IPMU 10-0101
G-minimal varieties are quantum minimal
SERGEY GALKIN

Abstract. We show that G-minimal Fano varieties are quantum minimal.

The aim of this note is to provide a conceptual explanation (theorem 15, remarks 23.24) for all discovered (sofar) phenomena of minimality in quantum cohomology. In particular we show that 8 known differential operators oftype D3 (see example 39) that do not correspond to any minimal Fano threefolds, do actually come from geometryof G-Fano threefolds.
Let X be a smooth (n − 1)-dimensional variety embedded by linear system L⊗k . Assume that X admits action ofq
some finite (possibly trivial) group G. Consider cohomology algebra H = H (X, Q), its subalgebra Ha of algebraic
cycles, its G-invariant subalgebra HaG , and subring HL generated by L = c1 (L): HL = Q[L]/Ln ⊂ H.
Inclusions HL ⊂ HaG ⊂ Ha ⊂ H imply inequalities(1)n = dim HL 6 dim HaG 6 dim Ha 6 dim H.

Definition 2. Variety X is called homologically minimal (or just minimal ) if its cohomology H is n-dimensional,
and algebraically G-minimal if dim HaG = n.
Remark 3. Variety X is called G-Fano if H 2 (X, Q)G is 1-dimensional, i.e. H 2 (X, Q)G = Qc1 (X) and anticanonicalline bundle is ample. Obviously, algebraically G-minimal Fano varieties are G-Fano, and G-Fano varieties ofdimension 6 3 are algebraically G-minimal.
qFor classes γi ∈ H (X) and β ∈ H2 (X, Z) let hγ1 , γ2 , γ3 iβ be 3-pointed genus-0 Gromov–Witten invariant (naivelyequal to number of rational curves of homology class β passing through three cycles Poincare-dual to classes γi if
β is effective class and 0 otherwise).
Definition 4. Very small (algebraic) quantum cohomology of X is a trivial Q[q] − module qHa (X) = Ha (X)[q]with multiplication ? defined byZX(5)(γ1 ? γ2 ) ∪ γ3 =hγ1 , γ2 , γ3 iβ q c1 (X)·β .
[X]

β∈H2 (X,Z)

Ring qH(X) is a commutative associative unital ring, and it is homogeneous with degree of q equal to 1 andqdegree of γ ⊗ 1 equal to half of usual degree of γ ∈ H (X).
Definition 6. Minimal quantum cohomology qHm (X) is a subring of very small quantum cohomology generated
by c1 (X) ⊗ 1 and q.
Definition 7. Quantum rank N (X) of X is the rank of qHm (X) as Q[q]-module.
Fano variety X is called quantum maximal if N (X) = dim Ha i.e. qHm (X) = qHa (X).
Fano variety X is called quantum minimal if N (X) = n.
Some reconstruction theory for quantum minimal varieties is developed in [10].
Remark 8. Upper bound for quantum rank is obvious and congruence c1 (X)?k = c1 (X)∪k mod q implies thatN (X) cannot be less than n.
Example 9. Variety X is algebraically minimal ⇐⇒ it is both quantum minimal and maximal.
Grassmannian G(2, 4) is quantum minimal, but not maximal.
Grassmannian G(2, 5) is quantum maximal, but not minimal.
Grassmannian G(2, 6) is neither quantum minimal, nor quantum maximal.
Date: May 29, 2012.
This work was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. This work
was supported by Grant-in-Aid for Scientific Research (10554503) from Japan Society for Promotion of Science. This is an update forthe first part of my earlier SFB/TR 45 preprint [3].
1More generally Grassmannians G(k, p) for prime values of p (e.g. G(2, 5)) are quantum maximal (see corollary 14)
and algebraically G-minimal varieties are quantum minimal (see theorem 15).
Definition 10 (see also [5]). Quantum characteristic polynomial of variety X is the characteristic polynomial of
the operator ?c1 (X) acting on the vector space qHa (X) ⊗ Q(q). (Anticanonical) spectrum of variety X is the set ofroots of its quantum characteristic polynomial.
Lemma 11. Quantum rank N (X) is not less than number of distinct non-zero elements in the spectrum of X.
Proof. By definition N (X) is the degree of the minimal polynomial for operator ?c1 (X) acting on qHa (X) ⊗ Q(q).
Corollary 12. If variety X is quantum minimal then number of distinct non-zero elements in its spectrum is not
more than n = dim X + 1.
Corollary 13. If the spectrum of variety X is simple i.e. all eigenvalues of ?c1 (X) are distinct and non-zero, thenvariety X is quantum maximal.
Corollary 14. Grassmannians G(k, p) are quantum maximal for prime values of p.
Proof. By [4] roots of quantum characteristic polynomial of G(k, p) are proportional to sums of the elements in all
different k-tuples of distinct p-th roots of unity. For prime values of p all these numbers are distinct, so previous
corollary 13 implies G(k, p) is quantum maximal. Theorem 15. If Fano variety X is algebraically G-minimal then X is quantum minimal.
This theorem holds because quantum multiplication respects the group action:
Lemma 16. For any g ∈ G, γ1 , γ2 ∈ qHa (X) we have g ∗ γ1 ? g ∗ γ2 = g ∗ (γ1 ? γ2 ), where action of G on Ha [q] isdefined by the base change i.e. g ∗ q = q.
Corollary 17. Vector space HaG [q] is closed with respect to ?-multiplication.
Proof of the corollary 17. If g ∗ γ1 = γ1 and g ∗ γ2 = γ2 then g ∗ (γ1 ? γ2 ) = g ∗ γ1 ? g ∗ γ2 = γ1 ? γ2 . Corollary 18. Minimal quantum cohomology qHm (X) is a subring of HaG [q].
Proof. Anticanonical class is algebraic and G-invariant, so c1 (X) ∈ HaG [q] and by corollary 17 the whole subringqHm (X) generated by c1 (X) lies in HaG [q]. 
Proof of the lemma 16. Since Gromov–Witten invariants are well defined and are indeed invariant with respect toq
the isomorphims for all classes β ∈ H2 (X) and γi ∈ H (X). one hashg ∗ γ1 , ..., g ∗ γn iβ = hγ1 , . . . , γn ig∗ β(19)Anticanonical class c1 (X) is G-invariant, so any automorphism preserves anticanonical degrees:(20)g ∗ c1 (X) = c1 (X)(21)

