Quantum cohomology of the odd symplectic
Grassmannian of lines
Clélia PechInstitut Fourier, Grenoble

June 28th, 2011

Introduction
Motivation
What are odd symplectic Grassmannians ?
Classical cohomology
Schubert varieties
Pieri and Giambelli formulas
Quantum cohomology
Enumerativity of GW invariants
Quantum Pieri rule
Quantum presentation

Introduction
Motivation

Quantum cohomology has been extensively studied forI homogeneous spaces ;I toric varieties.

Introduction
Motivation

Quantum cohomology has been extensively studied forI homogeneous spaces ;I toric varieties.

ButI very few explicit formulas for non-homogeneous non-toricvarieties ;I quasi-homogeneous varieties (e.g odd symplectic

Grassmannians) should provide interesting examples.

Introduction
What are odd symplectic Grassmannians ?

Studied by Mihai (2007).

Definition
ω antisymmetric form of maximal rank on C2n+1 .
IGω (m, 2n + 1) := {Σ ∈ G(m, 2n + 1) | Σ is isotropic for ω} .

Introduction
What are odd symplectic Grassmannians ?

Studied by Mihai (2007).

Definition
ω antisymmetric form of maximal rank on C2n+1 .
IGω (m, 2n + 1) := {Σ ∈ G(m, 2n + 1) | Σ is isotropic for ω} .

Remarks
1. independant of the form ω ;2. endowed with an action of the odd symplectic group :Sp2n+1 := {g ∈ GL(2n + 1) | ∀u, v ∈ V ω(gu, gv ) = ω(u, v )} ;3. odd symplectic Grassmannians of lines are the m = 2 case.

Introduction
What are odd symplectic Grassmannians ?

Properties (of IG(m, 2n + 1))of
1. smooth subvariety of dimension m(2n + 1 − m) − m(m−1)
2G(m, 2n + 1).
2. two orbits under the action of Sp2n+1 :closed orbit O := {Σ ∈ IG(m, 2n + 1) | Σ ⊃ K }, isomorphic toIG(m − 1, 2n) ;I open orbit {Σ ∈ IG(m, 2n + 1) | Σ 6⊃ K }, isomorphic to thedual of the tautological bundle over IG(m, 2n) ;I

where K = Ker(ω).

Classical cohomology
Schubert varieties for the symplectic Grassmannian

Schubert varieties of the symplectic Grassmannian IG(m, 2n)
I are subvarieties defined by incidence conditions with respect

to an isotropic flag ;I can be indexed by k-strict partitions(cf Buch-Kresch-Tamvakis), i.e
λ = (2n−m ≥ λ1 ≥ · · · ≥ λm ≥ 0) such that λj > k ⇒ λj > λj+1 ,with k = n − m ;I correspond to classes σλ ∈ H|λ| (IG, Z) generating thecohomology ring H∗ (IG, Z) as a Z-module.

Classical cohomology
Schubert varieties for IG(m, 2n + 1)

Embedding in the symplectic Grassmannian :I IG(m, 2n + 1) ,→ IG(m, 2n + 2) identifies IG(m, 2n + 1) witha Schubert variety of IG(m, 2n + 2) (Mihai) ;I hence “induced” Schubert varieties for IG(m, 2n + 1) andL
decomposition H∗ (IG(m, 2n + 1), Z) = λ Zσλ .

Classical cohomology
Schubert varieties for IG(m, 2n + 1)

Embedding in the symplectic Grassmannian :I IG(m, 2n + 1) ,→ IG(m, 2n + 2) identifies IG(m, 2n + 1) witha Schubert variety of IG(m, 2n + 2) (Mihai) ;I hence “induced” Schubert varieties for IG(m, 2n + 1) andL
decomposition H∗ (IG(m, 2n + 1), Z) = λ Zσλ .

For IG(2, 2n + 1), Schubert varieties are indexed by
I “usual” (n − 2)-strict partitions λ = (2n − 1 ≥ λ1 ≥ λ2 ≥ 0) ;I the “partition” λ = (2n − 1, −1) corresponding to the class ofthe closed orbit O.

Classical cohomology
Special Schubert classes ; Pieri and Giambelli formulasH∗ (IG(2, 2n + 1), Z) is generated as a ring by two sets of special
Schubert classes :1. “rows” σp for 1 ≤ p ≤ 2n − 1, plus the class σ2n−1,−1 ;2. “columns” σ1 and σ1,1 .

