Quantum Lefshetz for non-split bundle via “abelian/non-abelian correspondence”.

Bumsig Kim explained me how their beautiful theory provides a tool for computing J-series of V_{22} and many other Fano threefolds.

The computation for V_{22} can be reproduced by the following pari/gp code (I omit checking that mirror map is almost trivial):

o = O(t^(N+1))
h(n) = sum(k=1,n,1/k)
hh(n) = sum(k=1,n,sum(l=k+1,n,1/k/l))
gg(x, p) = (x!*(1+h(x)*p+hh(x)*p^2+O(u^4)))
simplemirrormap(F) = F * exp(-polcoeff(F,1,t)*t)
reg(F) = sum(n=0,N,t^n*polcoeff(F,n,t)*n!)+O(t^(N+1))
period(F) = reg(simplemirrormap(F))

gp > v22 = period( polcoeff( sum(a=0,N,sum(b=0,N,sum(c=0,N,o+ t^(a+b+c)* ( gg(a+b,u*(A+B)) * gg(a+c,u*(A+C)) * gg(b+c,u*(B+C)) )^3 / (gg(a,u*A) * gg(b,u*B) * gg(c,u*C) )^7 *(c-b+u*(C-B))*(c-a+u*(C-A))*(b-a+u*(B-A)) ))) + O(u^4) , 3,u) /(C-B)/(C-A)/(B-A) + o )

%2 = 1 + 12*t^2 + 60*t^3 + 636*t^4 + 5760*t^5 + 58620*t^6 + 604800*t^7 + 6447420*t^8 + O(t^9)

Indeed, period sequence 17.
Note that gg(x,p) = \frac{\Gamma(1 + x + \frac{p}{z})}{\Gamma(1 + \frac{p}{z})} + o(\frac{1}{z^3}) is the familiar Gamma-factor with u=\frac{1}{z}.


So, how does it works?

Consider 3-dimensional vector space U = \CC^3 with a fixed base, 7-dimensional vector space V = \CC^7, and space M = \CC^{21} of 3×7 matrices M = Hom(U,V) =Hom(\CC^3, \CC^7)

Group G = Aut(U) = GL(3) acts on M by left multiplication. It has a subgroup T = (\CC^*)^3 of diagonal matrices and one may restrict the action to this smaller subgroup.

Let M_{na} be the subset of matrices of maximal rank and M_{ab} be the subset of matrices with non-vanishing rows, M_{na} is an open subset in M_{ab}.

Consider quotient spaces X_{na} = M_{na} / G and X_{ab} = M_{ab} / T. Note that X_{na} = Gr(3,V) = Gr(3,7) and X_{ab} = (\PP(V))^3 = (\PP^6)^3.

Since T is a subgroup of G, there is a natural rational map \pi: X_{ab} -> X_{na}: a triple of points in \PP^6 is sent to their linear span.

Weyl group (symmetric group S_3) acts on X_{ab} = (\PP^6)^3 and hence it acts on the cohomology H(X_{ab}) = H((\PP^6)^3), so cohomology space is decomposed into representations of S_3.

[I’ll omit the part of the story with the partial flag space and non-holomorphic map].

1. It turns out that cohomology H(X_{na}) can be identified with antisymmetric part of H(X_{ab}) as a graded vector space (with grading shifted by 3).
Explicitly, H(X_{ab}) is generated by 3 pullbacks H_1, H_2, H_3 of hyperplane sections on \PP^6; cohomologies of Grassmanian are known to be quotient of symmetric polynomials. Vector space of anti-symmetric polynomials is obtained from vector space of symmetric polynomials via multiplication by anti-symmetric polynomial of the smallest degree Formula does not parse: \Delta  = \prod_{i<j} (s_i-s_j)[/latex]. So cohomology class in Grassmanian corresponding to symmetric function [latex]\sigma[/latex] is sent into [latex]\Delta \cup \sigma(H_1,H_2,H_3)[/latex].  <b>2.</b>Also we can compare vector bundles on [latex]X_{ab} and X_{na} by pulling them back to M, and considering as G-linearized.
It turns out that universal bundle U over Gr(3,7) decomposes into sum of 3 line bundles on X_{ab}:  U  <->  O(1,0,0) \oplus O(0,1,0) \oplus O(0,0,1).
So O_{Gr(3,7)}(1) <-> O(1,1,1) and U^*(1) <->  O(1,1,0) \oplus O(1,0,1) \oplus O(0,1,1).