c1 (X) · β = (g −1 )∗ c1 (X) · β = c1 (X) · g∗ β

For fixed γ1 , γ2 and arbitrary γ3 using equalities 19, 20, 21 we derive:
ZX
X
(22)
(g ∗ γ1 ? g ∗ γ2 ) ∪ γ3 =hg ∗ γ1 , g ∗ γ2 , γ3 iβ q c1 (X)·β =γ1 , γ2 , (g −1 )∗ γ3 g∗ β q c1 (X)·g∗ β =[X]ββZ=[X](γ1 ? γ2 ) ∪ g −1 )∗ γ3 =Z(g ∗ (γ1 ? γ2 )) ∪ γ3[X]This proves the lemma. 
Proof of the theorem 15. By corollary 18 minimal quantum cohomology qHm (X) is contained inside HaG [q]. Thatimplies N (X) 6 dim HaG , but by definition of algebraic G-minimality dim HaG = n, so N (X) 6 n i.e. X is quantum
minimal. Remark 23. Actually theorem 15 can be considered as a particular manifestation of monodromy action for family[X/G] → [pt/G] over stack [pt/G] with fibers X. Geometrically deformation invariance of Gromov–Witten isinvariance of correlators with respect to Gauss-Manin connection.
2

Remark 24. There are two frameworks for quantum cohomology — symplectic and algebraic. One may noticeneither of these definitions were used in the proof. Geometrical part is hidden behind the equalities 19, 20, 21 andthe fact that correlators are invariant with respect to algebraic or symplectic isomorphisms.
Moreover, one can even apply the theorem in the case of non-geometric action of the Galois group (or mixed
geometric and Galois action) on variety X and it’s cohomologies (e.g. Het (X, Ql )) if X is defined over Q (or oversome number field). This is true since in algebraic framework is motivic and everything is defined over the basefield of X: Mg,n (X, β), evaluation map ev : Mg,n (X, β) → X n , ψ-classes and the virtual fundamental class.
Remark 25. Since quantum cohomology is deformation-invariant we deduce from corollary 18 that minimal quantum
cohomology qHm (X) is contained in the intersection of all G-invariant algebraic cycles for all possible complex
structures and actions of groups on X.
We’d like to note that since quantum rank N (X) doesn’t depend on any action of group G, corollary 18 saysthat knowledge of quantum cohomology provides us with a lower bound for dimension of G-invariant cohomology
for any G. In small dimensions this gives bound for G-invariant Picard number ρG = dim H 2 (X, Q)G . In particular
Lemma 26. For any action G : X we have lower bound dim Ha (X)G > N (X).
For any action G : S of group on del Pezzo surface we have lower bound ρG (S) > N (X) − 2− 1.
For any action G : V of group on Fano threefold we have lower bound ρG (V ) > N (X)
2
In practice bound of lemma 26 is sharp in a sense that there exists some deformation and action of some group
G : X 0 with inequality becoming equality. Manin argues 1 that quantum cohomology should be promoted toquantum motive and this promotion turns every algebraic variety into a sort of homogeneous space (over operad of
motives of moduli stacks of curves). Motivated by this insight we propose the following conjecture (which should
be a “consequence of quantum Torelli”):
Conjecture 27. For any deformation class of Fano threefolds the bounds 26 are sharp i.e. there exists some familyof threefolds in this class over some base such that monodromy-invariant part of cohomology coincides with minimal
quantum cohomology (after change of scalars).
In collaboration [1] we compute minimal quantum cohomology of Fano threefolds and, in particular, verify theconjecture 27.
Quantum differential equation and its solution.
dLet t be a coordinate on Gm = Spec C[t, t−1 ] and D = t dt.
Definition 28. Quantum differential equation (QDE) is a trivial vector bundle over Gm with fibre H and connection(29)DΦ = L ? Φwhere Φ ∈ H[[t]].
Definition 30. Let GX (t) = [pt] +GX (0) = [pt]. Define G-series asP(31)GX (t) =n>1 Gn (X)tn

be the unique analytic solution of 29 with initial condition

ZGX (t) = 1 +[X]X

gn (X)tn

n>1

Define regularized G-series as Fourier-Laplace transform of G-series:
X(32)ĜX (t) = 1 +n! · gn (X)tnn>1

