Flag Varieties Notes from a Reading Course in Spring 2019 George H

Home , Flag (linear algebra), Tautological bundle

Flag Varieties Notes from a reading course in Spring 2019

George H. Seelinger 1. Introduction (presented by Weiqiang Wang) These notes will start with a quick introduction to vector bundles and Chern classes following [BT82, Sections 20–21,23]. Then, they will shift to a discussion of equivariant cohomology, loosely following [Ful07]. Note that citations may be incomplete because the author is basing these notes oﬀ of lectures presented in a reading course organized by Weiqiang Wang. 2. A Quick Introduction to Vector Bundles (not presented) This section will list some facts about vector bundles without proof. For a more comprehensive treatment, see references such as [BT82] and [MS74]. The reader may skip this section and refer back to it as necessary.

2.1. Definition. A rank n (complex) vector bundle ξ, consists of (a) a topological space E called the total space, (b) a topological space B called the base space, (c) and a continuous projection map ρ: E → B such that (i) for all b ∈ B, the preimage ρ−1(b), which is called the ﬁber over b is a vector space, and (ii) (local triviality) for all b ∈ B, there exists a neighborhood n U ⊆ B with b ∈ U and a homeomorphism h: U × C → ρ−1(U) (called a local trivialization) such that, for all x ∈ U, n −1 h|{x}×Cn : C → ρ (x) is a vector space isomorphism.

n h −1 U × C ρ (U) π1 ρ U

2.2. Remark. (a) One can just as easily define a real vector bundle by simply replacing C with R everywhere in the definition. −1 ∼ n −1 0 (b) One can define a vector bundle where the ρ (b) = C and ρ (b ) = m 0 C for n 6= m if b and b are in different connected components of B. However, for simplicity, we will usually keep the rank of every fiber the same. (c) If b, b0 ∈ B are in the same connected component, then the local −1 ∼ n ∼ −1 0 triviality condition forces that ρ (b) = C = ρ (b ).

2.3. Example. (a) Perhaps the most familiar example of a (smooth) real vector bundle is that of a “tangent bundle” of a manifold. That is, given an n-dimensional smooth manifold M, we have a 2n- n dimensional manifold TM := {(x, v) ∈ M × R | v ∈ TxM} where TxM is the tangent space of x and map π1 : TM → M given by π1(x, v) = x. Then, we have tangent bundle τ(M) given by

π1 M

−1 ∼ n with ﬁber π1 (x) = TxM = R . Similarly, the “normal bundle” ν(M) is a vector bundle.

3 (b) Given a space X, we have the trivial bundle n given by n X × C

π1 X −1 n with π1 (x) = C . If X is a single point, then the trivial bundle is n essentially just the vector space C . In this way, we see that vector bundles serve as a generalization of vector spaces. A particular example of a real trivial bundle would be

1 S × R

π1 S1 1 where S × R is a cylinder. (c) Given a M¨obiusstrip M, we have a real vector bundle M ρ

S1 where ρ: M → S1 sends a point in the M¨obiusstrip to the circle embedded along the “middle”. For instance, every point on the dashed line is sent by ρ to the point where the dashed line intersects the red line in the picture below.

−1 The fiber is given by ρ (x) = (0, 1) which is homeomorphic to R. Furthermore, this example is not equivalent to the trivial bundle 1 1 S × R → S . Informally, this can be seen by noticing that one can pick a consistent choice of orientation on the trivial bundle but not on this bundle. n n (d) Given a space B = CP , the points x ∈ CP are lines. Then, we 1 define the tautological line bundle γn to be given by total space n n+1 E = {(x, v) ∈ CP × C | v lies on x} and map ρ(x, v) = x. −1 ∼ Then, ρ (x) = {x} × Cv = C. 1 ∼ 1 (e) If we do the tautological line bundle construction over RP = S , we recover the Möbiusstrip example above. To see this in detail, see [MS74, pp 16–17] 2.4. Definition. A line bundle is a rank 1 vector bundle.

4 2.5. Proposition. Given a rank n vector bundle ξ with base space B and total space E and ρ: E → B, we can construct new vector bundles in a variety of ways. (a) Given some map f : B0 → B, we can construct the pullback bundle f ∗(ξ) over B0 by setting f ∗(E) = {(x0, v) ∈ B0 × E | f(x) = ρ(e)} and ρ0 : f ∗(E) → B to be given by ρ0(x0, e) = x0 so that (ρ0)−1(x0) = ρ−1(f(x0)). This is a speciﬁc version of a pullback square:

f˜ f ∗(E) E

ρ0 ρ f B0 B (b) We can take the Whitney sum of two vector bundles over B induced by taking the pullback bundle of the diagonal inclusion ι: B → B×B 0 −1 ∼ −1 that sends b 7→ (b, b) and this induces ﬁbers (ρ ) (b) = ρ1 (b) ⊕ −1 ρ2 (b). ∗ ˜ι i (E1 × E2) E1 × E2

ρ0 ρ1×ρ2 B ι B × B

We will denote this as ξ1 ⊕ ξ2. (c) Any operation that makes sense to be done on a vector space can also be done on two vector bundles over B to get a vector bundle over B. Flush out with (d) Given a complex vector bundle ξ of rank n, one can get a rank 2n some explcit real vector bundle ξR by simply forgetting the complex structure. constructions. 2.6. Definition. Given a vector bundle ξ, a section is a map s: B → E such that ρ ◦ s = IdB. 2.7. Example. (a) For every vector bundle, there exists a zero section s0 : B → E defined by s0(b) = (b, 0). (b) In general, there are uncountably many sections whose images in- tersect the image of the zero section. These are called vanishing sections and are relatively uninteresting. (c) Add something not boring. More generally, we wish to define a generalization of vector bundles that we will use sparringly. 2.8. Definition. A fiber bundle ξ with total space E, base space B, projection map ρ: B → E, and fiber F is defined analogously to a complex n vector bundle where every occurrence of C is replaced by F . See [Hat01, pp 376–377] for a more precise definition. One reason to study vector bundles is that they can give us insight into the cohomology

5 rings of the spaces involved. A general theorem which we will state without proof is 2.9. Theorem (Leray-Hirsch). [Hat01, Theorem 4D.1] Let ξ be a fiber bundle with ρ: E → B such that, for some coefficient ring R, (a) Hn(F ; R) is a finitely generated free R-module for each n. k ∗ (b) There exist classes ej ∈ H j (E; R) who restrictions u (ej) form a basis for H∗(F ; R) in each fiber F , where i: F → E is the inclusion. ∗ ∗ ∗ Then, the map Φ: H (B; R) ⊗R H (F ; R) → H (E; R) given by X ∗ X ∗ bi ⊗ i (ej) 7→ ρ (bi) ^ ej i,j i,j is an isomorphism of R-modules. In particular, the Leray-Hirsch theorem tells us that H∗(E; R) is a free ∗ ∗ H (B; R)-module with basis {ej} where we view H (E; R) as a module over the ring H∗(B; R) by defining the action b.e = ρ∗(b) ^ e for b ∈ H∗(B; R) and e ∈ H∗(E; R). 2.10. Definition. In the discussion that follows, for any vector space V , let V0 = V \{0} and for any total space E, let E0 = E \ s0(B) where s0 is the zero section. 2.11. Definition. [MS74, p 96] An orientation of a real vector bundle ξ is a function which assigns an orientation to each fiber F of ξ such that for every point b0 ∈ B, there is a neighborhood U containing b0 with local n −1 −1 trivialization h: U × C → ρ (U) such that for each fiber F = ρ (b) n for b ∈ B, the homomorphism hb : C → F given by hb(x) := h(b, x) is orientation preserving. This definition tells us that, for an orientable rank n complex vector ∼ n bundle ξ, for each fiber F = R , we can pick a preferred generator uF ∈ n H (F,F0; Z) = Z. Then, the local conditions in the definition of orientation imply that, for every b0 ∈ B, there is a neighborhood U and a cohomology n −1 −1 −1 class u ∈ H (ρ (U), ρ (U)0; Z) so that for every fiber F = ρ (b) over U, ∗ n ι (u) = uF ∈ H (F,F0; Z) where ι: ρ−1(b) ,→ ρ−1(U). 2.12. Theorem. [MS74, Theorem 9.1] Let ξ be an oriented real vector bundle of rank n with total space E. Then, i (a) H (E,E0; Z) = 0 for i < n and n (b) H (E,E0; Z) contains a unique cohomology class u, called the Thom class, whose restriction ∗ n ι (u) = uF ∈ H (F,F0; Z)

6 for every ﬁber F of ξ where ι: F,→ E is the standard inclusion. k (c) The correspondence y 7→ y ^ u maps H (E; Z) isomorphically to k+n H (E,E0; Z) for every integer k. 2.13. Definition. For an oriented real vector bundle ξ, the Thom isomorphism ∼ k k φ: H (B; Z) → H (E,E0; Z) is given by the formula φ(x) := ρ∗x ^ u

We will take for granted that this is an isomorphism of abelian groups. ∗ Now, the inclusion (E, ∅) ,→ (E,E0) induces a restriction H (E,E0; Z) → ∗ n H (E; Z) which we will denote by y 7→ y|E. Then, for u ∈ H (E,E0; Z), we can take n ∼ n u 7→ u|E ∈ H (E; Z) = H (B; Z) via the Thom isomorphism above. Thus, we deﬁne

2.14. Definition. The Euler class of an oriented rank n vector bundle ξ is the cohomology class

n e(ξ) ∈ H (B; Z) n ∼ n which corresponds to u|E under the Thom isomorphism H (B; Z) = H (E; Z). 2.15. Proposition. (a) [MS74, Property 9.2] If f : B → B0 is continuous and covered by an orientation preserving map f : ξ → ξ0, then e(ξ) = f ∗e(ξ0). (b) [MS74, Property 9.6] The Euler class of a Whitney sum is given by e(ξ ⊕ ξ0) = e(ξ) ^ e(ξ0). (c) e(ξ ⊗ ξ0) = e(ξ) + e(ξ0). (d) [MS74, Property 9.7] If the oriented real vector bundle ξ possesses a nowhere zero section, then e(ξ) must be zero

3. Chern Classes (presented by Thomas Sale) The material in this section will roughly correspond to [BT82, Sections 20–21]. Because of this, throughout this section all cohomology is de Rham cohomology and thus over R. All spaces can be assumed to be smooth manifolds.

