The Riemann-Roch Theorem
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1 The Riemann-Roch Theorem Paul Baum Penn State TIFR Mumbai, India 20 February, 2013
2 THE RIEMANN-ROCH THEOREM Topics in this talk : 1. Classical Riemann-Roch 2. Hirzebruch-Riemann-Roch (HRR) 3. Grothendieck-Riemann-Roch (GRR) 4. RR for possibly singular varieties (Baum-Fulton-MacPherson)
3 CLASSSICAL RIEMANN - ROCH M compact connected Riemann surface genus of M = # of holes = 1 2 [rankh 1(M; Z)]
4 D a divisor of M D consists of a finite set of points of M p 1, p 2,..., p l and an integer assigned to each point n 1, n 2,..., n l Equivalently D is a function D : M Z with finite support Support(D) = {p M D(p) 0} Support(D) is a finite subset of M
5 D a divisor on M deg(d) := p M D(p) Remark D 1, D 2 two divisors D 1 D 2 iff p M, D 1 (p) D 2 (p) Remark D a divisor, D is ( D)(p) = D(p)
6 Example Let f : M C { } be a meromorphic function. Define a divisor δ(f) by: 0 if p is neither a zero nor a pole of f δ(f)(p) = order of the zero if f(p) = 0 (order of the pole) if p is a pole of f
7 Example Let ω be a meromorphic 1-form on M. Locally ω is f(z)dz where f is a (locally defined) meromorphic function. Define a divisor δ(ω) by: 0 if p is neither a zero nor a pole of ω δ(ω)(p) = order of the zero if ω(p) = 0 (order of the pole) if p is a pole of ω
8 D a divisor on M { } meromorphic functions H 0 (M, D) := δ(f) D f : M C { } { } meromorphic 1-forms H 1 (M, D) := ω on M δ(ω) D Lemma H 0 (M, D) and H 1 (M, D) are finite dimensional C vector spaces dim C H 0 (M, D) < dim C H 1 (M, D) <
9 Theorem (RR) Let M be a compact connected Riemann surface and let D be a divisor on M. Then: dim C H 0 (M, D) dim C H 1 (M, D) = d g + 1 d = degree (D) g = genus (M)
10 HIRZEBRUCH-RIEMANN-ROCH M non-singular projective algebraic variety / C E an algebraic vector bundle on M E = sheaf of germs of algebraic sections of E H j (M, E) := j-th cohomology of M using E, j = 0, 1, 2, 3,...
11 LEMMA For all j = 0, 1, 2,... dim C H j (M, E) <. For all j > dim C (M), H j (M, E) = 0. χ(m, E) := n = dim C (M) n ( 1) j dim C H j (M, E) j=0 THEOREM[HRR] Let M be a non-singular projective algebraic variety / C and let E be an algebraic vector bundle on M. Then χ(m, E) = (ch(e) T d(m))[m]
12 Hirzebruch-Riemann-Roch Theorem (HRR) Let M be a non-singular projective algebraic variety / C and let E be an algebraic vector bundle on M. Then χ(m, E) = (ch(e) T d(m))[m]
13 EXAMPLE. Let M be a compact complex-analytic manifold. Set Ω p,q = C (M, Λ p,q T M) Ω p,q is the C vector space of all C differential forms of type (p, q) Dolbeault complex 0 Ω 0,0 Ω 0,1 Ω 0,2 Ω 0,n 0 The Dirac operator (of the underlying Spin c manifold) is the assembled Dolbeault complex + : j Ω 0, 2j j 0, 2j+1 Ω The index of this operator is the arithmetic genus of M i.e. is the Euler number of the Dolbeault complex.
14 K-theory and K-homology in algebraic geometry Let X be a (possibly singular) projective algebraic variety / C. Grothendieck defined two abelian groups: Kalg 0 (X) = Grothendieck group of algebraic vector bundles on X. K alg 0 (X) = Grothendieck group of coherent algebraic sheaves on X. Kalg 0 (X) = the algebraic geometry K-theory of X contravariant. K alg 0 (X) = the algebraic geometry K-homology of X covariant.
15 K-theory in algebraic geometry Vect alg X = set of isomorphism classes of algebraic vector bundles on X. A(Vect alg X) = free abelian group with one generator for each element [E] Vect alg X. For each short exact sequence ξ 0 E E E 0 of algebraic vector bundles on X, let r(ξ) A(Vect alg X) be r(ξ) := [E ] + [E ] [E]
16 K-theory in algebraic geometry R A(Vect alg (X)) is the subgroup of A(Vect alg X) generated by all r(ξ) A(Vect alg X). DEFINITION. K 0 alg (X) := A(Vect algx)/r Let X, Y be (possibly singular) projective algebraic varieties /C. Let f : X Y be a morphism of algebraic varieties. Then have the map of abelian groups f : K 0 alg (X) K0 alg (Y ) [f E] [E] Vector bundles pull back. f E is the pull-back via f of E.