3. On domain of \pi one may define a relative tangent bundle T_{\pi} (“traceless” part of Hom(U,U)). It turns out this vector bundle can be extended as a split vector bundle to whole X_{ab}: T_{\pi} = O(1,-1,0) \oplus O(-1,1,0) \oplus O(1,0,-1) \oplus O(-1,0,1) \oplus O(0,1,-1) \oplus O(0,-1,1).
Consider “square root” of relative tangent bundle t_{\pi} = O(0,-1,1) \oplus O(-1,0,1) \oplus O(-1,1,0).

4.Recall that Fano threefolds V_{22} = V_{na} are sections of homogeneous vector bundle E = 3 U^*(1) on Gr(3,7). Comparision (2) shows these threefolds has 9-dimensional abelianizations V_{ab} — complete intersections of $9$-dimensional split bundle E_{ab} = (O \oplus O \oplus O) \otimes (O(1,1,0) \oplus O(1,0,1) \oplus O(0,1,1)) on X_{ab}.

5. Abelian/non-abelian correspondence is similar for pairs X_{ab}/X_{na} and V_{ab}/V_{na}.
J-series for Gr(3,7) can be obtained as twisted by relative tangent bundle T_{\pi} I-series for e(t_{\pi}) \cup  I_{X_{ab},T_{\pi}} after the comparision of cohomologies described in (1).
Similarly, J-series for V_{22} can be obtained via mirror map from twisted by T_{\pi} + E_{ab} I-series e(t_{\pi}) \cup I_{X_{ab},T_{\pi}+E} after the “pullbacked” comparision (1).

6. The sign comes from considering closely the Gamma-factor for relative tangent bundle T_{\pi}. Note that fibers of abelian/non-abelian correspondence are in some sense holomorphic symplectic (relative tangent bundle contains both O(D) and O(-D)), so they behave like varieties with trivial canonical class.
Consider the factor \frac{\Gamma(1+D+d) \Gamma(1-D-d)}{\Gamma(1+D)\Gamma(1-D)}. Since \Gamma(1+x) \Gamma(1-x) = \frac{\pi x}{sin (\pi x)} and sin (\pi (x+d)) = (-1)^d sin(\pi x) we have \frac{\Gamma(1+D+d) \Gamma(1-D-d)}{\Gamma(1+D)\Gamma(1-D)} = (-1)^d \frac{d+D}{D}.

The same method can also be applied to complete intersections of homogeneous bundles in orthogonal isotropic and symplectic isotropic Grassmanians, since these Grassmanians themselves are just sections of some homogeneous bundles (wedge or symmetric powers of universal bundle) on ordinary Grassmanians of type A. Also this can be uprgaded to treat different blowups of these varieties. In particular, in the comments to this post I compute J-series for Fano threefolds V_5, #2.14, #2.17, #2.20, #2.21 and #2.22.

Also Bumsig points out that one can express the ab/non-ab twist as a differential operator applied to abelian multi-parameter J-function (basically, just Vandermonde \prod_{i>j} (\frac{d}{d q_i} - \frac{d}{d q_j}). This interpretation is more useful for dealing with Frobenius manifolds.