3.1. Remark. As one gets more accustomed to using vector bundles, typically one stops denoting the bundle by something like ξ and just denotes it by the total space E when the bundle structure is clear.

7 3.1. The first Chern class of a line bundle. 3.2. Definition. Given a complex line bundle ξ, we define the first Chern class of ξ to be c1(ξ) := e(ξR). 3.3. Proposition. Given line bundles ξ, ξ0, 0 0 (a) c1(ξ ⊗ ξ ) = c1(ξ) + c1(ξ ). ∗ ∗ ∗ (b) 0 = c1(ξ ⊗ ξ ) = c1(ξ) + c1(ξ ) =⇒ c1(ξ ) = −c1(ξ). Proof. The first part is immediate from 2.15(c) and the second follows from the fact that ξ ⊗ ξ∗ = Hom(ξ, ξ) always has a nonvanishing section sending b ∈ B to the identity map in Hom(E,E) and so e(LR) = 0. 3.2. Projectivized vector bundles and Chern clases. 3.4. Definition. (a) Given a rank n complex vector bundle ξ, we define the projectivized bundle P(ξ) to be the bundle P(E) π B −1 −1 ∼ n ∼ n−1 where π (b) = P(ρ (b)) = P(C ) = CP . Note that π ◦ P = ρ −1 and that a point x ∈ P(E) is a line in ρ (π(x)). E ρ P π P(E) B (b) We define the pullback vector bundle π∗(ξ) of the projectivization 0 −1 −1 map π : P(E) → M and the fiber (ρ ) (x) = ρ (π(x)). π∗(E) π˜ E

ρ0 ρ π P(E) B (c) The universal sub-bundle of π∗(ξ) is deﬁned by taking the total space S = {(x, v) ∈ π∗(E) | v ∈ x} and is a copy of the tautological line bundle over P(E) (see 2.3(d)) and so S is a rank 1 vector bundle. (d) The universal quotient bundle Q of π∗(E) is deﬁned by taking the short exact sequence 0 → S → π∗E → Q → 0 Then, Q is a rank n − 1 vector bundle.

8 (e) If we restrict the unviversal sub-bundle S to be over a ﬁber π−1(b) =∼ n−1 CP , then we have a bundle with total space S˜ which is a copy n−1 of the tautological line bundle over CP . ∗ −1 ∗ n−1 ∼ n Now, recall the fact that H (P(π (b)); R) = H (CP ; R) = R[a]/(a ) ˜∗ ˜ 2 −1 as rings. If we set x = c1(S ) = −c1(S) ∈ H (P(E)) and i: P(π (b)) ,→ ∗ 2 n−1 P(E), then −i (x) = a ∈ H (CP ; R) . Since this works for every ﬁber Explain why simultaneously, we can apply ∗ 3.5. Corollary (Leray-Hirsch (special case)). The cohomology of H (P(E); R) ∗ n−1 k is a free module over H (B; R) with basis {1, x, . . . , x } with action y.x = π∗(y) ^ x.

n ∗ From this fact, we can write x ∈ H (P (E); R) as a linear combination with coefficients in H∗(B). 3.6. Definition. (a) The Chern classes of the bundle ξ with data ∗ ρ: E → B are elements ci(ξ) ∈ H (B; R) which are the unique coefficients such that n ∗ n−1 ∗ ∗ x + π (c1(ξ))x + ··· + π (cn(ξ)) = 0 ∈ H (P(E); R) 2i Thus, ci(ξ) ∈ H (B; R). (b) We define the total Chern class to be ∗ c(ξ) := 1 + c1(ξ) + ··· + cn(ξ) ∈ H (B; R) 3.7. Proposition (Main properties of Chern classes). (a) Given a map f : B0 → B, Chern classes have naturality c(f ∗(ξ)) = f ∗(c(ξ)). As an immediate corollary, this menas that Chern classes are an invariant of vector bundles, that is, ξ =∼ ξ0 =⇒ c(ξ) = c(ξ0). 1 n (b) Given γ to be the tautological complex line bundle over P , then ∗ n ∼ 1 1 n+1 H (P ; R) = R[c1(γ )]/((c1(γ ))) . (c) c(ξ ⊕ ξ0) = c(ξ) ^ c(ξ0), called the Whitney product formula. (d) If ξ has rank n, then ci(ξ) := 0 for i > n. (e) If ξ has a non-vanishing section, then cn(ξ) = 0. (f) The top Chern class of a complex vector bundle ξ is the Euler class of ξR. In other words, cn(ξ) = e(ξR). 3.8. Corollary. As a consequence of the above properties, the total Chern class of a line bundle λ is always of the form

c(λ) = 1 + c1(λ) where c1(λ) = e(λR) by deﬁnition. 3.3. The Splitting Principle. 3.9. Definition. Let ξ with data τ : E → M be a smooth complex vector bundle of rank n over a manifold M. Then, we deﬁne a split manifold of τ to be a space F (E) with map σ : F (E) → M such that

9 (a) σ∗ξ is a direct sum of line bundles and ∗ ∗ ∗ (b) the induced map on cohomology σ : H (M; R) → H (F (E); R) is injective. 3.10. Example. Given a rank 2 complex vector bundle ξ with ρ: E → ∗ M, we can take F (E) = P(E) because π (ξ) decomposes as a direct sum of S ⊕ Q, its universal sub and quotient bundles. E π∗(E) =∼ S ⊕ Q ρ

M π P(E) 3.11. Example. Given a rank 3 complex vector bundle ξ with ρ: E → M, we can iterate the construction above where the 2-dimensional quotient bundle QE can be split into a direct sum of line bundles when pulled back to P(QE). ∗ ∼ ∗ ∼ ∗ E α (E) = SE ⊕ QE β (SE ⊕ QE) = β Se ⊕ SQE ⊕ QQE ρ

M (E) (Q ) α P β P E

Thus, P(QE) is the split manifold F (E). This leads us to the general idea 3.12. Proposition (The Splitting Principle). [BT82, p 275] To prove a polynomial identity in Chern classes, it suﬃces to assume that the vector bundles are direct sums of line bundles. Proof. Given a polynomial f, we want to show f(c(E)) = 0. Then, with setup E σ∗E

M σ F (E) we get σ∗(f(c(E))) = f(c(σ∗E)) and so, because σ∗ is injective by deﬁnition of F (E), f(c(σ∗(E))) = 0 =⇒ f(c(E)) = 0. Since σ∗(E) is a direct sum of line bundles by construction, we are done. 4. Chern class computations, ﬂag manifolds, and the Grassmannian (presented by George H. Seelinger) 4.1. Proof of the Whitney Product Formula.

4.1. Lemma. Let ξ be a direct sum of line bundles, that is ξ = λ1⊕· · ·⊕λn where each λi is a line bundle over B. Then,

c(ξ) = c(λ1)c(λ2) ··· c(λn)

10 Proof. First, let us take vector bundle ξ to be a a direct sum of line bundles. Then, if we take pullback bundle ξ by the projectivization map ∗ (see 3.4) π : P(E) → M, π ξ is a direct sum of line bundles, let us say ∗ π ξ = L1 ⊕ · · · ⊕ Ln.

