Đề tài "Quantum Riemann-Roch, Lefschetz and Serre " pptx

Annals of Mathematics Quantum Riemann-Roch, Lefschetz and Serre By Tom Coates and Alexander Givental* Annals of Mathematics, 165 (2007), 15–53 Quantum Riemann-Roch, Lefschetz and Serre By Tom Coates and Alexander Givental* To Vladimir Arnold on the occassion of his 70 th birthday Abstract Given a holomorphic vector bundle E over a compact K¨ahler manifold X, one defines twisted Gromov-Witten invariants of X to be intersection numbers in moduli spaces of stable maps f :Σ→ X with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle H 0 (Σ,f ∗ E)  H 1 (Σ,f ∗ E). Using the formalism of quantized quadratic Hamiltonians [25], we express the descendant potential for the twisted theory in terms of that for X. This result (Theorem 1) is a consequence of Mumford’s Grothendieck-Riemann-Roch theorem applied to the universal family over the moduli space of stable maps. It determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invari- ants. When E is concave and the C × -equivariant inverse Euler class is chosen as the characteristic class, the twisted invariants of X give Gromov-Witten invariants of the total space of E. “Nonlinear Serre duality” [21], [23] expresses Gromov-Witten invariants of E in terms of those of the super-manifold ΠE: it relates Gromov-Witten invariants of X twisted by the inverse Euler class and E to Gromov-Witten invariants of X twisted by the Euler class and E ∗ . We derive from Theorem 1 nonlinear Serre duality in a very general form (Corollary 2). When the bundle E is convex and a submanifold Y ⊂ X is defined by a global section of E, the genus-zero Gromov-Witten invariants of ΠE coin- cide with those of Y . We establish a “quantum Lefschetz hyperplane section principle” (Theorem 2) expressing genus-zero Gromov-Witten invariants of a complete intersection Y in terms of those of X. This extends earlier results [4], [9], [18], [29], [33] and yields most of the known mirror formulas for toric complete intersections. *Research is partially supported by NSF Grants DMS-0072658 and DMS-0306316. 16 TOM COATES AND ALEXANDER GIVENTAL Introduction The mirror formula of Candelas et al. [10] for the virtual numbers n d of degree d =1, 2, 3, holomorphic spheres on a quintic 3-fold Y ⊂ X = CP 4 can be stated [20] as the coincidence of the 2-dimensional cones over the following two curves in H even (Y ; Q)=Q[P ]/(P 4 ): J Y (τ)=e Pτ + P 2 5  d>0 n d d 3  k>0 e (P +kd)τ (P + kd) 2 and I Y (t)=  d≥0 e (P +d)t (5P + 1)(5P +2) (5P +5d) (P +1) 5 (P +2) 5 (P + d) 5 . The new proof given in this paper shares with earlier work [9], [18], [21], [29], [33], [35] the formulation of sphere-counting in a hypersurface Y ⊂ X as a problem in the Gromov-Witten theory of X. Gromov-Witten invariants of a compact almost-K¨ahler manifold X are defined as intersection numbers in moduli spaces X g,n,d of stable pseudo- holomorphic maps f :Σ→ X. Most results in this paper can be stated and hold true in this generality (see Appendix 2 in [11]): the only exceptions are those discussed in Sections 9 and 10 which depend on equation (19). We prefer however to stay on the firmer ground of algebraic geometry, where the majority of applications belong. Given a holomorphic vector bundle E over a compact projective complex manifold X and an invertible multiplicative characteristic class c of complex vector bundles, we introduce twisted Gromov-Witten invariants as intersection indices in X g,n,d with the characteristic classes c(E g,n,d ) of the virtual bundles E g,n,d =“H 0 (Σ,f ∗ E)H 1 (Σ,f ∗ E)”. The “quantum Riemann-Roch theorem” (Theorem 1) expresses twisted Gromov-Witten invariants (of any genus) and their gravitational descendants via untwisted ones. The totality of gravitational descendants in the genus-zero Gromov-Witten theory of X can be encoded by a