Gromov-Witten Invariants for Abelian and Nonabelian Quotients by Aaron Bertram, Ionut Ciocan-Fontanine, Bumsig Kim
The Abelian/Nonabelian Correspondence and Frobenius Manifolds by Ionut Ciocan-Fontanine, Bumsig Kim, Claude Sabbah
Quantum cohomology of the Grassmannian and alternate Thom-Sebastiani by Bumsig Kim, Claude Sabbah


  1. Tom says:

    Very nice! Is it possible to upgrade this to treat blow-ups of complete intersections in Grassmannians, e.g. #14 on the rank 2 list (the blow-up of V_5 \subset \PP^6 with center an elliptic curve that is a complete intersection of two hyperplace sections)?

  2. Sergey says:

    Abelianizations for V_5 and 2.14

    #2.14 is a relatively easy case (just the hypersurface in V_5 \times \PP^1), so can be reduced to V_5 easily, and it can also be done via abelianization:

    V_5 is an intersection of 3 hypersurfaces in Gr(2,5), abelianization of Gr(2,5) is \PP^4 \times \PP^4,
    so abelianization of #14 is an intersection of 4 hypersurfaces in \PP^4 \times \PP^4 \times \PP^1
    of degrees (1,1,0), (1,1,0), (1,1,0), (1,1,1). Relative abelian/non-abelian tangent bundle is T_{\pi} = O(1,-1,0) + O(-1,1,0), and its “square root” (Vandermonde) is O(-1,1,0).

    Here is the code for V_5 and #2.14.

    v5 = period( polcoeff( sum(a=0,N,sum(b=0,N, o + t^(2*(a+b)) * gg(a+b,u*(A+B))^3 / (gg(a,u*A) * gg(b,u*B))^5 * (b-a+u*(B-A))*(-1)^(b-a) )) + O(u^2) , 1,u) /(B-A) + o )
    mm214 = period( subst(polcoeff( sum(a=0,N,sum(b=0,N,sum(c=0,N, o + t^(a+b+c) * ( gg(a+b,u*(A+B))^3 * gg(a+b+c,u*(A+B)+C) ) / (gg(a,u*A) * gg(b,u*B))^5 / gg(c,C)^2 * (b-a+u*(B-A))*(-1)^(b-a) ))) + O(u^2) , 1,u) /(B-A) + o, C,0) )

    %3 = 1 + 6*t^2 + 114*t^4 + 2940*t^6 + 87570*t^8 + O(t^10)
    %4 = 1 + 16*t^2 + 90*t^3 + 1104*t^4 + 11460*t^5 + 133990*t^6 + 1588860*t^7 + 19463920*t^8 + 242996040*t^9 + O(t^10)

    Note there is a sign (-1)^{b-a} as explained in 6. In case of Gr(3,7) there were 3 signs (-1)^{b-a} \cdot (-1)^{c-a} \cdot (-1)^{c-b} that cancelled altogether.

  3. Sergey says:

    Abelianizations for 2.17, 2.20, 2.21 and 2.22

    It is also possible to compute period sequence 46 (that should correspond to #2.20) in a similar way:
    consider non-abelianization of complete intersection of line bundles with degrees
    (1,0,1), (0,1,1) and thrice (1,1,0) on (\PP^4)^2 \times \PP^2 (we consider only involution permuting two four-spaces).

    This non-abelianization is a complete intersection of three hyperplane sections O_{Gr(2,5)}(1) and (dual?) universal bundle twisted by O_{\PP^1}(1) on Gr(2,5) \times \PP^1.


    Abelianization for #2.17 seem to be complete intersection in
    (\PP^3)^2 \times \PP^3 (involution exchanges first two 3-spaces)
    of line bundles O(1,0,1),O(0,1,1),O(1,1,0),O(1,1,1).

    Abelianization for #2.21 seem to be complete intersection in
    (\PP^3)^2 \times \PP^4 (involution exchanges first two 3-spaces)
    of line bundles O(1,0,1) twice, O(0,1,1) twice and O(1,1,0).

    Abelianization for #2.22 seem to be complete intersection in
    (\PP^4)^3 \times \PP^3 (S_3 permutes three 4-spaces)
    of line bundles O(1,0,0,1),O(0,1,0,1),O(0,0,1,1) and thrice O(1,1,1,0).

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