∗ ∼ π˜ S π E = L1 ⊕ ... ⊕ Ln E

ρ0 ρ π P(E) M

Now, let si : S → Li be the projection of vector bundles. Thus, si induces 1 a section of the bundle Hom(S, Li) → M. Furthermore, since the ﬁber of S over every point y ∈ P(E) is a 1-dimensional subspace of the ﬁber of y in ∗ 0 −1 π ξ, that is (ρ ) (y), the projections s1, . . . , sn cannot be simultaneously zero and so the open sets

Ui := {y ∈ P(E) | si(y) 6= 0} form an open cover of P(E). ∼ ∗ Now, it is useful to note that the bundle Hom(S, Li) = S ⊗Li. Then, for ∗ 2 2 each i, c1(S ⊗ Li) ∈ H (P(E)) restricts to zero in H (Ui) by construction ∗ of Ui (S ⊗Li is a trivial line bundle over Ui), and so we can lift this element 2 to H (P(E),Ui) by a long exact sequence argument. Thus, if we take the product, n Y ∗ c1(S ⊗ Li) i=1 2 2 we can lift to H (P(E),U1 ∪ · · · ∪ Un) = H (P(E), P(E)) = 0 by using a relative cup product. Therefore, n n Y ∗ 3.3(a) Y ∗ 0 = c1(S ⊗ Li) = (c1(S ) + c1(Li)) i=1 i=1 ∗ and if we set x = c1(S ), we get n Y n n−1 0 = (x+c1(Li)) = x +e1(c1(L1), . . . , c1(Ln))x +···+en(c1(L1), . . . , c1(Ln)) i=1 where ei is the ith symmetric polynomial. However, this is precisely how we deﬁned ci(ξ), so we get

∗ ∗ Y Y π ci(ξ) = ei(c1(L1), . . . , c1(Ln)) =⇒ π c(E) = (1 + c1(Li)) = c(Li) Q and so c(E) = c(λi).

1 This is a special case of a canonical fact. The map si induces a vector space isomorphism on each ﬁber over b, say (si)b, and so the canonical associated section of the Hom bundle sends each b to the map (si)b.

11 4.2. Corollary (Corollary of proof). Given a complex vector bundle ξ of rank n, n n ∗ Y X π c(E) = (1 + c1(Li)) = ej(c1(L1), . . . , c1(Ln)) i=1 j=1 ∗ where π ξ = L1 ⊕ · · · ⊕ Ln is a direct sum of line bundles and, as before, π : P(E) → M. Proof of 3.7(c). We now use the lemma above and the splitting principle 3.12 to prove the formula in general. Given complex vector bundles ξ, ξ0 over M of rank n and m, respectively, we can iterate the splitting constructions to get a direct sum of line bundles for which the Chern class identites are equivalent. 0 ∗ 0 0 E ⊕ E L1 ⊕ · · · ⊕ Ln ⊕ π E L1 ⊕ · · · ⊕ Ln ⊕ L1 ⊕ · · · ⊕ Lm

M F (E) F (π∗E) π π0 Now, if σ = π0 ◦ π, then σ∗(c(ξ ⊕ ξ0)) = c(σ∗(ξ ⊕ ξ0)) By naturality of Chern classes 3.7(a) 0 0 = c(L1 ⊕ · · · ⊕ Ln ⊕ L1 ⊕ · · · ⊕ Lm) By splitting principle construction above 0 0 = c(L1) ··· c(Ln)c(L1) ··· c(Lm) By Lemma 4.1 0 0 = c(L1 ⊕ · · · ⊕ Ln)c(L1 ⊕ · · · ⊕ Lm) By Lemma 4.1 = c(σ∗ξ)c(σ∗ξ0) By splitting principle construction above = σ∗c(ξ)c(ξ0) By naturality of Chern classes However, also by construction (since we have a split manifold 3.9), σ∗ is 0 0 injective, and thus c(ξ ⊕ ξ ) = c(ξ)c(ξ ). 4.3. Remark. (a) Given a short exact sequence of smooth complex vector bundles 0 → A → B → C → 0 one can prove that c(B) = c(A)c(C) using the fact that B =∼ A ⊕ C as smooth bundles and the Whitney product formula. (b) Using the splitting principle and the Whitney product formula, one can directly prove 3.7(f), that is cn(ξ) = e(ξR). See [BT82, p 278] for a complete proof. 4.2. Computing some Chern classes. Recall from Corollary 4.2 that the Chern classes of a rank n complex vector bundle ξ are precisely the elementary symmetric functions in the ﬁrst Chern classes of the line bundles into which ξ splits when pulled back to the splitting manifold F (E). Thus, by the fundamental theorem of symmetric functions, any symmetric polynomial in these variables is a polynomial in c1(ξ), . . . , cn(ξ).

12 4.4. Example. Given V a complex vector space with basis {v1, . . . , vn}, p (a) ∧ V has a basis {vi1 ∧ · · · ∧ vip }1≤i1<···

Note, however, that this is symmetric in the xi’s! Thus, by the Fundamental Theorem of Symmetric Polynomials and the fact that ci(ξ) = ei(x1, . . . , xn), we can write p c (∧ ξ) = Q(c1(ξ), . . . , cn(ξ)) where Q is some polynomial. Note that this result holds for any ξ by the splitting principle and Q depends only on n and p. (b) c (Symp ξ) = Q (1 + x + ··· + x ) 1≤i1≤···≤ip≤n i1 ip 0 Q (c) c(ξ ⊗ ξ ) = 1≤i≤n,1≤j≤m(1 + xi + yj)n ∗ (d) Since c1(L ) = −c1(L) for a line bundle L by deﬁnition,

ξ = λ1 ⊕ · · · ⊕ λn =⇒ c(ξ) = (1 + c1(λ1) ··· (1 + c1(λn))) ∗ ∗ ∗ ξ = λ1 ⊕ · · · ⊕ λn =⇒ c(ξ) = (1 − c1(λ1) ··· (1 − c1(λn))) ∗ and so, setting the homogeneous parts equal to each other, cq(ξ ) = q (−1) cq(ξ). 4.3. Flag manifolds. 4.5. Definition. (a) Given a complex vector space V of dimension n, a (complete) ﬂag in V is a sequence of subspaces

A1 $ A2 $ ··· $ An = V, dimC Ai = i Note that a flag on V induces a C basis on V by picking basis vector bi ∈ Ai/Ai−1 (take A0 = {0}). (b) Let F`(V ) be the collection of all flags of V , called the flag manifold.

4.6. Remark. In fact, F`(V ) is a manifold because GL(n, C) acts transitively on F`(V ) and the stabilizer of a ﬂag is the closed subgroup of all upper triangular matrices (with respect to the induced basis), say H, and so ∼ F`(V ) = GL(n, C)/H as a G-space, but GL(n, C)/H can be given a manifold structure since H is a closed subgroup and GL(n, C) is a Lie group. For those with some Lie theory knowledge, H is a “Borel subgroup.” 4.7. Definition. Given a vector bundle ξ with ρ: E → M, one can form the associated ﬂag bundle F`(ξ) with f : F`(E) → M where f −1(x) = F`(ρ−1(x)) with the natural transition maps.

13 Since the fiber is not a vector space, the associated flag bundle is not a vector bundle, but it is still a fiber bundle. 4.8. Proposition. [BT82, Proposition 21.15] The associated flag bundle F`(ξ) of a vector bundle ξ is the split manifold constructed in the examples after 3.9.

Proof. Let V → {∗} be a rank 3 vector bundle over a point (ie a 3 dimensional vector space.) Then, we have

V SV ⊕ QV SV ⊕ SQV ⊕ QQV

{∗} P(V ) P(QV ) = F (V ) However, we have correspondences

A point in x ∈ P(V ) a line `x in V A point in SV ⊕ QV (x, (v1, v2)) such that x ∈ P(V ), v1 ∈ `x, v2 ∈ V/`x 0 A point in P(QV ) a line ` in V and a line ` in V/`. However, this means that we can similarly regard `0 as a 2-dimensional plane in V containing `, so 0 F (V ) = P(QV ) = {` ⊆ ` ⊕ ` ⊆ V } = F`(V ) More generally, given rank n vector bundle E → M, the split manifold F (E) is obtained by a sequence of n − 1 projectivizations (see 3.4). Then,

−1 A point in P(E) (p, `) where p ∈ M and ` is a line in ρ (p) −1 A point in P(Q1) over (p, `1) ∈ P(E) (p, `1, `2) where `2 ∈ ρ (p)/`1 −1 A point in P(Q2) over (p, `1, `2) ∈ P(Q1) (p, `1, `2, `3) where `3 ∈ ρ (p)/(`1 ⊕ `2) . . . . A point in F (E) = P(Qn−1) over (p, `1, `2, . . . , `n−1) (p, `1, . . . , `n) −1 where `n ∈ ρ (p)/(`1 ⊕ · · · ⊕ `n−1)

But this last point naturally identiﬁes with the ﬂag −1 (p, `1 ⊆ `1 ⊕ `2 ⊆ · · · ⊆ ρ (p)) Thus, F (E) and F`(E) are the same under this equivalence. Recall from before (3.6) that we essentially constructed the Chern classes of M such that ∗ ∼ ∗ n n−1 ∗ H (P(E)) = H (M)[x]/(x + c1(ξ)x + ··· + cn(ξ)), x = c1(S ) as rings. However, we have another useful characterization given by

14 4.9. Proposition. [BT82, Proposition 21.16] Given the setup

E 0 → S → π∗E → Q → 0

M π P(E) we have ∗ ∗ ∗ H (P(E); R) = H (M; R)[c1(S), c1(Q), . . . , cn−1(Q)]/(c(S)c(Q) = π c(E))