semi-infinite cone L X in the cohomology alge- bra of X with coefficients in the field of Laurent series in 1/z (see §6). Another such cone corresponds to each twisted theory. Let L E be the cone correspond- ing to the total Chern class c(·)=λ dim(·) + c 1 (·)λ dim(·)−1 + + c dim(·) (·). Theorem 1 specialized to this case says that the cones L X and L E are related by a linear transformation. It is described in terms of the stationary phase asymptotics a ρ (z) of the oscillating integral 1 √ 2πz  ∞ 0 e −x+(λ+ρ)lnx z dx QUANTUM RIEMANN-ROCH, LEFSCHETZ AND SERRE 17 as multiplication in the cohomology algebra by  i a ρ i (z), where ρ i are the Chern roots of E. Assuming E to be a line bundle, we derive a “quantum hyperplane section theorem” (Theorem 2). It is more general than the earlier versions [4], [29], [18], [33] in the sense that the restrictions t ∈ H ≤2 (X; Q) on the space of parameters and c 1 (E) ≤ c 1 (X) on the Fano index are removed. In the quintic case when X = CP 4 and ρ =5P , the cone L X is known to contain the curve J X (t)=  d≥0 e (P +zd)t/z (P + z) 5 (P + zd) 5 , and Theorem 2 implies that the cone L E contains the curve I E (t)=  d≥0 e (P +zd)t/z (λ +5P + z) (λ +5P +5dz) (P + z) 5 (P + dz) 5 . One obtains the quintic mirror formula by passing to the limit λ =0. The idea of deriving mirror formulas by applying the Grothendieck- Riemann-Roch theorem to universal stable maps is not new. Apparently this was the initial plan of M. Kontsevich back in 1993. In 2000, we had a chance to discuss a similar proposal with R. Pandharipande. We would like to thank these authors as well as A. Barnard and A. Knutson for helpful conversations, and the referee for many useful suggestions. The second author is grateful to D. van Straten for the invitation to the workshop “Algebraic aspects of mirror symmetry” held at Kaiserslautern in June 2001. The discussions at the workshop and particularly the lectures on “Variations of semi-infinite Hodge structures” by S. Barannikov proved to be very useful in our work on this project. 1. Generating functions Let X be a compact projective complex manifold of complex dimension D. Denote by X g,n,d the moduli orbispace of genus-g, n-pointed stable maps [7], [31] to X of degree d, where d ∈ H 2 (X; Z). The moduli space is compact and can be equipped [8], [34], [38] with a (rational-coefficient) virtual fundamental cycle [X g,n,d ] of complex dimension n +(1− g)(D − 3) +  d c 1 (TX). The total descendant potential of X is a generating function for Gromov- Witten invariants. It is defined as D X (t 0 ,t 1 , ) := exp    g≥0  g−1 F g X (t 0 ,t 1 , )   ,(1) 18 TOM COATES AND ALEXANDER GIVENTAL where F g X is the genus-g descendant potential, F g X (t 0 ,t 1 , ):=  n,d Q d n!  [X g,n,d ] ( ∞  k 1 =0 (ev ∗ 1 t k 1 )ψ k 1 1 ) ( ∞  k n =0 (ev ∗ n t k n )ψ k n n ).(2) Here ψ i is the first Chern class of the universal cotangent line bundle over X g,n,d corresponding to the i th marked point, the map ev i : X g,n,d → X is evaluation at the i th marked point, t 0 ,t 1 , ∈ H ∗ (X; Q) are cohomology classes, and Q d is the representative of d ∈ H 2 (X; Z) in the semigroup ring of degrees of holomorphic curves in X. Let E be a holomorphic vector bundle over X. We associate to it an ele- ment E g,n,d in the Grothendieck group K 0 (X g,n,d ) of orbibundles 1 over X g,n,d as follows. Consider the universal stable map X g,n+1,d ev n+1 −−−−→ X π    X g,n,d formed by the operations of forgetting and evaluation at the last marked point. We pull E back to the universal family and then apply the K-theoretic push-forward to X g,n,d . This means the following: there is a complex 0 → E 0 g,n,d → E 1 g,n,d → 0 of bundles on X g,n,d with cohomology sheaves equal to R 0 π ∗ (ev ∗ n+1 E) and R 1 π ∗ (ev ∗ n+1 E) respectively. Moreover, the difference E g,n,d := [E 0 g,n,d ] − [E 1 g,n,d ] in