Lehn's conjecture for Segre classes on Hilbert schemes of points of surfaces and generalizations Lecture 3 : Formulas and proof for K3 Rahul Pandharipande (1) Our original question was the following. Let p: S -> M be a family of nonsingular projective surfaces over a base M, and let E be a K-theory class on the total space S. Question: What is h_*( Seg(E[n]) ) in A*(M) ? An answer was given in term of universal formulae in tautological classes via the associated localization vertex. We ask here for much more explicit answers. There are 3 measures of difficulty for the question (a) Codimension in M --- the higher the codimension, the the more complicated the formula. Codimension 0 concerns integration over a Hilbert scheme of points of a fixed surface and is the simplest. (b) Rank of the K-theory class E --- ranks equal to 0 or near 0 are the simplest. (c) Complexity of the surfaces in the family --- Abelian surfaces tend to be simplest, followed by K3 surfaces. Toric surfaces are of no advantage for the formulas (but toric surfaces can help in proofs). (2) The complete answer to the question in codimension 0, all rank r, for K3 surfaces is known via [MOP]. (a) Let S be a K3 surface and let E in K*(S) be of rank r. \sum_{n=0}^{infty} q^n \int_{S[n]} Seg_{2n}(E[n]) = A(q)^(c1(E)^2-c2(E)) * B(q)^(\chi(c1(E))) * C(q) where \chi(c1(E)) = (1/2) c1(E)^2 + (1/2) c1(E)c1 + (1/12)(c1^2+c2) = (1/2) c1(E)^2 + 2 A = (1+(r+1)x)^(r+1) * (1+ (r+2)x)^(-r) B = (1+(r+1)x)^(-r-2) * (1+ (r+2)x)^(r+1) C = (1+(r+1)x)^((r+2)^2) * (1 + (r+2)x)^(-(r+1)^2) * (1+(r+2)(r+1)x)^(-1) q = x (1+(r+1)x)^(r+1) (b) If we write the solution in the language of Lectures 1 and 2, we must take the logarithm: c1(E)^2 Coeff of F_2 = Log A + 1/2 * Log B c2(E) Coeff of F_2 = -Log A (st) Coeff of F_2 = (1/12) * Log B + (1/24) * Log C . The K3 results give no information on the (s+t)^2 Coeff of F_2 and the (s+t)*c1(E) Coeff of F_2 . (3) Voisin's proof in codimension 0, rank r=1, for K3 surfaces is elegant. I give her complete argument here. (a) Let (S,L) be a K3 surface with a line bundle L with L^2 = 2l. Using the general structure result (see Theorem 3 of Lecture 1), \sum_{n=0}^{infty} q^n \int_{S[n]} Seg_{2n}(L[n]) = A(q)^{24} * B(q)^{2l} for q-series A and B. By calculating the n=1 case by hand, B(q) = 1 + q + ... . We therefore see that \int_{S[n]} Seg_{2n}(L[n]) is a polynomial of degree n in l with leading coefficient 2^n/n!. The result of [MOP] is that \int_{S[n]} Seg_{2n}(L[n]) = 2^n * Binom (l-2n+2,n) which is in fact a polynomial of degree n in l with the correct leading coefficient. In order to prove the [MOP] evaluation, Voisin needs only to prove the vanishing for n>0 \int_{S[n]} Seg_{2n}(L[n]) = 0 for the n values l = 2n-2, 2n-1, ... ,3n-3, since the degree n polynomial is uniquely determined by these n roots and the leading coefficient. (b) We switch notation now to L^2 = 2g-2, so g = l+1 matches the notation of part (3.a) above. Let (S,L) further satisfy three more conditions (i) L is primitive, (ii) Pic(S) is rank 1 generated by L, (iii) g>0. By (iii), L* has no sections. (c) Let P(L[n]*) -> S[n] be the projective bundle, and Let O_P(1) -> P(L[n]*) be the tautological bundle of the polarization. An elementary exercise shows H^0(P(L[n]*),O_P(1)) = H^0(S[n],L[n]) = H^0(S,L) and these spaces are of dimension g+1. (i) We obtain a rational map w: P(L[n]*) - - -> P^g via the complete space of sections of O_P(1). (ii) The Segre class is recovered via the intersection product \int_{P(L[n]*)} c_1(O_P(1))^{3n-1} = \int_{S[n]} Seg_{2n}(L[n]). (iii) Main Claim: if g>2n-2, then L[n] is generated by global sectons. Let us assume the Main Claim for now to see how the vanishing is implied. If g>2n-2 => L[n] is generated by global sections => w: P(L[n]*) -> P^g is an actual morphism => if 3n-1>g, then \int_{S[n]} Seg_{2n}(L[n])=0 by (ii). So \int_{S[n]} Seg_{2n}(L[n])=0 for precisely the range g = 2n-1, 2n, ... , 3n-2 which is exactly the [MOP] vanishing range. (d) Proof of the Main Claim (closely related to older results of Lazarsfeld). Let g>2n-2 and assume L[n] is not globally generated. We construct a contradiction. Let [Z] in S[n] be a witness to the failure of global generation. Then, H^0(S,L) -> H^0(S,L|_Z) is not surjective. Hence H^1(S,L \tensor I_Z) is not zero. Using Serre duality, Ext^1(I_Z,L*) is also not zero, so we have a non-split extension 0 -> L* -> E -> I_Z -> 0 . We first note that Hom(E,L*)= 0. If h is in Hom(E,L*) then h the composition L* -> E ---> L* must be zero (or else the