Proof. We simply eliminate the generators c1(Q), . . . , cn−1(Q) using the relation c(S)c(Q) = π∗c(E) by equating terms of equal degrees in ∗ ∗ ∗ (1 − c1(S ))(1 + c1(Q) + ··· + cn−1(Q)) = 1 + π c1(E) + ··· + π cn(E) gives us ∗ ∗ c1(Q) − c1(S ) = π c1(E) ∗ ∗ c2(Q) − c1(S )c1(Q) = π c2(E) ∗ ∗ c3(Q) − c1(S )c2(Q) = π c3(E) . . ∗ ∗ cn−1(Q) − c1(S )cn−2(Q) = π cn−1(E) ∗ ∗ −c1(S )cn−1(Q) = π cn(E) ∗ This shows that ci(Q), 1 ≤ i ≤ n − 1 can be expressed in terms of c1(S ) and elements of H∗(M) and so they can be eliminated as generators. The one remaining relation transformed by the other relations will give ∗ ∗ ∗ n ∗ ∗ n−1 ∗ −c1(S )cn−1(Q) = π cn(E) ⇐⇒ c1(S ) +π c1(E)c1(S ) +···+π cn(E) = 0 and so we have established the equivalence. 4.10. Proposition. [BT82, Proposition 21.17 (enhanced)] Let E → M be a complex rank n vector bundle. Then, cohomology ring of the ﬂag manifold F`(E) is n ! ∗ Y H (F`(E); R) = R[x1, . . . , xn]/ (1 + xi) = c(E) i=1 = R[x1, . . . , xn]/ (ej(x1, . . . , xn) = cj(E) ∀j) where xi = c1(Si) for 1 ≤ i ≤ n − 1 and xn = c1(Qn−1) where the Si’s and Qi’s come from iterating the projectivization construction to acheive the splitting principle. Furthermore, in the special case where our vector bundle is V → {∗}, we get the identity above with c(E) = 1, which says that all the non-trivial elementary symmetric polynomials in x1, . . . , xn would be zero.

15 Proof. Since the flag bundle is obtained by a sequence of n − 1 projectivizations, we can compute first ∗ ∗ H (P(Q1); R) = H (P(E); R)[c(S2), c(Q2)]/(c(S2)c(Q2) = c(Q1)) ∗ = H (M; R)[c(S1), c(Q1), c(S2), c(Q2)]/(c(S1)c(Q1) = c(E), c(S2)c(Q2) = c(Q1)) ∗ = H (M; R)[c(S1), c(S2), c(Q2)]/(c(S1)c(S2)c(Q2) = c(E)) Then, using induction, ∗ ∗ H (P(Qn−2); R) = H (M; R)[c(S1), . . . , c(Sn−1), c(Qn−1)]/(c(S1) ··· c(Sn−1)c(Qn−1) = c(E)) Then, since c(Si) = (1 + xi) and c(Qn−1) = (1 + xn), we are done. 4.11. Definition. Given a manifold M, the Poincaréseries of a manifold M is ∞ X i i Pt(M) := dim H (M; R)t i=0 L∞ More generally, if A = i=0 Ai is a graded algebra over a field K, then the Poincaréseries is given by ∞ X i Pt(A) := (dimK Ai)t i=0

4.12. Remark. Pt(A) is sometimes usually called the “graded dimension of A” by algebraists and so the Poincar´eseries of a manifold is the graded dimension of its cohomology ring. ∗ n−1 n 4.13. Example. Since H (CP ; R) = R[x]/(x ) with deg x = 2, we get immediately that 1 − t2n P ( n−1) = 1 + t2 + ··· + t2(n−1) = t CP 1 − t2 4.14. Lemma. Since ∗ ∼ ∗ ∗ n−1 H (P(E); R) = H (M; R) ⊗ H (CP ; R) as modules, we have that 1 − t2n P ( (E)) = P (M) t P t 1 − t2 Proof. The result follows immediately from dimension counting of the tensor product and the example above. 4.15. Corollary. Let V be a complex complex vector space of dimension n. Then, (1 − t2)(1 − t4) ··· (1 − t2n) P (F`(V )) = t (1 − t2)n and, more generally, if E → M is a rank n vector bundle, then (1 − t2)(1 − t4) ··· (1 − t2n) P (F`(E)) = P (M) t t (1 − t2)n

16 Proof. The ﬂag manifold is constructed by a sequence of projectivizations and, since each time we projectivize a rank k vector bundle we multiply the Poincar´epolynomial by (1 − t2k)/(1 − t2), we get 1 − t2n 1 − t2n 1 − t2n−2 1 − t2 P (F`(E)) = P ( (Q )) = ··· = ··· P (M) t 1 − t2 t P n−3 1 − t2 1 − t2 1 − t2 t 4.16. Remark. One can reformulate the above using notions of “quantum factorial” and “quantum binomial” as follows. n−1 1−t2n (a) Pt(CP ) = 1−t2 =: [n]t2 (b) Pt(F`(V )) = [n]t2 [n − 1]t2 ··· [1]t2 =: [n]t2 !

4.4. The Grassmannian.

4.17. Definition. The Grassmannian of a complex vector space V , denoted Gk(V ) is the set of all subspaces of codimension k in V . 4.18. Example. Note that for V a complex vector space of dimension ∼ n−1 n, Gn−1(V ) = P(V ) = CP . 4.19. Remark. The Grassmannian is a manifold because it can be rep- resented as the space U(n) G (V ) = k U(k) × U(n − k) The argument follows a similar one to 4.6. In Lie theoretic language, a Grassmannian can be thought of as GL(V )/P for P a parabolic subgroup of GL(V ).

4.20. Definition. Given a complex vector space V of dimension n, we deﬁne the following vector bundles over Gk(V ). (a) The universal subbundle which has total space

S := {(x, v) ∈ Gk(V ) × V | v ∈ x}

and thus the fiber of x ∈ Gk(V ) is the plane x defines in V . (b) The product bundle Vˆ := Gk(V ) × V . (c) The universal quotient bundle Q defined by the exact sequence

0 → S → Vˆ → Q → 0

4.21. Proposition. [BT82, Proposition 23.1] The cohomology of the complex Grassmannian Gk(V )has Poincar´epolynomial (1 − t2) ··· (1 − t2n) P (G (V )) = t k (1 − t2) ··· (1 − t2k)(1 − t2) ··· (1 − t2(n−k))

17 Proof. Consider the setup over the Grassmannian S ⊕ Q f ∗Q

G (V ) F (S) F (f ∗Q) k f Then, we consider

A point in F (S) (P,L1 ⊆ · · · ⊆ P ) for (n − k) plane P ⊆ V and flag in P ∗ A point in F (f Q) A point (P,L1 ⊆ · · · ⊆ P ) in F (S) together with a flag in V/P ie (P,L1 ⊆ · · · ⊆ Ln−k−1 ⊆ P ⊆ Ln−k+1 ⊆ · · · ⊆ V ) Thus, F (f ∗Q) is the flag manifold F (V ) and is obtained from the Grass- mannian by 2 flag constructions. Thus, by our computations of the flag manfold Poincarépolynomials 4.15, (1 − t2) ··· (1 − t2(n−k))(1 − t2) ··· (1 − t2k) P (F`(V )) = P (F (f ∗Q)) = P (G (V )) t t t k (1 − t2)n 4.22. Remark. Thus, once again using the “quantum factorial language”, [n]t2 ! n Pt(Gk(V )) = =: [n − k]t2 ![k]t2 ! k t2 4.23. Proposition. [BT82, Proposition 23.2] Let V be a complex vector space of dimension n. (a) As a ring, [c (S), . . . , c (S), c (Q), . . . , c (Q)] H∗(G (V ); ) = R 1 n−k 1 k k R (c(S)c(Q) = 1)

(b) The Chern classes c1(Q), . . . , ck(Q) of the quotient bundle generate ∗ the cohomology ring H (Gk(V )). (c) For a ﬁxed k and a ﬁxed i, there are no polynomial relations of degree i among c1(Q), . . . , ck(Q) if the dimension of V is large enough. To prove this, we will need the following lemma

4.24. Lemma. [BT82, p 297] If I is an ideal in A = R[x1, . . . , xn−k, y1, . . . , yk] generated by the homogeneous terms of

(1 + x1 + ··· + xn−k)(1 + y1 + ··· + yk) − 1 where deg xi = 2i and deg yi = 2i, then the Poincar´eseries of A/I is given by (1 − t2) ··· (1 − t2n) P (A/I) = t (1 − t2) ··· (1 − t2(n−k))(1 − t2) ··· (1 − t2k)

18 Proof of Proposition. Consider once again the setup

S ⊕ Q f ∗Q

G (V ) F (S) F (f ∗Q) = F`(V ) k f Then, we have by repeated application of 4.10 for H∗(F (f ∗Q)) that

∗ ∗ Y H (F`(V )R) = H (F (S); R)[y1, . . . , yk]/ (1 + yj) = c(Q)

∗ Y Y = H (Gk(V ); R)[x1, . . . , xn−k, y1, . . . , yk]/ (1 + xi) = c(S), (1 + yj) = c(Q) but also by the special case of 4.10, we have directly that

∗ Y Y H (F`(V ); R) = R[x1, . . . , xn−k, y1, . . . , yk]/ (1 + xi) (1 + yj) = 1

∗ and so c(S), c(Q) have no other relations besides c(S)c(Q) = 1 in H (Gk(V ; R)), otherwise they would be present in the presentation above. Then, it follows that there is an injection of algebras where the left algebra is thought of as a formal polynomial ring with relations.

