Chern characters Recall for an n n matrix A = (a ij ), its trace is defined by tr(a) = n a jj. j=1 It has properties: tr(a + B) = tr(a) + tr(b), tr(ab) = tr(ba). In particular, if B is non-singular n n matrix, tr(bab 1 ) = tr ( (BA)B 1) = tr ( B 1 (BA) ) = tr(a). This means that we can define trace for a linear transformation L : V V, x xa. Recall the exponential series e x = 1 + x + 1 2! x2 + 1 3! x3 +... which converges for every complex number x. If A is any n n matrix, Then for any non-singular n n matrix B, e A = I + A + 1 2! A2 + 1 3! A3 +... e BAB 1 = I + BAB 1 + 1 2! (BAB 1 ) 2 + 1 3! (BAB 1 ) 3 +... = B(I + A + 1 2! A2 + 1 3! A3 +...)B 1 = Be A B 1 so that tr(e A ) = tr(e BAB 1 ). It gives invariant polynomial of degree k: tr(e A ) = P 0 (A) + P 1 (A) + p 2 (A) +... Then we define the kth Chern character ( ) ch k (E) := P 1 k 2π Ω and the total Chern character where ch 0 (E) = r = rank(e). ch(e) = ch 0 (E) + ch 1 (E) +... 116
In terms of Chern classes, the total Chern character can be expressed by ch(e) = k + c 1 (E) + 1 2 (c 1(E) 2 2c 2 (E)) +... Todd classes We can define invariant polynomial in a different way: det(tb) det(i e tb ) = k ˆP k (B)t k Then we define the Todd class ( ) td k (E) = ˆP 1 k 2π Ω and the total Todd class td(e) = td 0 (E) + td 1 (E) + td 2 (E) +... where k = rank(e). In terms of Chern classes, the total Todd character can be expressed by td(e) = 1 + 1 12 c 1(E) + 1 12 (c 1(E) 2 + c 2 (E)) +... Splitting principle Let ξ : E X be a complex vector bundle of rank n over a manifold X. There exists a space Y = Fl(E), called the flag bundle associated to E, and a map p : Y X such that 1. the induced cohomology homomorphism is injective, and 2. the pull-back bundle p ξ : p E Y breaks up as a direct sum of line bundles: p (E) = L 1 L 2... L l. By Property 18.2, c(e) = c(l 1 L 2... L l ) = c(l 1 ) c(l 2 )... c(l l ). This means the construction of Chern class for vector bundles can be reduced into construction of Chern class for line bundles. 117
Characterizing Chern class by axioms In fact, Chern class can be characterized by the following axiomes: Given a complex vector bundle V over a topological space X, the Chern classes of V are a sequence of elements of the cohomology of X. The kth Chern class of V, which is usually denoted c k (V ), is an element of H 2k (X, Z). The Chern classes satisfy the following four axioms: 1. c 0 (V ) = 1 for all V. 2. (Functoriality) For every smooth map f : Y X, c(f E) = f (c(e)). 3. (Whitney Sum Formula) If V and W are complex vector bundles over X, then c(v W) = c(v ) c(w). 4. (Normalization) The total Chern class of the tautological line bundle O CP n( 1) over CP n is 1 H, where H is Poincaré-dual to the hyperplane CP n 1 CP n, i.e., H is the generator of H 2 (CP, Z). [Example] There is an exact sequence of vector bundles 0 C O CP n(1) C n+1 T CP n 0 over CP n. From the Splitting Principle and the Whitney Sum Formula, c(t CP n) = c(o CP n(1) C n+1 ) = (1 + H) n+1. For example, c(t CP 1) = 1 + 2H and c(t CP 2) = 1 + 3H + 3H 2. Remark There is a one-to-one correspondance between equivalence classes of complex rank k vector bundles over a manifold M and homotopy classes of maps φ : M G k where G k is Grassmannian. The correspondance associates the bundle E = φ U k where U k is the unitary group to a classifying map φ. The Chern classes of E may now be defined as the pullbacks of the universal Chern classes under φ. 118
19 Hirzebruch-Riemann-Roch Theorem Hirzebruch-Riemann-Roch theorem over complex curves Hirzebruch-Riemann- Roch theorem applies to any holomorphic vector bundle E on a compact complex manifold X. It asserts χ(e) = ch(e)td(e) where n = dim X, χ(e) = h 0 (X, E) h 1 (X, E) + h 2 (X, E) h 3 (X, E) +..., h j (X, E) := dim H j (X, E), is the Eulier characteristic, ch(e) is the total Chern character of E and td(e) is the total Todd class of E. Riemann-Roch theorem over complex curves When dim X = 1 and E = O(D) for a divisor D, Hirzebruch-Riemann-Roch theorem becomes h 0 (X, O(D)) h 1 (X, O(D)) = c 1 (O(D)) + c 1(T X ), 2 which is equivalent to the calssical Riemann-Roch theorem. c 1 (T X ) = 2 2g where g is genius. Also, if dim X = 1 and E is any holomorphic vector bundle over X, we have Weil s Riemann-Roch theorem: where g is the genus. h 0 (X, E) h 1 (X, E) = c 1 (E) + rank(e)(1 g) Hirzebruch-Riemann-Roch theorem for line bundles When E is a holomorphic line bundle O(L k) over a compact complex manifold X of dimension n, it produces the Hilbert polynomial χ(x, L k ) = P(k) which is a polynomial of degree n with leading term k n n! c 1(L) n. (71) Remark Grothendieck s version of the Riemann-Roch theorem was originally conveyed in a letter to Serre around 1956-7. It was made public at the initial Bonn Arbeitstagung, in 1957. Serre and Armand Borel subsequently organized a seminar at Princeton to understand it. The final published paper was in effect the Borel-Serre exposition. 119
The significance of Grothendieck s approach rests on several points. First, Grothendieck changed the statement itself: the theorem was, at the time, understood to be a theorem about a variety, whereas after Grothendieck, it was known to essentially be understood as a theorem about a morphism between varieties. In short, he applied a strong categorical approach to a hard piece of analysis. Moreover, Grothendieck introduced K-groups which paved the way for algebraic K theory. Let X be a smooth quasi-projective scheme over a field. Under these assumptions, the Grothendieck group K 0 (X) of complexes of coherent sheaves is canonically isomorphic to the Grothendieck group K(X) of complexes of finite-rank vector bundles. Using this isomorphism, consider the Chern character (a rational combination of Chern classes) as a functorial transformation Now consider a proper morphism f : X Y between smooth quasi-projective schemes and a bounded complex of sheaves F. The Grothendieck-Riemann-Roch theorem relates the push forward maps with the the formula f! : K 0 (X) K 0 (Y ) ch(f! F ) = f (ch(f )td(x)). Here td(x) is the Todd genus of (the tangent bundle of) X. 120