory on simplicial complete toric varieties which may not be smooth! Recall that for general varieties, we required smoothness in order to use a moving lemma to ensure that the intersection product is well-defined. If X Σ is smooth, multiplicities of all cones are 1, so we get a nice Chow ring that s in fact defined over the integers. Page 3 / 8

4 4 Hirzebruch-Riemann-Roch 4.1 Definitions HRR shows how to compute the Euler characteristic of a coherent sheaf F on a complete smooth variety X: it says that χ(x, F ) = X ch(f ) T d(x), where ch(f ) H (X, Q) is the Chern character, and T d(x) := T d(t X ) H (X, Q) is the Todd class of the tangent bundle T X. Equivalently, we could work in the rational Chow ring of X, as for smooth complete X this is isomorphic to the rational cohomology ring. The Chern character of F is defined by defining it for vector bundles E first: ch(e) = r e ξ i, i=1 for ξ i a collection of virtual Chern roots of E; from this we can see that ch(e F ) = ch(e) + ch(f ), and ch(e F ) = ch(e) ch(f ). The Todd class of E is given by T d(e) = r i=1 ξ i 1 e ξ i, where each factor is understood as the power series it defines; the first few terms are x 1 e x = x + ( 1) i 1 B i (2i)! x2i = x x x4 +, i=1 where B k are the Bernoulli numbers. Note that this is a symmetric polynomial in ξ i in Q coefficients; by the theory of symmetric polynomials, this is therefore a polynomial in c i (E) with Q coefficients. Example 1. For a surface, T d(x) = ( ξ )( 1 2 ξ ξ ) 1 2 ξ2 2 = c 1(T X ) (c 1(T X ) 2 + c 2 (T X )). 4.2 Some computations The Euler short sequence of a smooth complete X Σ (which we encountered in Irit s talk on the canonical sheaf; see Chapter 9 of [DS11]) 0 Ω 1 X Σ ρ Σ(1) together with the above intersection theory result, gives 1. c(t X ) = ρ (1 + [D ρ]) = σ Σ [V (σ)]. 2. c 1 (T X ) = ρ [D ρ]. 3. c n (T X ) = Σ(n) [pt]. O X ( D ρ ) P ic(x Σ ) Z O XΣ 0, Page 4 / 8

5 in either either in A (X Σ ) or H (X Σ, Z). Part (3) leads to (yet another) proof that χ(x Σ ) = Σ(n). (another proof secretly cropped up in our spectral sequence calculation). We also have the following expression for the Todd class: T d(x) = ρ Σ(1) [D ρ ] 1 e [Dρ]. Note that this is not a sum over the virtual Chern roots of T X! But it s still of the same form. 4.3 Why the Todd class? The Todd class looks very strange, but a section in Chapter 13 of [DS11] gives a nice reason to believe that a human being could come up with it. Let f : X Y be a proper morphism between smooth complete varieties. In this case, we can define pushforwards on the Grothendieck group of coherent sheaves f! : K(X) K(Y ) and on cohomology f! : H (X, Q) H (Y, Q). It is also true that the Chern character defines a group homomorphism ch : K(X) H (X, Q), which implants the hope that the following diagram commutes: ch K(X) H (X, Q) f! f! ch K(Y ) H (Y, Q) It doesn t, in general. But suppose we are wedded to the idea that commutativity isn t far off that there is a single polynomial T d that works for all varieties that we just need to wedge with to get the compositions to match up: ch T d(x) K(X) H (X, Q) f! f! ch T d(y ) K(Y ) H (Y, Q) Working out what this means for specific cases can give us ideas about what T d(x) looks like. A nice idea is to look at an inclusion of a hypersurface i : X Y, and look at the images of (the class of) the structure sheaf O X. The northeast path gives i! (ch([o X ])T d(x)) = i! T d(x), while the southwest path gives ch([i O X ])T d(y ), since i is a closed embedding, hence affine, meaning higher direct images vanish. Put i O X into the short exact sequence for a closed subscheme 0 O Y ( X) O Y i O X 0 to see that ch([i O X ]) = ch([o Y ] [O Y ( X)]) = 1 e [X]. Page 5 / 8

6 From this, it s a short way, via the property of the Gysin map that i! i α = α [X], that it s desirable if the Todd class satisfies ( ) [X] i T d(y ) = T d(x) i 1 e [X]. Using purported naturality of our mystery object and the facts that (1) i T Y = T X N Y/X and (2) that c 1 (N Y/X ) = i [X], we are led to suppose that, generally, T d ought to satisfy T d(e L) = T d(e) c 1 (L ) 1 e c 1(L ) for a line bundle L. This is consistent with and suggests the formula that was given. 5 Application: mixed volumes This is all from 5.4 of [Ful93]. 5.1 Recollection: the Ehrhart polynomial E P Several weeks ago, Alex talked about the nice computability properties of the cohomology of line bundles on toric varieties. As we saw, those properties, together with HRR, supplied for us a neat proof of the fact that, for P M R a polytope in a lattice, the function E P defined for positive integers ν as E P (ν) := νp M was in fact a polynomial with rational coefficients. Serre duality together with knowledge of the canonical sheaf ω XP showed that E P also counted interior points of dilations of P at negative integer arguments. In this section, we will extend this discussion and talk about mixed volumes. Suppose M is n-dimensional, and let D denote the divisor on X P that P defines. Recall that the volume of the polytope lim ν E P (ν) ν n = Dn n! is the (normalized) self-intersection number of D. 5.2 Minkowski sums and mixed volumes Let P, Q be compact convex subsets of Euclidean space; their Minkowski sum P + Q is also compact and convex, and for a positive integer ν, the ν-dilation of P is equal to its ν-fold Minkowski sum. For n = 2, define V (P, Q), the mixed volume (area) of P and Q, to be V (P, Q) = 1 2( V ol(p + Q) V ol(p ) V ol(q) ). Nice properties: 1. V (P, P ) = V ol(p ) 2. V (P, Q) is symmetric and multilinear 3. For B ε the ε-ball centered at 0, the perimeter of P can be expressed as 2 lim ε 0 ε V (P, B ε) Page 6 / 8