Definition 33. Scalar QDE is a differential operator of minimal degree (in D) annihilating GX (t), and scalar
RQDE (scalar regularized QDE) is a differential operator of minimal degree annihilating ĜX ).
Proposition 34. Degree of scalar QDE(X) equals to quantum rank N (X).
Examples.
Let Sd be a del Pezzo surface of degree d = 1, . . . , 9 which is a blowup of projective plane in (9 − d) points ingeneric position, and Q = P1 × P1 ⊂ P3 — a smooth quadric surface. Anticanonical linear system maps Sd to Pd
and for d > 3 this map is embedding. (−1)-curves are exceptional curves of blowups and strict transforms of lines
passing through 2 points of blowup, conics passing through 5 points of blowup, etc
1in his lecture at IHES dedicated to Grothendieck’s anniversary
3Example 35. Surface S8 = PP1 (OP1 ⊕ OP1 (1)) has unique (−1)-curve C. Uniqueness implies C is G-invariant for
any action G : S8 , hence space H 2 (S8 , Q)G is two-dimensional since it contains two linearly independent elements
c1 (S8 ) and C.
Example 36. Similarly, intersection graph for (−1)-curves on S7 is A3 and so the middle (−1)-curve C (onethat intersects other two (−1)-curves) is G-invariant for any action G : S7 , and space H 2 (S7 , Q)G contains twodimensional space generated by KS7 and C. Action of Z/2Z on S7 induced from line-interchanging involution on
P1 × P1 swaps two (−1)-curves different from C, so dim H Z/2Z (S7 ) = 4.
Theorem 37 (see [2]). Del Pezzo surface Sd of degree d is G-Fano ⇐⇒ S is P2 , Q or d 6 6.
Corollary 38. Del Pezzo surfaces S1 , S2 , S3 , S4 , S5 , S6 , P1 × P1 and P2 are quantum minimal.
Example 39. There are 8 deformation classes of G-Fano threefolds Y with dim H 2 (Y, Q) > 1, 6 has index one:
Y20 of degree 20, Y24 of degree 24, Y28 of degree 28, Y30 of degree 30, U12 and V12 of degree 12. and two has index
equal to two: P1 × P1 × P1 and W ⊂ P2 × P2 of degree 48.
They produce 8 equations of type D3 [6].
d.
Let t be a coordinate on Gm = Spec C[t, t−1 ] and D = t dt
Scalar RQDE(Y20 ) has the form(40) DY20 = D3 − t · 2D(D + 1)(2D + 1) − t2 · 112(D + 1)3 −− t3 · 184(D + 1)(D + 2)(2D + 3) − t4 · 336(D + 1)(D + 2)(D + 3)
Acknowledgement. This article is an extended first part of an older preprint [3] that was prepared during
my visit in 2008 to Johannes Gutenberg Universitaet by SFB/TR 45 program, and informal visit to IHES. I thank
Duco van Straten and Vasily Golyshev for invitations, support and inspiring discussions on differential equations.
Thanks to Ivan Cheltsov, Yuri Prokhorov and Constantin Shramov for the discussions of G-Fano varieties.
References[1] Tom Coates, Alessio Corti, Sergey Galkin, Vasily Golyshev, Alexander Kasprzyk: Quantum differential equations of Fano threefolds,
in preparation: http://coates.ma.ic.ac.uk/fanosearch/[2] Igor Dolgachev and Vasily Iskovskikh: Finite subgroups of the plane Cremona group, arXiv:math/0610595.
[3] Sergey Galkin: Two instances of fake minimal Fano threefolds (2008), http://sergey.ipmu.jp/fakefano.pdf[4] Sergey Galkin and Vasily Golyshev: Quantum cohomology of Grassmannians and cyclotomic fields, Russian Mathematical Surveys(2006), 61(1):171[5] Vasily Golyshev: Spectra and strains, arXiv:0801.0432.
[6] Vasily Golyshev and Jan Stienstra: Fuchsian equations of type DN, arXiv:math/0701936.
[7] Kenji Matsuki: Weyl groups and birational transformations among minimal models, 1995
[8] Shigefumi Mori and Shigeru Mukai: Classification of fano 3-folds with b2 > 2, Manuscr. Math., 36:147–162 (1981). Erratum 110:407 (2003).
[9] Yu. G. Prokhorov, G-Fano threefolds I, II, arXiv:1012.4959, arXiv:1101.3854.
[10] Victor Przyjalkowski: Minimal Gromov–Witten ring, arXiv:0710.4084.

4