the Grothendieck group of bundles does not depend on the choice of the complex. These facts are based on some standard results about local complete intersection morphisms, and are discussed further in Appendix 1. A rational invertible multiplicative characteristic class of complex vector bundles takes the form c(·) = exp  ∞  k=0 s k ch k (·)  (3) where ch k are components of the Chern character and s 0 ,s 1 ,s 2 , are arbi- trary coefficients or indeterminates. Given such a class and a holomorphic vector bundle E ∈ K 0 (X)overX, we define the (c,E)-twisted descendant potentials D c,E and F g c,E by replacing the virtual fundamental cycles [X g,n,d ] in (1) and (2) with the cap-products c(E g,n,d ) ∩ [X g,n,d ]. For example, the Poincar´e intersection pairing arises in Gromov-Witten theory as an intersec- tion index in X 0,3,0 = X, and in the twisted theory therefore takes on the 1 We will usually omit the prefix orbi. QUANTUM RIEMANN-ROCH, LEFSCHETZ AND SERRE 19 form (a, b) c(E) :=  [X 0,3,0 ] c(E 0,3,0 )ev ∗ 1 (a)ev ∗ 2 (1) ev ∗ 3 (b)=  X c(E) ab.(4) We will often assume that all vector bundles carry the S 1 -action given by fiberwise multiplication by the unitary scalars. In this case the ch k should be understood as S 1 -equivariant characteristic classes, and all Gromov-Witten invariants take values in the coefficient ring of S 1 -equivariant cohomology the- ory. We will always identify this ring H ∗ (BS 1 ; Q) with Q[λ], where λ is the first Chern class of the line bundle O(1) over CP ∞ . 2. Quantization formalism Theorem 1 below expresses D c,E in terms of D X via the formalism of quantized quadratic Hamiltonians [25], which we now outline. Consider H = H ∗ (X; Q) as a super-space equipped with the nondegenerate symmetric bi- linear form defined by the Poincar´e intersection pairing (a, b)=  X ab. Let H = H(( z −1 )) denote the super-space of Laurent polynomials in 1/z with coef- ficients in H, where the indeterminate z is regarded as even. We equip H with the even symplectic form Ω(f, g):= 1 2πi  (f(−z), g(z)) dz = −(−1) ¯ f ¯ g Ω(g, f ). The polarization H = H + ⊕H − defined by the Lagrangian subspaces H + = H[z], H − = z −1 H[[z −1 ]] identifies (H, Ω) with the cotangent bundle T ∗ H + . The standard quantization convention associates to quadratic Hamiltoni- ans G on (H, Ω) differential operators ˆ G of order ≤ 2 acting on functions on H + . More precisely, let {q α } be a Z 2 -graded coordinate system on H + and {p α } be the dual coordinate system on H − , so that the symplectic structure in these coordinates assumes the Darboux form Ω(f, g)=  α [p α (f)q α (g) − (−1) ¯p α ¯q α q α (f)p α (g)]. For example, when H is the standard one-dimensional Euclidean space then f =  q k z k +  p k (−z) −1−k is such a coordinate system. In a Darboux coor- dinate system the quantization convention reads (q α q β )ˆ := q α q β  , (q α p β )ˆ := q α ∂ ∂q β , (p α p β )ˆ :=  ∂ 2 ∂q α ∂q β . The quantization gives only a projective representation of the Lie algebra of quadratic Hamiltonians on H as differential operators. For quadratic Hamil- tonians F and G we have [ ˆ F, ˆ G]={F, G}ˆ+C(F, G), 20 TOM COATES AND ALEXANDER GIVENTAL where {·, ·} is the Poisson bracket, [·, ·] is the super-commutator, and C is the cocycle C(p α p β ,q α q β )=  (−1) ¯q α ¯p β if α = β, 1+(−1) ¯q α ¯p α if α = β, C = 0 on any other pair of quadratic Darboux monomials. We associate the quadratic Hamiltonian h T (f)=Ω(T f, f)/2 to an infinitesimal symplectic transformation T , and write ˆ T for the quantization ˆ h T .IfA and B are self-adjoint operators on H then the operators f → (A/z)f and f → (Bz)f on H are infinitesimal symplectic transformations, and C(h A/z ,h Bz ) = str(AB)/2. In what follows, we will often apply symplectic transformations exp T in the quantized form exp ˆ h T to various generating functions for