17 K-homology in algebraic geometry S alg X = set of isomorphism classes of coherent algebraic sheaves on X. A(S alg X) = free abelian group with one generator for each element [E] S alg X. For each short exact sequence ξ 0 E E E 0 of coherent algebraic sheaves on X, let r(ξ) A(S alg X) be r(ξ) := [E ] + [E ] [E]
18 K-homology in algebraic geometry R A(S alg (X)) is the subgroup of A(S alg X) generated by all r(ξ) A(S alg X). DEFINITION. K alg 0 (X) := A(S alg X)/R Let X, Y be (possibly singular) projective algebraic varieties /C. Let f : X Y be a morphism of algebraic varieties. Then have the map of abelian groups f : K alg 0 (X) K alg 0 (Y ) [E] Σ j ( 1) j [(R j f)e]
19 f : X Y morphism of algebraic varieties E coherent algebraic sheaf on X For j 0, define a presheaf (W j f)e on Y by U H j (f 1 U; E f 1 U) U an open subset of Y Then (R j f)e := the sheafification of (W j f)e
20 f : X Y morphism of algebraic varieties f : K alg 0 (X) K alg 0 (Y ) [E] Σ j ( 1) j [(R j f)e]
21 SPECIAL CASE of f : K alg 0 (X) K alg 0 (Y ) Y is a point. Y = ɛ: X is the map of X to a point. Kalg 0 ( ) = Kalg 0 ( ) = Z ɛ : K alg 0 (X) K alg 0 ( ) = Z ɛ (E) = χ(x; E) = Σ j ( 1) j dim C H j (X; E)
22 X non-singular = K 0 alg (X) = K alg 0 (X) Let X be non-singular. Let E be an algebraic vector bundle on X. E denotes the sheaf of germs of algebraic sections of E. Then E E is an isomorphism of abelian groups Kalg 0 (X) Kalg 0 (X) This is Poincaré duality within the context of algebraic geometry K-theory&K-homology.
23 Grothendieck-Riemann-Roch Theorem (GRR) Let X, Y be non-singular projective algebraic varieties /C, and let f : X Y be a morphism of algebraic varieties. Then there is commutativity in the diagram : K 0 alg (X) K0 alg (Y ) ch( ) T d(x) ch( ) T d(y ) H (X; Q) H (Y ; Q)
24 WARNING!!! The horizontal arrows in the GRR commutative diagram K 0 alg (X) K0 alg (Y ) ch( ) T d(x) ch( ) T d(y ) are wrong-way (i.e. Gysin) maps. H (X; Q) H (Y ; Q) K 0 alg (X) = K alg 0 (X) f K alg 0 (Y ) = K 0 alg (Y ) H (X; Q) = H (X; Q) f H (Y ; Q) = H (Y ; Q) Poincaré duality Poincaré duality
25 K-homology is the dual theory to K-theory. How can K-homology be taken from algebraic geometry to topology? There are three ways in which this has been done: Homotopy Theory K-homology is the homology theory determined by the Bott spectrum. Geometric Cycles K-homology is the group of K-cycles. C* algebras K-homology is the Kasparov group KK (A, C).
26 Riemann-Roch for possibly singular complex projective algebraic varieties Let X be a (possibly singular) projective algebraic variety / C Then (Baum-Fulton-MacPherson) there are functorial maps α X : K 0 alg (X) K0 top(x) K-theory contravariant natural transformation of contravariant functors β X : K alg 0 (X) K top 0 (X) K-homology covariant natural transformation of covariant functors Everything is natural. No wrong-way (i.e. Gysin) maps are used.
27 α X : K 0 alg (X) K0 top(x) is the forgetful map which sends an algebraic vector bundle E to the underlying topological vector bundle of E. α X (E) := E topological
28 Let X, Y be projective algebraic varieties /C, and let f : X Y be a morphism of algebraic varieties. Then there is commutativity in the diagram : K 0 alg (X) K0 alg (Y ) α X α Y K 0 top(x) K 0 top(y ) i.e. natural transformation of contravariant functors
29 Let X, Y be projective algebraic varieties /C, and let f : X Y be a morphism of algebraic varieties. Then there is commutativity in the diagram : K 0 alg (X) K0 alg (Y ) α X α Y K 0 top(x) K 0 top(y ) ch ch H (X; Q) H (Y ; Q)
30 Let X, Y be projective algebraic varieties /C, and let f : X Y be a morphism of algebraic varieties. Then there is commutativity in the diagram : K alg 0 (X) K alg 0 (Y ) β X β Y K top 0 (X) Ktop 0 (Y ) i.e. natural transformation of covariant functors
31 Let X, Y be projective algebraic varieties /C, and let f : X Y be a morphism of algebraic varieties. Then there is commutativity in the diagram : K 0 alg (X) K0 alg (Y ) α X α Y K 0 top(x) K 0 top(y ) ch ch H (X; Q) H (Y ; Q)
32 Let X, Y be projective algebraic varieties /C, and let f : X Y be a morphism of algebraic varieties. Then there is commutativity in the diagram : K alg 0 (X) K alg 0 (Y ) β X β Y K top 0 (X) Ktop 0 (Y ) ch ch H (X; Q) H (Y ; Q)
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