extension splits), so h induces a map I_Z -> L*. But the latter would give a section of L* (by Hartogs) which is impossible, so h=0. The contradiction will come by constructing a non-trival map E -> L*. We calculate chi(E,E) by Riemann-Roch to be 2g+6-4n = 2(g-2n+2)+2 > 2. Therefore, there exists a non-trivial element f in Hom(E,E) which is not proportional to the identity Id: E -> E. By a standard eigenvalue argument, we may assume the sheaf Im(f) in E has generic rank equal to 1. Let F = Saturation of Im(f) in E. Then we obtain a second exact sequence 0 -> F -> E -> Q -> 0 where both F and Q are torsion free of generic rank 1. Consider first F. Since F is torsion free of rank 1, we have an injection F -> F** where F** = L^k is locally free (remember L generates Pic(S)). Since E -> F -> F**=L^k, we have a non-trivial map E -> L^k. Since Hom(E,L*)=0, we conclude k>=0. Then F= I_W \tensor L^{k>=0}. Next, we use the extension 0 -> L* -> E -> I_Z -> 0 . Since F sits in E and does not map to L* (since k>=0), F must have a non-trival map to I_Z. Hence, k<=0, so k=0. Putting the above together, we have the extension 0 -> I_W -> E -> Q -> 0. Finally, Q is also torsion free of rank 1, so we have an injection Q -> Q** and a non-trival map E -> Q -> Q**. What is the line bundle Q**? Since det(E) = L* by the original extension 0 -> L* -> E -> I_Z -> 0 and det(I_W) is trivial, Q** must be L*. So we have constructed a non-trivial map E -> L* which is a contradiction. QED (4) The formulas of Lehn's conjecture concern codimension 0, rank r=1, and all S. Let S be a nonsingular projective surface and let L in K*(S) be of rank 1. Given the series already determined in (2) from the K3 surface specialized to r=1, we need only the Coeffs of F_2 corresponding to (s+t)^2 and (s+t)*c1(L). In total we have: c1(L)^2 Coeff of F_2 = (1/2) * Log ( 1 + 2x ) c2(L) Coeff of F_2 = - 2 * Log ( 1 + 2x ) + Log ( 1 + 3x ) (st) Coeff of F_2 = (3/24) * Log (1 + 2x ) - (1/24) * Log ( 1 + 6x ) (s+t)^2 Coeff of F_2 = (15/24) * Log (1 + 2x ) + (11/24) * Log ( 1 + 6x ) -2 * Log ( Sqrt(1+2x) + Sqrt(1+6x) ) + Log(4) (s+t)*c1(L) Coeff of F_2 = Log ( 1 + 2x ) - Log ( Sqrt(1+2x) + Sqrt(1+6x) ) + Log(2) q = x (1+2x)^2 Comments: (a) The above formulas are strictly more than Lehn's conjecture of 1999 which only considered the case where L is a line bundle on the surface S. The 5 series solve the Segre class question for every rank 1 element L of K*(S). For a line bundle L, c2(L) = 0 so the second series does not appear in Lehn's proposal. (b) The proof use K3 + an extension of Voisin's argument for K3 to the blow-up of K3 in a single point (see her paper). (c) Can we find a straight localization proof? (5) Lehn's conjecture is the complete answer for a fixed surface S and E in K*(S) of rank 1 We also have complete answers for a fixed surface S and E in K*(S) of rank -2,-1, and 0. (a) In rank -2, three of the 5 functions come from the K3 result. The remaining 2 functions come from the following simple geometry. If E=[-B] where B -> S is a rank 2 bundle with a transverse section, then \sum_{n=0}^{infty} q^n \int_{S[n]} Seg(E[n]) = \sum_{n=0}^{infty} q^n \int_{S[n]} c(B[n]) = (1 + q)^(c2(B)). The above geometric calculation holds for all rank 2 bundles B and all S, so both c1*c1(E) and c1^2(E) can be probed. (b) In rank -1, three of the 5 functions come from the K3 result. The remaining 2 functions come from the observation that when E=[-L] where L -> S is a line bundle, \sum_{n=0}^{infty} q^n \int_{S[n]} Seg(E[n]) = \sum_{n=0}^{infty} q^n \int_{S[n]} c(L[n]) = 1 since the integrals over S[n] vanish for n>0 for dimension reasons. The above geometric calculation holds for all line bundles L and all S, so both c1*L and c1^2 can be probed. (c) In rank 0, three of the 5 functions come from the K3 result. One of the remaining 2 functions come from the observation that when E=[L-L], we have \sum_{n=0}^{infty} q^n \int_{S[n]} Seg(E[n]) = 1 for dimension reason. Determining the last function in rank 0 is via Mellit's localization analysis (much less trivial than the geometric constraints discussed above). In rank 2, a complete conjecture was found by MOP (unpublished), but at the moment requires either further geometry or further localization analysis (both approaches are likely to succeed). The last 2 functions in rank 2 are much more complicated than the in the rank -2,-1,0, and 1 cases --- and suggest closed forms for all ranks will be difficult to find explicitly. I would hope for some recursive structure instead.