Recall for an n n matrix A = (a ij ), its trace is defined by. a jj. It has properties: In particular, if B is non-singular n n matrix,

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1 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! x ! x 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! A ! A e BAB 1 = I + BAB ! (BAB 1 ) ! (BAB 1 ) = B(I + A + 1 2! A ! 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)

2 In terms of Chern classes, the total Chern character can be expressed by ch(e) = k + c 1 (E) (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) = c 1(E) (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

3 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

4 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 It was made public at the initial Bonn Arbeitstagung, in 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

5 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