R[a1, . . . , an−k, b1, . . . , bk] ∗ ,→ H (Gk(V ); R) Pn−k Pk 1 + i=1 ai 1 + j=1 bj = 1

ai 7→ ci(S)

bj 7→ cj(Q) However, by the lemma above,   2 n R[a1, . . . , an−k, b1, . . . , bk] (1 − t ) Pt   = = Pt(Gk(V )) Pn−k Pk (1 − t2) ··· (1 − t2(n−k))(1 − t2) ··· (1 − t2k) 1 + i=1 ai 1 + j=1 bj = 1 Thus, the injection is an isomorphism by dimension counting, proving part (a). For part (b), we write

N ! (a) 1 1 Y X c(S) = = = yr c(Q) Q(1 + y ) j j j r=0

N since there exists an N such that yj = 0 for all j since the Grassmannian is ﬁnite dimensional. Thus, c(S) can be written in terms of ci(Q)’s and it follows that the ci(S) terms are unnecessary generators. However, this process also induces polynomial relations of degrees 2(n − k + 1),..., 2n among the c1(Q), . . . , ck(Q). So, if 2(n − k + 1) > i, there exists no polynomial relations of degree i among the cj(Q)’s.

19 4.25. Remark. (a) A more careful proof of statement (a) for the cohomology of Gk(V ) with Z coefficeints can be found in [Hat01, pp 435–437]. (b) In these notes, we have chosen to look at Grassmannians with codimension k planes. The whole theory translates into Grassmannians with dimension k planes by dualizing the whole picture. Then, the roles of the universal sub-bundle and universal quotient bundle are reversed. (c) The Grassmannian plays a special role in the theory of fiber bundles because the universal quotient bundle over the Grassmannian is a “universal vector bundle” in the sense that every vector bundle is n a pullback for sufficiently large n in Gk(C ). More precisely, if ξ is a rank k complex vector bundle over a compact base space, then n there exists an n ∈ N and a map f : B → Gk(C ) such that ξ is the pullback under f of the universal quotient bundle. This is essentially [BT82, Proposition 23.9]. This idea is also covered in [MS74, Section 5], although for real vector bundles.

5. Equivariant cohomology and Chern classes (presented by Liron Speyer) In this talk, we will introduce notions of equivariant cohomology and equivariant Chern classes while simultaneously abandoning the “differential” assumptions of the earlier chapters since we will move away from [BT82] as a reference and move instead to lecture notes by Anderson and Fulton.2 If a Lie group G acts on a space X, we want to construct a cohomology theory for X that reflects the G-action. A natural guess might be that H∗(X/G) would reflect this, but if the G-action is not free, X/G could be quite ugly (eg not Hausdorff). So, instead, we need to modify our picture to get a free G-action.

5.1. Definition. Let G be a Lie group acting on the left on a space X. Then, we deﬁne abstractly spaces (a) EG a contractible space on which G acts freely (on the right) and (b) BG, the classifying space, which is given by BG := EG/G = EG/(x = x.g, ∀g ∈ G) (c) EG ×G X := EG × X/((e.g, x) ∼ (e, g.x))

Since EG is contractible, EG ×G X is homotopy-equivalent to X on which G acts freely, which it may not have done on just X.

2Author did not attend this talk.

20 5.2. Definition. The equivariant cohomology of X with respect to a Lie group G is given by i i G HG(X) := H (EG × X) where Hi(−) is ordinary (singular) cohomology. G ∗ ∗ 5.3. Proposition. (a) Since BG = EG× {pt}, HG({pt}) = H (BG). ∗ (b) The (unique) map X → {pt} induces pullback on cohomology HG({pt}) → ∗ HG(X), so the equivariant cohomology of X has the structure of a ∗ HG({pt})-module. × 1 5.4. Example. (a) Let G = C = S . Then, G acts freely on ∞ ∞ C \{0}, which is contractible, so we will take EG = C \{0}. (Remember we do not yet have a way to construct EG in general). ∞ Then, BG = CP and so H∗ ({pt}) = H∗( ∞) = [t] C× CP Z ∞ where t = c1(L) for L the tautological line bundle over CP . Thus, ∗ H × (X) is always a Z[t]-module. C ∞ ∞ (b) Let G = GLn(C). Then, BG = Gr(n, C ) = {W ⊆ C | dim W = n} (we will prove this later). So, then, H∗ ({pt}) = GLn(C) ∗ ∞ H (Gr(n, C )) = Z[t1, . . . , tn] for ti = ci(S) where S is the univer- ∞ sal sub-bundle over Gr(n, C ) (see 4.23). 5.5. Remark. Equivariant cohomology can recover ordinary cohomology ∗ by taking G to be the trivial group. Then, H ({pt}) = Z and we recover ∗ the fact that H (X; Z) always has the structure of a Z-module. 5.6. Definition. (a) A (right) G-bundle is a fiber bundle ρ: E → B with G acting on E on the right preserving fibers. In otherwords, for x ∈ B, y ∈ ρ−1(x), then y.g ∈ ρ−1(x) for all g ∈ G. (b)A principal G-bundle is a G-bundle on which G acts freely and transitively on fibers. 5.7. Remark. It is immediate that a principal G-bundle has fiber ρ−1(x) = G.x ⊆ E since the G-action is transitive and G.x is isomorphic to G as a G-set since the G-action is free. 5.8. Proposition. The bundle EG → BG is a principal G-bundle with the universal property that, if E → B is a principal G-bundle, then there exists a map f : B → BG such that E → B is the pullback bundle of EG → BG. f ∗(EG) = E EG

f B BG In general, EG is infinite-dimensional, so it is not necessarily an algebraic variety. However, we will find finite-dimensional approximation spaces

21 by constructing principal G-bundles EGm → BGm for m ∈ N such that i πi(EGm) = 0 = H (EGm) for 0 < i < k(m) where k(m) → ∞ as m → ∞. For this to work, we need a corollary to Leray-Hirsch 2.9. 5.9. Corollary. (a) If Hi(F ) = 0 for 0 < i ≤ m, then Hi(B) → Hi(E) is an isomorphism for i ≤ m. (b) If E → B and E0 → B0 are two principal right G-bundles with Hi(E) = Hi(E0) = 0 for 0 < i ≤ k, then there is a canonical isomorphism Hi(E ×G X) =∼ Hi(E0 ×G X) i < k Proof. We prove the second part granting the ﬁrst. If we let G act diagonally on E × E0, we get the diagram E × X E × E0 × X E0 × X

E ×G X (E × E0) ×G X E0 ×G X where each vertical map is a G-bundle, the horizontal maps on the left are bundles with fiber E0, and the horizontal maps on the right are bundles with fiber E. Then, the induced maps Hi(E ×G X) Hi((E × E0) ×G X) Hi(E0 ×G X) are isomorphisms for i < k by the first part of the corollary. 5.10. Definition. (a) We say ξ is a G-equivariant complex vector bundle if the total space and base spaces have G-actions and the G-action commutes with the projection map. In other words, for ρ: E → B, we have ρ(g.x) = g.ρ(x) and g.ρ−1(x) ⊆ ρ−1(g.x). (b) To a G-equivariant complex vector bundle ξ given by ρ: E → B, we associate a complex vector bundle ξ0 given by ρ0 : EG ×G E → EG ×G B. This is nonstandard terminology. (c) If ξ is a G-equivariant complex vector bundle over B, then it has equivariant Chern classes G 2i ci (ξ) ∈ HG (B) G 0 0 where ci (ξ) = ci(ξ ) for ξ the associated complex vector bundle to ξ.

5.11. Remark. (a) Note, we can replace EG with EGm for m suf- ﬁciently large. ∼ n (b) If we have a G-equivariant vector bundle E → {pt}, then E = C is an n-dimensional representation of G. In other words, the G-action ∼ n on E = C can be encoded by a group homomorphism ρ: G → n GLn(C) such that, for x ∈ E = C , g.x = ρ(g)x

22 Thus, we will often write Eρ → {pt} to indicate that G acts on E via ρ or that Eρ is the vector space corresponding to representation ρ.

× 5.12. Example. (a) Let G = C act on the line bundle C = L → {pt} via multiplication: g.z = gz. Then, we have the picture

× ∞ × L EG ×C L (C \{0}) ×C L

× ∞ {pt} EG ×C {pt} CP Now, we know we have a map on the total spaces m−1 m EGm × L →CP × C (z1, . . . , zm) × z 7→[z1, . . . , zm] × (z1z, . . . , zmz)

whose image gives the tautological bundle since (z1z, . . . , zmz) ∈ × [z1, . . . , zm]. Now, if we quotient by the C action, we have

(z1g, . . . , zmg) × z [z1, . . . , zm] × (z1z, . . . , zmz)

(z1, . . . , zm) × gz [z1, . . . , zm] × (z1gz, . . . , zmgz) and so our map passes nicely to the quotient, giving us an isomorphism of vector bundles

× ∞ × EG ×C L (C \{0}) ×C L

× ∞ EG ×C {pt} CP

× × C C 2i ∞ Thus, ci (L) = ci(EG× L) ∈ H (CP ) = Z[t] where t = c1(S). × C So, by pulling back over our isomoprhism, we get that ci (L) = t ∈ Z[t]. × (b) More generally, if we let G = C act on the line bundle C = L → a {pt} via g.z = g z for a ∈ Z. Let us denote such a bundle La → {pt}. Then we have the picture

× ∞ × La EG ×C L (C \{0}) ×C L

× ∞ f ∞ {pt} EG ×C {pt} = CP CP where f is the map such that we have a pullback. In particular, we have that Come back to this example!