7 4. The isoperimetric inequality (that Vol(P ) 1 4π Perim(P )2 ) is the statement that V (B ε, B ε )V (P, P ) V (P, B ε ) 2. We would like to find analogs of the above for general compact convex subsets in higher dimensions. We will restrict our consideration to n-dimensional compact convex lattice polytopes; to get more general results, we would dilate the lattice and use polytope approximations to the compact convex subset. We can find a fan Σ that is compatible with P, in which case P defines a T -Cartier divisor D, whose basis of global sections is given by P M. Given P 1,..., P s polytopes in M, we may find a fan (smooth, if desired, by subdividing more) compatible with all P i. Denoting by E i the divisor corresponding to P i, we see that the divisor associated to the polytope ν 1 P 1 + +ν s P s is ν 1 E ν s E s. By earlier, we know that V ol(ν 1 P ν s P s ) = 1 n! (ν 1E ν s E s ) n ; this implies Minkowski s fundamental result that this volume is a polynomial in the dilation constants. For n polytopes P 1,..., P n, we define the mixed volume to be the coefficient of ν 1 ν n : with the closed form V (P 1,..., P n ) = 1 n! V (P 1,..., P n ) := 1 n! (E 1 E n ), ( V ol(p P n ) i + i<j V ol(p 1 + ˆP i + + P n ) (3) V ol(p ˆP i + + ˆP j + + P n ) ± + ( 1) n 1 i From either of these formulations, we easily glean some properties that we compile here: Proposition 6. Mixed volume enjoys the following properties: 1. V (P,..., P ) = V ol(p ) 2. mixed volume is symmetric and multilinear (for non-negative dilations) 3. mixed volume is invariant under shifts of polytopes by lattice vectors: V (P 1 + u 1,..., P n + u n ) = V (P 1,..., P n ). ) V ol(p i ). (4) 4. for A : M M a group homomorphism, we have V (AP 1,..., AP n ) = det A V (P 1,..., V n ). 5. mixed volumes are non-negative! (as intersection numbers of globally-generated divisors E i ) 6. in fact, a stronger result holds: for Q i P i subpolytopes, V (Q 1,..., Q n ) V (P 1,..., P n ). Page 7 / 8

8 This is because we get the divisors associated to Q s by subtracting effective divisors from P s. In more detail, let F 1 + Y = E 1 for Y the effective divisor; wiggle E i s in their classes to ensure that each restricts to a Cartier divisor on Y ; these E i Y are all globally generated, thus the inequality. As a corollary, if P is the convex hull of n i=1 P i, then V (P 1,..., P n ) V ol(p ). As a second corollary, if all interiors of the P i are nonempty (and hence the volumes are positive), then, by translating the P i such that they all intersect in a polytope Q (and by scaling the lattice so that this is possible), we get that 0 < V ol(q) V (P 1,..., P n ). Last thing, which generalizes form of the isoperimetric inequality we wrote down for n = 2 using mixed volumes: Theorem 1. (Alexandrov-Fenchel Inequality) V (P 1,..., P n ) 2 V (P 1, P 1, P 3,..., P n ) V (P 2, P 2, P 3,..., P n ). Proof. This is equivalent to the statement that (E 1 E n ) 2 (E 1 E 1 E 2 E n )(E 2 E 2 E 3 E n ). Let ϕ i : X P P n i be the projective morphisms with n-dimensional images corresponding to E i. Bertini s theorem says that, for generic hyperplanes H i P n i, Y = ϕ 1 3 (H 3) ϕ 1 n (H n ) is a smooth projective surface in X P. Thus it suffices to prove the inequality (D 1 D 2 ) 2 D1 2D2 2 for D i the restriction of E i to Y. This follows from the Hodge index theorem (see [Rei96]), which says that, on a smooth projective surface, if A, B are divisors such that A 2 > 0 and AB = 0, then B 2 0. References [Ful93] William Fulton. Introduction to Toric Varieties. Princeton, NJ: Princeton Univ. Press, [Rei96] Miles Reid. Chapters on Algebraic Surfaces. In: (1996). doi: alg-geom/ pdf. [DS11] John Little David Cox and Henry Schenck. Toric Varieties. AMS, Page 8 / 8