Gromov-Witten invariants — that is to certain formal functions of q = q 0 +q 1 z+q 2 z 2 + ∈H + and  — which we refer to as asymptotic elements of the Fock space. In fact the quantized symplectic transformations that we will use do not have a con- venient common domain that includes all the formal functions which we will need. We will therefore not describe any “Fock space”, but instead regularly indicate those special circumstances that make the application of particular quantized symplectic transformations to particular generating functions well- defined. Such special circumstances usually involve -adic convergence with respect to some auxiliary formal parameters (such as s k in Corollary 3, 1/λ in (12), Q in (13), etc.). The key point here is that our formulas provide unam- biguous rules for transforming generating functions (and their coefficients): the description of these rules as symplectic transformations or their quantizations remains merely a convenient interpretation 2 . Let us begin by setting up notation for such an interpretation. We will assume that the ground field Q of constants is extended to the Novikov ring Q[[Q]], or to Q[[Q]] ⊗ Q(λ) in the S 1 -equivariant setting, and will denote the ground ring by Λ. The potentials F g X (t 0 ,t 1 , ) are naturally defined as formal functions on the space of vector polynomials t(z)=t 0 + t 1 z + t 2 z 2 + where t 0 ,t 1 ,t 2 , ∈ H. The total descendant potential D X is simply the formal expression exp   g−1 F g X defined by these formal functions. It cannot be viewed as a formal function of  and t because of the presence of  −1 - and  0 -terms in the exponent. The reader uncomfortable with this situation could note that the formal functions F 0 X and F 1 X when reduced modulo Q contain only terms which are respectively at-least-cubic and at-least-linear in 2 This approach, somewhat resembling the terminology in the theory of formal groups, is not the only one possible. We refer to Section 8 in [26] where the class of tame asymptotic functions (convenient for the purposes of that paper) is introduced. QUANTUM RIEMANN-ROCH, LEFSCHETZ AND SERRE 21 the variables t i , and that D X can therefore be considered as a formal function of , t/ and Q/. This point of view will, however, play no role in what follows. We regard the total descendant potential (1) as an asymptotic element of the Fock space via the identification q(z)=t(z) − z(5) which we call the dilaton shift. The twisted descendant potentials D c,E can be similarly considered as asymptotic elements of Fock spaces corresponding to the super-space H equipped with the twisted inner products (4). Alternatively, we can identify the inner product spaces (H, (·, ·) c(E) ) with (H, (·, ·)) by means of the maps a → a  c(E), hence considering the twisted descendant potentials D c,E as asymptotic elements of the original Fock space via the twisted dilaton shift: q(z)=  c(E)(t(z) − z).(6) We thus obtain a formal family D s := D c,E of asymptotic elements of the Fock space depending on the parameters s =(s 0 ,s 1 ,s 2 , ) from (3). Note that, due to the dilaton shift, D s is a formal function of q defined near the shifted origin q(z)=−  c(E)z, which varies with s. 3. Quantum Riemann-Roch Let us identify z with the first Chern class of the universal line bundle L and denote by · the one-dimensional subspace spanned by the asymptotic element “·” of the Fock space. Theorem 1. D c,E  = ˆ D X , where  : H→His the linear symplectic transformation defined by the asymp- totic expansion of  c(E) ∞  m=1 c(E ⊗ L −m ). This should be interpreted as follows. Let ρ 1 , ρ r be the Chern roots of E, and let s(·)=  k≥0 s k ch k (·). 