23 × n m n (c) If we let G = (C ) , we have that EGm = (C \{0}) and BGm = m−1 n (CP ) . Furthermore, we let Lξi be the 1-dimensional represen- G tation of G with character χi(z1, . . . , zn) = zi. Then, C1 (Lξi ) := G ci(EGm × Lξi ) and so, on bundles,

G ∼ ∗ EG × Lξi pi (S) S

∼ m−1 n pi m−1 BGm (CP ) CP

where pi is the projection onto the i-th coordinate. Thus, we get G ∗ ∗ c1(EGm × Lξi ) = c1(pi (S)) = ti ∈ Z[t1, . . . , tn] = HG({pt}). 0 (d) Let G = GLnC. Then, for m ≥ n, we take EGm = Mm,n = {A ∈ Mm,n | A has rank n} with G acting on the right by matrix 0 multiplication. We will take as given that πi(Mm,n) = 0 for 0 < 0 ∼ m i ≤ 2(m − n). Then, BGm = Mm,n/G = Gr(n, C ) by mapping a m matrix A to its image, which deﬁned an n-plane in C . We already know that H∗(Gr(n, V )) is generated by the Chern classes of the tautological subbundle of rank n with relations in ∗ degrees m − n + 1, . . . , m. Thus, HG({pt}) = Z[c1, . . . , cn] since these relations disappear as m → ∞. Now, we have a G-equivariant vector bundle V → {pt} and this yields isomorphisms

G ∼ EGm × V S

∼ m BGm Gr(n, C ) where the map is given by φ × v 7→ φ(v) ∈ im(φ), which all works out since φ · g × v and φ × gv both map to φ(gv). 0 (e) For subgroups G ≤ GLnC, we can use the same EGm = Mm,n. So, × n if we take the torus G = (C ) ⊆ GLnC, then we get 0 × n BGm = Mm,n/(C ) = {(L1,...,Ln) | dim Lk = 1,Li ∩ Lj = {0}, ∀i 6= j}

comes equipped with tautological line bundles L1,...,Ln. Further- ∗ more, c1(Li) = ti ∈ Z[t1, . . . , tn] = HG({pt}) . why? + Similary, if G = B , the upper-triangular matrices in GLnC, then 0 m m Mm,n/G = {0 ⊆ L1 ⊆ L2 ⊆ · · · ⊆ Ln ⊆ C | dim Li = i} = F`(1, . . . , n; C )

where the map is given by sending a matrix A to (L1 ⊆ · · · ⊆ Ln) where Li is the span for the ﬁrst i columns of A. Thus, we can construct a tautological sequence of bundles S1 ⊆ · · · ⊆ Sn and m get that the cohomology ring of F`(1, . . . , n; C ) is generated by ti = c1(Si/Si−1). why? Connect to earlier lecture.

24 5.13. Proposition. Let Lie group G act on X and Lie group G0 act on X0. Furthermore, let φ: G → G0 be a continuous Lie group homomorphism and let f : X → X0 be continuous and equivariant with respect to φ, ie f(g · x) = φ(g) · f(x), x ∈ X, g ∈ G Then, there is a degree-preserving ring homomorphism ∗ 0 ∗ HG0 X → HGX and this map is functorial. Proof. One can ﬁnd a continuous map EG → EG0 that is equivariant for the right actions of G and G0 so that we get commutative diagram EG EG0

BG BG0 In fact, these maps are well-deﬁned up to homotopy, and so we can get an induced map 0 EG ×G X → EG0 ×G X and thus we get our map on cohomology via functoriality. 6. Localization in Equivariant Cohomology (presented by Chris Chung) 6.1. Background. 6.1. Definition. Throughout this talk, we will denote the tautological n−1 line bundle OPn−1 (−1) and its dual OPn−1 (1), dropping the P when it is clear by context. 6.2. Remark. This notation is identifying line bundles with the struc- n−1 ture sheaf of P , denoted OPn−1 , combined with a grading shift or “twist- ing” given by the number in parentheses. It is not important to understand this notion for this talk. × n n 6.3. Example. Let T = (C ) act on C via the standard action. Then, ∗ HT ({pt}) = Z[t1, . . . , tn] where ti = c1(O(−1)). This induces an action of T n−1 n on P = P(C ).

6.4. Proposition. Both OPn−1 (−1) and OPn−1 (1) are T -equivariant line bundle. T 6.5. Definition. Let ζ := c1 (OPn−1 (1)). 6.6. Proposition. ∗ n−1 ∼ n n−1 HT (P ) = Z[t1, . . . , tn][ζ]/(ζ + e1(t)ζ + ··· + er(t)) n ! ∼ Y = Z[t1, . . . , tn][ζ]/ ζ + ti i=1

25 n Proof. Let T act on C → {pt}. Then, we have associated approxi- T n m n mation space E = ETm × C → BTm where ETm = (S ) and BTm = m−1 n T n−1 (CP ) . Thus, we have ETm × P = P(E) and so, abusing notation T ∼ slightly, we have ETm × OPn−1 (1) = OP(E)(1). Next, recall that

∗ ∗ n n−1 H (P(E)) = H (BT )[ζ]/(ζ + c1(E)ζ + ··· + cn(E))

T T n Then, ζ := c1 (OPn−1 (1)) = c1(OP(E)(1)). Thus, we get ci (C ) = ci(E) = ei(t). × n 6.7. Theorem (A Generalization). Let T = (C ) act on vector space V of dimension r by characters (weights) χ1, . . . , χr, ie

t.(v1, . . . , cr) = (χ1(t)v1, . . . , χr(t)vr)

Then, we have that

r ! ∗ Y HT (P(V )) = Z[t1, . . . , tn][ζ]/ (ζ + χi) i=1

T where, in the expression above, we think of χi as the pullback of c1 (Lχi ) across the splitting map.

T So, if r = n and χi = ti = c1 (O(1)), we recover the standard action in the proposition above. Now, consider the setup

X EG ×G X

{pt} BG where G acts on X. Then, ∗ ∗ ∗ (a) We have a canonical map HG(X) → H (X) compatible with HG({pt}) → ∗ ∗ Z. Furthermore, under nice conditions, we get HG(X) → H X is ∗ surjective with kernel given by ker(HG({pt}) → Z). (b) Let XG be the G ﬁxed points of X. Then, the “ﬁxed locus” G ∗ ∗ ∗ G ι: X ,→ X induces a “restriction” ι : HGX → HGX . Further- more, under nice conditions, ι∗ is injective.

6.8. Example. We give an example where these nice condtions are not × met. Let G = C = X with G acting on X via left multiplication. Then,

× 1 × 1 ∞ C × 1 ∞ HC× (C ) = H (S × C ) = H (S ) = 0

1 × 1 1 G but H (C ) = H (S ) = Z. Also, X = ∅. Thus, both nice situations fail.

26 6.2. Restriction and Gysin Maps. Throughout this section, let T = × n ∗ (C ) and ΛT = HT ({pt}) = Z[t1, . . . , tn]. 6.9. Definition. For T acting on X and p ∈ XT a T -action ﬁxed point, let ιp : {p} ,→ X be the inclusion map of this point.

∗ ∗ Thus, we have induced map ιp : HT (X) → ΛT . 6.10. Example. (a) Let E → X be a rank r T -equivariant vector T bundle with p ∈ X and ﬁber at p denoted Ep → {p}. Such a ﬁber is a representation of T with weights, say χ1, . . . , χr. Then, ∗ T T ιp(ci (E)) = ci (Ep) = ei(χ1, . . . , χr) In particular, if i = r, then ∗ T ιp(cr (E)) = χ1 ··· χr n−1 (b) If X = CP and T acts on X by the standard action

(z1, . . . , zn).[x1, . . . , xn] = [z1x1, . . . , znxn]

The ﬁxed points of this action are pi = [0,..., 0, 1, 0,..., 0] where the 1 is in the ith coordinate for i = 1, . . . , n. The ﬁber of O(−1) at pi is given by Ci, the “coordinate line”. One can check that T

acts on O(−1)pi by the character ti and on O(1)pi by the character −1 −ti (where we write characters additively, ie (z1, . . . , zn).x = xzi ). T ∗ Thus, if ζ = c1 (O(1)), then ιpi ζ = −ti. Hence, on cohomology, the map