Classical cohomology
Special Schubert classes ; Pieri and Giambelli formulasH∗ (IG(2, 2n + 1), Z) is generated as a ring by two sets of specialSchubert classes :1. “rows” σp for 1 ≤ p ≤ 2n − 1, plus the class σ2n−1,−1 ;2. “columns” σ1 and σ1,1 .

Definition
I A Pieri formula is a rule for multiplying any Schubert classwith a special class ;I A Giambelli formula is a rule expressing any Schubert class asa polynomial in special classes.
Knowing both formulas, all cup-products of Schubert classes canbe computed.

Classical cohomology
A Pieri formula for IG(2, 2n + 1)
I i : IG(m, 2n + 1) ,→ IG(m, 2n + 2) induces a restriction map

in cohomology, which happens to be surjective.
I For m = 2 the map and its “inverse” can be explicitelycomputed.
I So Pieri rules for IG(2, 2n + 2) (cf Pragasz-Ratajski,BKT) can be “pulled back” to IG(2, 2n + 1), hence

Proposition (Pieri formula for IG(2, 2n + 1))(
σa+1,b + σa,b+1
if a + b 6= 2n − 3,σa,b · σ1 =
σa,b+1 + 2σa+1,b + σa+2,b−1 if a + b = 2n − 3.
(σa+1,b+1if a + b 6= 2n − 4, 2n − 3,σa,b · σ1,1 =
σa+1,b+1 + σa+2,b if a + b = 2n − 4 or 2n − 3.

Classical cohomology
Giambelli formula and presentation

To find a Giambelli formula for IG(2, 2n + 1) :I use the well-known Giambelli formula on G(2, 2n + 1) ;I “pull it back” to IG(2n + 1) by the natural embeddingIG(2, 2n + 1) ,→ G(2, 2n + 1) ;I get an explicit formula.

Proposition (Presentation of H∗ (IG(2, 2n + 1), Z))
The ring H∗ (IG(2, 2n + 1), Z) is generated by the classes σ1 , σ1,1and the relations aredet (σ11+j−i )1≤i,j≤2n = 0
1det (σ11+j−i )1≤i,j≤2n+1 = 0
σ1Quantum cohomology
Definition

Goal : compute the small quantum productXXσα,β ? σγ,δ =Id (σα,β · σγ,δ · σ̌,ζ ) σ,ζ q d ,{z}|d≥0Gromov-Witten invariant

where
I q is the quantum parameter and has degree 2n ;I σ,ζ runs through the Schubert classes ; σ̌,ζ runs through thecorresponding dual basis.

Quantum cohomology
Definition

Goal : compute the small quantum productXXσα,β ? σγ,δ =Id (σα,β · σγ,δ · σ̌,ζ ) σ,ζ q d ,{z}|d≥0Gromov-Witten invariant

whereI q is the quantum parameter and has degree 2n ;I σ,ζ runs through the Schubert classes ; σ̌,ζ runs through thecorresponding dual basis.
Idea : to compute the GW invariants, use their enumerativeinterpretation.

Quantum cohomology
Enumerativity of GW invariants

What does it mean ?
Id (γ1 , γ2 , γ3 ) = number of degree d rational curves through Γ1 , Γ2 , Γ3 ,where Γi ’s are cycles Poincaré dual to the classes γi .
What are the obstructions ?
1. moduli space may not have the expected dimension ;2. maybe Γi ’s can’t be made to intersect transversely ;3. stable maps with reducible source may contribute ;4. a curve may cut one of the Γi ’s in several points, contributing
several times to the invariant ;5. similarly a curve may cut one of the Γi ’s with multiplicities.

Quantum cohomology
The moduli spaces M0,2 (IG, 1) and M0,3 (IG, 1)

Proposition
The moduli spaces M0,2 (IG, 1) and M0,3 (IG, 1) are smooth (asstacks) and of the expected dimension.

Quantum cohomology
The moduli spaces M0,2 (IG, 1) and M0,3 (IG, 1)

Proposition
The moduli spaces M0,2 (IG, 1) and M0,3 (IG, 1) are smooth (asstacks) and of the expected dimension.
Idea of proof : We prove that H1 (f ∗ T IG) = 0 for each stable f .
I If no irreducible component of the source of f

is entirely
mapped into O, use the generic global generation of f ∗ T IG
due to the transitive Sp2n+1 -action on IG \ O ;I Else use the tangent exact sequence of the closed orbit andprove that H1 (f ∗ T NO ) = 0.

Quantum cohomology
Graber’s lemma

For homogeneous varieties, enumerativity of GW invariants comesfrom Kleiman’s lemma. For quasi-homogeneous spaces there is aversion by Graber :Lemma
I G a connected algebraic group ;I X a quasi-G -homogeneous variety ;I f : Z → X a morphism from an irreducible scheme ;I Y ⊂ X intersecting the orbit stratification properly.