22 TOM COATES AND ALEXANDER GIVENTAL Then ln   c(E) ∞  m=1 c(E ⊗ L −m )  = r  i=1  s(ρ i ) 2 + ∞  m=1 s(ρ i − mz)  = r  i=1  1 1 − e −z∂ x − 1 2  s(x)     x=ρ i ∼ r  i=1    m≥0 B 2m (2m)! (z∂ x ) 2m−1 s(x)         x=ρ i =  m≥0  l≥0 B 2m (2m)! s l+2m−1 ch l (E)z 2m−1 . Here the B 2m are Bernoulli numbers: t 1 − e −t = t 2 +  m≥0 B 2m (2m)! t 2m . The operator of multiplication by ch l (E) in the cohomology algebra H of X is self-adjoint with respect to the Poincar´e pairing. Consequently, the operator of multiplication by ch l (E)z 2m−1 in the algebra H is an infinitesimal symplectic transformation of H and so is ln . Theorem 1 therefore is derived from the following more precise version. Theorem 1  . (7) exp  − 1 24  l>0 s l−1  X ch l (E)c D−1 (T X )   sdet  c(E)  − 1 24 D c,E = exp    m>0  l≥0 s 2m−1+l B 2m (2m)! (ch l (E)z 2m−1 )ˆ   exp   l>0 s l−1 (ch l (E)/z)ˆ  D X . Here sdet(·) = exp str ln(·) is the Berezinian. Remarks. (1) The variable s 0 is present on the RHS of (7) only in the form exp(s 0 ρ/z)ˆ where ρ =ch 1 (E). For any ρ ∈ H 2 (X) the operator (ρ/z)ˆ is in fact a divisor operator: the total descendant potential satisfies the divisor equation  ρ z  ˆ D X =  ρ i Q i ∂ ∂Q i D X − 1 24  X ρc D−1 (T X ) D X .(8) Here Q i are generators in the Novikov ring corresponding to a choice of a basis in H 2 (X) and ρ i are coordinates of ρ with respect to the dual basis. For ρ =ch 1 (E) the c D−1 -term cancels with the s 0 -term on the LHS of (7). Thus the action of the s 0 -flow reduces to the change Q d → Q d exp(s 0  d ρ)in QUANTUM RIEMANN-ROCH, LEFSCHETZ AND SERRE 23 the descendant potential D X combined with the multiplication by the factor exp (s 0 (dim E)χ(X)/48) coming from the super-determinant. (2) If E = C then E g,n,d = C − E ∗ g , where E g is the Hodge bundle. The Hodge bundles satisfy ch k (E g )=−ch k (E ∗ g ). In view of this, Theorem 1 in this case turns into Theorem 4.1 in [25] and is a reformulation in terms of the formalism explained in Section 2 of the results of Mumford [36] and Faber- Pandharipande [16]. The proof of Theorem 1 is based on a similar applica- tion of Mumford’s Grothendieck-Riemann-Roch argument to our somewhat more general situation. The argument was doubtless known to the authors of [16]. The main new observation here is that the combinatorics of the result- ing formula, which appears at first sight rather complicated, fits nicely with the formalism of quantized quadratic Hamiltonians. A verification of this — somewhat tedious but straightforward — is presented in Appendix 1. 4. The Euler class The S 1 -equivariant Euler class of E is written in terms of the (nonequiv- ariant) Chern roots ρ i as e(E)=  i (λ + ρ i ). Using the identity (λ + x) = exp(ln λ −  k (−x) k /kλ k ) we can express it via the components of the nonequivariant Chern character: e(E) = exp  ch 0 (E)lnλ +  k>0 ch k (E) (−1) k−1 (k −1)! λ k  .(9) Denote by D e the asymptotic element D s of the Fock space corresponding to s k =  ln λk=0 (−1) k−1 (k−1)! λ k k>0. (10) Substituting these values of s k into (7), replacing ch l (E)by  ρ l i /l! and using the binomial formula (1 + x) 1−2m =  l≥0 (−1) l (2m − 2+l)! (2m − 2)! l! x l for m>0. we arrive at the following conclusion. [...]... 0, 0, ; ±Q) 34 TOM COATES AND ALEXANDER GIVENTAL Remark Corollary 11 generalizes the “nonlinear Serre duality” from [23] (Theorem 5.2) obtained there by fixed point localization and applicable to torus-equivariant bundles E with isolated fixed points Theorem 2, Corollary 9 and the mirror formulas of Section 9 have Serredual partners obtained by replacing e and E with with e−1 and E ∗ We assume that the... (30) and (31) γred − ˜ Xg+ ,n+ +•,d+ ×X X0,1+•+◦,0 ×X Xg− ,n− +◦,d− − → Zg,n+1,d , 43 QUANTUM RIEMANN-ROCH, LEFSCHETZ AND SERRE where g+ + g− = g, n+ + n− = n, and d+ + d− = d The composition rule says that images of the virtual fundamental classes under the gluing ˜ maps add up to the virtual fundamental class [Zg,n+1,d ] Properties (ii) and (iii) are part of the axioms in [7] proved in [6], and (i)... section reproduce genus-zero mirror theorems (Theorem 4.2 and Corollary 5.1) from [23] We illustrate some results of the present section in an example where X = CP n−1 , E is a line bundle of degree 0 < l ≤ n, and the J-functions and their hypergeometric modifications are restricted to the small parameter plane 35 QUANTUM RIEMANN-ROCH, LEFSCHETZ AND SERRE t0 + tP Then Qd edt JX (t0 + tP, z; Q) = z e(t0... sheaf of relative differentials of π : Xg,n+1,d → Xg,n,d and Td∨ is the dual Todd class To derive (24) from (27) we follow Mumford [36] and Faber-Pandharipande [16] We begin by expressing the sheaf Ωπ of relative differentials in terms of universal cotangent lines Assume first that Xg,n,d , Xg,n+1,d , and Zg,n+1,d are smooth and of the expected dimension, and that the image π(Zg,n+1,d ) of the nodal locus... symplectic space (H, Ωc(E) ) defined 0 by the genus-zero descendant potential Fc,E , and Lc∗ ,E ∗ be the Lagrangian cone in the symplectic space (H, Ωc∗ (E ∗ ) ) defined by the genus-zero descendant QUANTUM RIEMANN-ROCH, LEFSCHETZ AND SERRE 33 0 potential Fc∗ ,E ∗ Let Jc,E and Jc∗ ,E ∗ denote the J-functions of the cones Lc,E and Lc∗ ,E ∗ respectively: Jc,E (τ, −z) := Lc,E ∩ (−z + τ + H− ), Jc∗ ,E ∗ (τ ∗... TOM COATES AND ALEXANDER GIVENTAL The equality (vii) is obvious since X0,3,0 = X and [X0,3,0 ] is the usual fundamental class of X; (viii) and (ix) follow from the well-known facts: • X1,1,0 = X × M1,1 • [X1,1,0 ] is the cap product of the fundamental class of X × M1,1 with e(p∗ TX ⊗ p∗ E−1 ) Here p1 and p2 are the projections to the first and 1 2 1 second factors of X × M1,1 respectively and E1 is... positive line bundle L and let the exact sequence 0 → Ker → H 0 (X; E ⊗ LN ) ⊗ L−N → E → 0 take on the role of 0 → B → A → E → 0 Then H 0 (Σ; f ∗ A) and H 0 (Σ; f ∗ B) vanish for QUANTUM RIEMANN-ROCH, LEFSCHETZ AND SERRE 39 any nonconstant map f : Σ → X and sufficiently large N , so that the following sequence is exact: 0 → H 0 (Σ; f ∗ E) → H 1 (Σ; f ∗ B) → H 1 (Σ; f ∗ A) → H 1 (Σ; f ∗ E) → 0 This construction... 0, H 0 (Σ; f ∗ A) and H 0 (Σ; f ∗ B) are nonzero but also have constant rank, so that the construction of Eg,n,d easily extends to this case as well 3 40 TOM COATES AND ALEXANDER GIVENTAL The universal family of stable maps lifts naturally to form G = GLdim W equivariant families p : Y → B and ev : Y → X of nodal curves and their stable maps The map p is an l.c.i morphism of schemes, and factors properly... infinitesimal t-variations of zL are in L, and it is in zH− since JX ∈ z + t + H− 29 QUANTUM RIEMANN-ROCH, LEFSCHETZ AND SERRE Further analysis reveals that Aγ are structure constants of the quantum αβ cohomology algebra φα • φβ = Aγ φα In particular, z∂1 JX = JX since αβ 1• = id We use the notation ∂v for the directional derivative in the direction of v ∈ H and take here v = 1 We can interpret (16)... QUANTUM RIEMANN-ROCH, LEFSCHETZ AND SERRE Td∨ (Ωπ ) = 1 + Td∨ (Ln+1 ) − 1 (29) n + 1 1 −1 + −1 ∨ Td (ODi ) Td (OZg,n+1,d ) ∨ i=1 The first two terms yield Td∨ (Ln+1 ) = ∗ σi (−Di ) Using we find ψn+1 = exp ψn+1 − 1 r≥0 Br r ψ r! n+1 = ψi and the exact sequence 0 → O(−Di ) → O → ODi → 0, 1 − 1 = Td∨ (O(−Di )) − 1 Td (ODi ) Br = (−Di )r r! ∨ r≥1 = −(σi )∗ r≥1 Br r−1 ψ r! i The codimension-2 summand in . Riemann-Roch, Lefschetz and Serre By Tom Coates and Alexander Givental* Annals of Mathematics, 165 (2007), 15–53 Quantum Riemann-Roch, Lefschetz and Serre By Tom Coates and Alexander. that paper) is introduced. QUANTUM RIEMANN-ROCH, LEFSCHETZ AND SERRE 21 the variables t i , and that D X can therefore be considered as a formal function of , t/ and Q/. This point of view will,. Replacing ch l (E) with (−1) l ch l (E), and s k with (−1) k+1 s k in (7) preserves all terms except the super-determinant. QUANTUM RIEMANN-ROCH, LEFSCHETZ AND SERRE 25 Corollary 3. Consider the dual