∗ ∗ n−1 Y ∗ T M ∗ ⊕n ι : HT P = ΛT [ζ]/ (ζ + ti) →HT X = HT (pi) = ΛT T pi∈X ζ 7→(−t1,..., −tn) n−1 is injective. This idea generalized to T acting on P by characters χ1, . . . , χn and the map above would then send ζ 7→ (−χ1,..., −χn). × 2 (c) Let T = C act on P with characters 0, z, 2z, ie 2 z.[x1, x2, x3] = [x1, zx1, z x2] (remember that we are using additive notation for characters.) Then, the ﬁxed points are [1, 0, 0], [0, 1, 0], and [0, 0, 1], which we will call p1, p2, and p3, respectively. Furthermore, every closed T -invariant 1 curve is isomorphic to P and contains two of {p1, p2, p3}. So, for instance, if x1 = 0, then z.[0, 1, b] = [0, z, z2b] = [0, 1, zb]

has character ±z. Similarly, x2 = 0 has character ±2z and x3 = 0 has character ±z. Also, for λ 6= 0, we have a family of conics passing through p1, p3 via 2 x2 − λx1x3 = 0

27 with character ±t. Now, from above work, we see

∗ 2 ⊕3 HT P = ΛT [ζ]/(ζ(ζ + z)(ζ + 2z)) ,→ ΛT ζ 7→ (0, −z, −2z)

Furthermore, one can work out that this image consists of triples (u1, u2, u3) satisfying conditions (i) x | u2 − u1, 2z | u3 − u1, z | u3 − u2 2 (ii)2 z | u1 − 2u2 + u3

6.11. Definition. For X,Y non-singular G-varieties and a proper equivariant map f : Y → X, we can deﬁne

i i+2d f∗ : HG(Y ) → HG (X) where d = dim X − dim Y (not necessarily non-negative) called the Gysin pushforward.

We will not actually construct this map here. For the curious, see [Ful07, Appendix A].

6.12. Proposition (Properties of Gysin maps). (a) Functoriality: given q : Z → Y another proper equivariant map,

(f ◦ q)∗ = f∗ ◦ q∗

(b) Naturality: given g : Y 0 → X another proper equivariant map and 0 ∗ g(Y ) ∩ f(Y ) = ∅, then g ◦ f∗ = 0. (c) Closed embedding: Given a G-invariant closed embedding ι: Y,→ X of codimension d, then we have induced map

j j+2d ι∗ : HG(Y ) → HG (X) that satisﬁes G G (i) ι∗(1) = [Y ] = [EG × Y ] ∗ ∗ G (ii) (self-intersection), for α ∈ HG(Y ) we have ι ι∗(α) = cd (NY |X )α. ∗ In other words, ι ι∗ is multiplication by the Euler class of the normal bundle. normal or conor- (d) Equivariant integration: given X a complete nonsingular variety of mal? dimension n, then for ρ: X → {pt},

j j−2n ρ∗ : HT (X) → HT ({pt}) 1 6.13. Example. Let T act on P with weights χ1 6= χ2 so

z.[x1, x2] = [χ1(z)x1, χ2(z)x2]

Set χ := χ1 − χ2, p1 = [0, 1], and p2 = [1, 0].

28 1 Tp1 P z.[a, 1] = [χ1(z)a, χ2(z)] = [(χ − χ )(z)a, 1] [1, 0] 1 2 = [χ(z)a, 1] 1 So, Tp1 P has weight χ. Similarly, 1 Tp2 P has weight −χ. [0, 1]

∗ ∗ 1 From an earlier example 6.10, ιp1 (ζ) = −χ1 = χ − χ2. So, in HT (P ), T T [p1] = ζ + χ2, [p2] = ζ + χ1 n−1 The general consequence (for P ) is that T n−1 ∗ T Y cn (Tpi (P )) = ιpi ([pi] ) = (χj − χi) j6=i

6.3. Localization. Let T act on X, a nonsingular variety with XT ﬁnite. Consider ∗ M ∗ T ι∗ ∗ ι M ΛT = HT (X ) → H (X) → ΛT p∈XT p∈XT

∗ ⊕|XT | ⊕|XT | By naturality (6.12(b)), ι ◦ ι∗ :Λ → Λ is diagonal and is multi- T T plication by cn (TpX) on the summand for p ∈ X . 6.14. Theorem (First Locatlization Theorem). Let S be a multiplica- Q T tively close set containing c := p∈XT cn (TpX). −1 ∗ −1 ∗ −1 ∗ T ∗ ∗ (a) The map S ι : S HT X → S HT X is surjective and coker(ι : HT X → ∗ T HT X ) is annihilated by c. ∗ (b) If, in addition, HT X is a free ΛT -module with rank less than or equal to |XT |, then S−1ι∗ is an isomorphism.

∗ −1 ∗ Proof. For the ﬁrst part, observe det(ι ◦ ι∗) = c and S (ι ◦ ι∗) = −1 ∗ −1 (S ι ) ◦ (S ι∗). After localizing at S, which contains c, we get that c is a −1 ∗ −1 ∗ unit, giving us that S (ι ◦ ι∗) is surjective. Thus, S ι is surjective. −1 The second part follows since S ΛT is Noetherian. One may wonder when the conditions in the 6.14(b) hold.

29 ∗ 6.15. Definition. we say that X is equivariantly formal if HT (X) is a ∗ free ΛT -module and has a basis that restricts to a Z-basis for H (X). 6.16. Proposition. If X is a non-singular projective variety with XT ﬁnite, then X is equivariantly formal. ∗ ∗ ∗ 6.17. Corollary. For such a T -variety, HT (X) H (X) and HT (X) ,→ ∗ T HT (X ) 6.18. Remark. This corollary is peculiar to T -actions. 6.19. Theorem (Atiyah-Bott-Berline-Vergne, ABBV Integration For- mula). Let X be a compact nonsingular T -variety of dimension n, XT ﬁnite. Let ρ: X → {pt} and thus j j−2n ρ∗ : HT (X) → HT ({pt}) ∗ Then, for any α ∈ HT (X), ∗ X ιpα ρ∗α = cn(TpX) p∈XT Proof. ∗ T ι∗ ∗ ι∗ ∗ T L ∗ HT (X ) HT (X) HT (X ) = p∈XT HT ({p})

ρ∗

L ∗ addition ∗ p∈XT HT ({p}) HT ({pt}) where the square commutes by the functoriality of the Gysin map for the composition {p} ,→ X {pt}. It suﬃces to show the theorem after we Q T ∗ invert c = p∈XT cn (TpX) so that ι , ι∗ become isomorphisms by 6.14. So, we reduce to α = (ιp)∗(β) for ﬁxed p. Then,

ρ∗(α) = ρ∗(ιp)∗(β) = (ρ ◦ ιp)∗(β) = β On the other hand, ∗ ∗ X ιq(ιp)∗(β) ιp(ιp)∗(β) T = T = β cn (TqX) cn (TpX) q∈XT n−1 6.20. Example. Let T act on P with distinct characters χ1, . . . , χn k 2k and ﬁxed points p1, . . . , pn. For ζ ∈ HT (X), we get ( k 0 if k < n − 1 ρ∗(ζ ) = 1 if k = n − 1 However, by ABBV, n k ( k X (−χi) 0 if k < n − 1 ρ∗(ζ ) = = Q (χ − χ ) 1 if k = n − 1 i=1 j6=i j i

30 7. Schubert Calculus (presented by Weinan Zhang) This presentation mostly follows the ﬁrst section of [Bri05] with some additional proofs and examples.

7.1. The Grassmannian.

n 7.1. Definition. Let V = C . Then, we will deﬁne the Grassmannian of V to be Grd(V ) := {E ⊆ V | dim W = d}. 7.2. Proposition. There is an embedding of the Grassmannian, called the Pl¨ucker embedding given by

d Grd(V ) → P ∧ V

E = hv1, . . . , vdi 7→ [v1 ∧ · · · ∧ vd]

Thus, Grd(V ) has the structure of a projective variety.

7.3. Definition. Let (e1, . . . , en) be the standard basis of V . For I = (i1, . . . , id), we deﬁne

EI := hei1 , . . . , eid i 7.4. Definition. Given the subgroup of upper triangular matrices B ≤ GLnC (a Borel subgroup),

(a) the Schubert cells in Grd(V ) are CI := BEI (b) the Schubert varieties in Grd(V ) are XI := CI .

7.5. Remark. Grd(V ) is a disjoint union of Schubert cells if one insists that i1 < i2 < ··· < id. This is a straightforward consequence of the next proposition.

7.6. Example. (a) Let d = 1. Then Gr1(V ) = P(V ), Cj = Bej, ∼ j−1 and Xj = P . So, we have

X1 ⊆ X2 ⊆ · · · ⊆ Xn = P(V )

(b) Notice that, for any d ≥ 1, X1,2,...,d = C1,2,...,d = BE1,2,...,d = {E1,2,...,d} = {E1,2,...,d} is a single point in the Grassmannian. In an opposite scenario, Xn−d+1,n−d+2,...,n = Grd(V ).

7.7. Proposition. Let I = (i1, . . . , id) with i1 < i2 < ··· < id. ∼ |I| Pd (a) CI = C are locally closed subsets of Grd(V ) where |I| := j=1(ij− j). (b) The XI are irreducible closed subvarieties of Grd(V ). (c) E ∈ CI ⇐⇒ dim(E ∩ he1, . . . , eji) = |{k | 1 ≤ k ≤ d, ik < j}|. (d) E ∈ XI ⇐⇒ dim(E ∩ he1, . . . , eji) ≥ |{k | 1 ≤ k ≤ d, ik < j}|. S (e) XI = J≤I CJ where “≤” means jk ≤ ik for all k.