Then there exists a dense open subset U of G such that ∀g ∈ U,
f −1 (gY ) is either empty or has pure dimension
dim Y + dim Z − dim X .

Quantum cohomology
Enumerativity theorem

TheoremI r = 2 or 3 ;I Y1 , . . . , Yr cycles in IG representing γ1 , . . . , γr andintersecting O generically transversely ;I deg γi ≥ 2 for all i ;IPr

i=1 deg γi = dim M0,r (IG, 1).

Then there exists a dense open subset U ⊂ Spr2n+1 such that for
all g1 , . . . , gr ∈ U, the Gromov-Witten invariant I1 (γ1 , . . . , γr ) isequal to the number of lines of IG incident to the translates
g1 Y1 , . . . , gr Yr .

Quantum cohomology
Enumerativity theorem

TheoremI r = 2 or 3 ;I Y1 , . . . , Yr cycles in IG representing γ1 , . . . , γr andintersecting O generically transversely ;I deg γi ≥ 2 for all i ;IPr

i=1 deg γi = dim M0,r (IG, 1).

Then there exists a dense open subset U ⊂ Spr2n+1 such that for
all g1 , . . . , gr ∈ U, the Gromov-Witten invariant I1 (γ1 , . . . , γr ) isequal to the number of lines of IG incident to the translates
g1 Y1 , . . . , gr Yr .
Idea of proof : We get rid of the last three obstructions toenumerativity using Graber’s lemma.

Quantum cohomology
Finding subvarieties with transverse intersection

Problem :I To compute an invariant with the enumerativity theorem weneed transverse cycles.
I Schubert varieties can never be made to intersect transversely.

Quantum cohomology
Finding subvarieties with transverse intersectionProblem :I To compute an invariant with the enumerativity theorem weneed transverse cycles.
I Schubert varieties can never be made to intersect transversely.

Solution :I Use pullbacks of the Schubert varieties of the type A

Grassmannian G(2, 2n + 1) ;I They can be made to intersect transversely on thehomogeneous space G(2, 2n + 1) ;I Corresponding pullbacks to IG(2, 2n + 1) stay transverse.

Quantum cohomology
Quantum Pieri rule for IG(2, 2n + 1)Theorem
σa+1,b + σa,b+1

σa,b+1 + 2σa+1,b + σa+2,b−1
σ1 ? σa,b =σ2n−1,b+1 + qσb
q(σ2n−1,−1 + σ2n−2 )σ1,1 ? σa,b =if a + b 6= 2n − 3 and a 6= 2n − 1,
if a + b = 2n − 3,
if a = 2n − 1 and 0 ≤ b ≤ 2n − 3,
if a = 2n − 1 and b = 2n − 2.


σa+1,b+1
if a + b 6= 2n − 4, 2n − 3 and a 6= 2n − 1,
σa+1,b+1 + σa+2,b
if a + b = 2n − 4 or 2n − 3,
qσb+1
if a = 2n − 1 and b 6= 2n − 3,q(σ2n−1,−1 + σ2n−2 ) if a = 2n − 1 and b = 2n − 3.

Quantum cohomology
Quantum Hasse diagrams

Figure: Quantum Hasse diagrams of IG(2, 6) and IG(2, 7)

Quantum cohomology
Quantum presentation

Proposition (Presentation of QH∗ (IG(2, 2n + 1), Z))
The ring QH∗ (IG(2, 2n + 1), Z) is generated by the classes σ1 ,σ1,1 and the quantum parameter q. The relations aredet (σ11+j−i )1≤i,j≤2n = 01det (σ11+j−i )1≤i,j≤2n+1 + q = 0σ1Quantum cohomology
Quantum presentation

Proposition (Presentation of QH∗ (IG(2, 2n + 1), Z))
The ring QH∗ (IG(2, 2n + 1), Z) is generated by the classes σ1 ,σ1,1 and the quantum parameter q. The relations aredet (σ11+j−i )1≤i,j≤2n = 0
1
det (σ11+j−i )1≤i,j≤2n+1 + q = 0
σ1Corollary
1. QH∗ (IG(2, 2n + 1), Z)q6=0 is semisimple ;2. hence Dubrovin’s conjecture holds for IG(2, 2n + 1).

Conclusion

Main results :I Enumerativity of GW invariants ;I Quantum Pieri formula ;I Quantum presentation and semisimplicity.

Next step :I The m > 2 case ?