31 Proof. Take B = UT where U is the subgroup of unipotent matrices and T is the subgroup of diagonal matrices. Then, the stabalizer of EI is

UEI := {A ∈ U | aij = 0 if i 6∈ I and j ∈ I} Let us also deﬁne its logical complement, I U := {A ∈ U | aij = 0 if i ∈ I or j 6∈ I} I So, U = U × UEI . Both T and UEI ﬁx EI and since any g ∈ GLnC can be I decomposed in U × UEI × T , we need only concern ourselves with matrices lying in U I . Now, the map I U → Grd(V ) = G/P

g 7→ gEI ∼ |I| is a locally closed embedding with image CI , so CI = C . The second part follows because the closure of an irreducible set is irreducible and CI is irreducible. I Since U provides an embedding of CI with respect to the standard basis, we see that E ∈ CI (under this embedding) will take the form  ∗ · · · 1 0 ··· 0 ···   ∗ · · · 0 ∗ · · · 1 0 ···    E = rowspace  . . . .   . . . .  ∗ · · · 0 ∗ · · · 0 ··· 1 0 ··· 0 where the rightmost 1 in row j occurs in column ij. Thus, this immediately yields our alternative characterization of the elements of CI . However, we need not be so stringent with respect to our embedding, so if we realize instead our E ∈ CI with the form  ∗ · · · ∗ 0 ··· 0 ···   ∗ · · · ∗ ∗ · · · ∗ 0 ···    E = rowspace  . . . .   . . . .  ∗ ··· ∗ ∗ ··· ∗ ··· ∗ 0 ··· 0 where the rightmost ∗ in row j occurs in column ij and is nonzero but is not necessarily equal to 1. Then, it becomes clear that CJ ∈ XI = CI if jk ≤ ik for all k, which also gives us characterization (d) of XI . 7.8. Example. Some elements of the proof are easier seen by example. For example, if I = (2, 4) and dim V = 4, then  1 ∗ 0 ∗  (2,4)  0 1 0 0  U =    0 0 1 ∗  0 0 0 1

32 4 Then, to see the embedding CI ,→ Gr2(C ),   1 a1,2 0 a1,4  0 1 0 0  4 ∼ 3   he2, e4i = ha1,2e1 + e2, a1,4e1 + a3,4e3 + e4i ∈ G2(C ) = C  0 0 1 a3,4  0 0 0 1 We can rewrite this as ∗ 1 0 0 E ∈ C(2,4) ⇐⇒ E = rowspace ∗ 0 ∗ 1 Then, we see that  dim(E ∩ he1i) = 0  dim(E ∩ he1, e2i) = 1 dim(E ∩ he1, e2, e3i) = 1  dim(E ∩ he1, e2, e3, e4i) = 2 From looking at this example, one may notice that the proposition forces these dimension intersections to jump by at most 1 and that those jumps happen precisely at the indices 2 and 4. 7.9. Remark. Note, there is a correspondence between the indices I with strictly increasing entries and partitions λ = (λ1, . . . , λd) where λj = ij − j (listed in increasing order). Furthermore, such a partition must have length ≤ d and each part must be bounded above by n − d. Also, many texts take the third and fourth parts of this proposition to define the Schubert cells and varieties with respect to a fixed flag.

7.10. Definition. For a partition λ, we will deﬁne Xλ := XI under the correspondence described in the remark above.

7.2. Flag Varieties. Recall from earlier sections that F`(V ) is the collection of all ﬂags. In this section, we give F`(V ) the structure of a projective variety (instead of a manifold) which is realized via n n Y Y i N F`(V ) ⊆ Gri(V ) ⊆ P(∧ V ) ⊆ P i=0 i=0

GLn acts on F`(V ) (for dim V = n) and, since the standard Borel subgroup of GLn stabalizes the coordinate ﬂag, 0 ⊆ he1i ⊆ he1, e2i ⊆ · · · ⊆ V , we can identify F`(V ) = GLn/B.

7.11. Definition. For w ∈ Sn, (a) We deﬁne the ﬂag

Fw := 0 ⊆ hew(1)i ⊆ hew(1), ew(2)i ⊆ · · · ⊆ V

(b) We deﬁne Schubert cells in F`(V ) by Cw := BFw = BwB/B. (c) We deﬁne Schuber varieties in F`(V ) by Xw = Cw.

33 7.12. Remark. To state the second equality of part (b), we made use of the general “Bruhat decomposition” G G = BσB− σ∈W for G any connected, reductive algebraic group (eg G = GLn). In fact, if G = GLn, this is precisely the statement of the existence of an LU-decomposition. ∼ `(w) 7.13. Proposition. (a) Cw = C are locally closed subsets in F`(V ). (b) Xw are irreducible subvarieties in F`(V ). S (c) Xw = v≤w Cv where ≤ is the (strong) Bruhat order.

Proof. We want to show Fv ∈ Xw if v ≤ w. Without loss of generality, assume v = w · (j, k) such that j < k and w(j) > w(k). We then deﬁne two bases   f := e if i 6= j, k f˜ = e = e if i 6= j, k  i w(i)  i w(i) v(i) 1 ˜ fj := ew(j) + t ew(k) , fj = ew(k) + tew(j) = ev(j) + tev(k)   ˜ fk := ew(k) fk = ew(j) = ev(k)

Let F (t) be the flag spanned by f1, . . . , fn or equivalently spanned by f˜1,..., fñ. Then, we note F (t) ∈ Xw by the first description. The second description gives us that t→0 F (t) −→ Fv =⇒ Fw ∈ Xw since Xw is closed. Thus, Cv ⊆ Xw if v ≤ w. Show at least the other direc- tion. 7.14. Definition. For w ∈ Sn, w − (a) we define the opposite Schubert cell C := B Fw = w◦Cw◦w where − w◦ is the longest element of Sn and B is the group of all lower − triangular matrices, ie B = w◦Bw◦. w w (b) we define the opposite Schubert variety X := C = w◦Xw◦w. w S v 7.15. Corollary. (a) X = v≥w C (b) codim(Xw) = `(w)

w◦ 7.16. Remark. Under these definitions, Xw◦ = F`(V ) and X = {∗}. v 7.17. Theorem. Xw ∩ X 6= ∅ ⇐⇒ v ≤ w. v Proof. If v ≤ w, then Fσ ∈ Xw ∩ X for all v ≤ σ ≤ w. v v − Suppose Xw ∩ X 6= ∅. Then, Xw ∩ X is stabalized by B ∩ B = T . Then, since the flags Fw for w ∈ Sn are precisely the fixed points of the action of T on F`(V ) (by direct computation), it must be that there is a σ v such that Fσ ∈ Xw ∩ X =⇒ v ≤ σ ≤ w.

34 v 7.18. Remark. In fact, dim(Xw ∩ X ) = `(w) − `(v). In particular, this w means Xw ∩ X = {Fw} since this intersection is transverse. 7.19. Definition. Let M be an oriented compact conected n-dimensional real manifold. Then, the fundamental class of M (in homology) is a canon- ∼ ical generator of Hn(M) = Z. Recall that the fundamental class gives us an isomorphism between cohomology and homology via the cap product j H (X) → Hn−j(X) α 7→ α ∩ [X] This then allows us to have the following deﬁnition.

7.20. Definition. Let m = dimR F`(V ) = n(n − 1) for dim V = n. (a) Given Schubert variety Xw ⊆ F`(V ), we deﬁne its Schubert class m−2`(w) [Xw] ∈ H to be the Poincare dual to the representative in homology of Xw. In other words, it is given by sending the unique generator of H2`(w)(Xw) through these two maps: ι∗ m−2`(w) H2`(w)(Xw) → H2`(w)(F`(V )) → H (F`(V )) (b) Given an opposite Schubert variety Xw, we deﬁne its Schubert class [Xw] ∈ H2`(w) to be its representative in cohomology following the same procedure as above. w ∗ 7.21. Proposition. Both {[Xw]} and {[X ]} form Z-bases of H (F`(V )). Proof. Since F`(V ) is a disjoint union of its Schubert classes and the real dimensions of all its cells are concentrated in even degree, its integral homology groups will not exhibit any torsion. Thus, the Schubert classes form a basis of the integral cohomology which is a free Z-module with rank n!.

35 Bibliography

[BT82] R. Bott and L. W. Tu, Diﬀerential Forms in Algebraic Topology, 1982. [Bri05] M. Brion, Lectures on the Geometry of Flag Varieties, Topics in Cohomological Studies of Algebraic Varieties (2005), 33–85. [Ful07] W. Fulton, Equivariant Cohomology in Algebraic Geometry (2007). Notes by Dave Anderson. Available at https://people.math.osu.edu/anderson.2804/ eilenberg/. [Hat01] A. Hatcher, Algebraic Topology, 2001. [MS74] J. W. Milnor and J. D. Stasheﬀ, Characteristic Classes, 1974.