Lefschetz fibrations on adjoint orbits - PMA

<strong>Lefschetz</strong> <strong>fibrations</strong> on adjoint orbits 

Elizabeth Gasparim (Unicamp) 

Maringá – November 2012 

Elizabeth Gasparim <strong>Lefschetz</strong> <strong>fibrations</strong> on adjoint orbits

Constructing Letschetz <strong>fibrations</strong> via Lie theory 

This is a report of joint work with Lino Grama and Luiz A. B. 

San Martin. We show that adjoint orbits have the structure of 

<strong>Lefschetz</strong> <strong>fibrations</strong>. 


Motivation 

I want to explain the motivation carefully, and this will take a little 

detour away from Lie theory. Our main motivations is the 

Mirror Symmetry Conjecture 

which proposes that every object has a mirror. This leaves us with 

the obvious question: 








◮ What is a mirror? 

or, more interestingly, 








◮ What is a mirror? 

or, more interestingly, 

◮ Where is the mirror? 


Good news 

Lie theory is useful to help uncover what is behind the mirrors. 


Mirror Symmetry for Calabi–Yaus 

To every Calabi–Yau variety X there corresponds a mirror 

Calabi–Yau variety ˆ X such that the Hodge diamond of ˆ X is 

obtained from the Hodge diamond of X by reflection on the 

mirror. 


Mirror Symmetry for Calabi–Yaus 

To every Calabi–Yau variety X there corresponds a mirror 

Calabi–Yau variety ˆ X such that the Hodge diamond of ˆ X is 

obtained from the Hodge diamond of X by reflection on the 

mirror. 

So we need to define the Hodge diamond. 


This is NOT the Hodge diamond 

Toric diagram of a CY threefold. 


Basic Hodge Theory 

Theorem (Hodge Decomposition Theorem) 

Let X be a compact Kähler manifold of dimension n. Then there 

exists a decomposition 

H k (X , C) = 

H p,q (X ) 

p+q=k 






H k (X , C) = 

H p,q (X ) 

Classical Symmetries: 

p+q=k 






H k (X , C) = 

H p,q (X ) 


p+q=k 

1. Conjugation H p,q (X ) ∼ = H q,p (X ) 






H k (X , C) = 

H p,q (X ) 


p+q=k 


2. Hodge ⋆−operator H p,q (X ) ∼ = H n−q,n−p (X ) 






H k (X , C) = 

H p,q (X ) 


p+q=k 



3. Serre duality H p,q (X ) ∼ = H n−p,n−q (X ) 






H k (X , C) = 

H p,q (X ) 


p+q=k 



3. Serre duality H p,q (X ) ∼ = H n−p,n−q (X ) 


Definition 

Hodge numbers 

h p,q (X ) := dim(H p,q (X )) 


The Hodge Diamond 

h n,n 

h n,n−1 h n−1,n 

h n,n−2 h n−1,n−1 h n−2,n 

hn,0 · · · 

. .. 

. 

. .. 

 

Serre · · · h0,n ↕ H 

. 

h 2,0 h 1,1 h 0,2 

h 1,0 h 0,1 

h 0,0 

←→ 

Conjugation 


The Hodge Diamond 

h n,n 

h n,n−1 h n−1,n 

h n,n−2 h n−1,n−1 h n−2,n 

. 

. .. 

hn,0 · · · hm,m · · · h0,n ↕ H 

. .. 

. 

h2,0 h1,1 h0,2 h 1,0 h 0,1 

h 0,0 

←→ 



Amusing consequence of the classical symmetries 

Corollary (Odd Betti numbers are even) 

Let X be a compact Kähler manifold. Then b2k+1 ∈ 2Z. 


Amusing consequence of the classical symmetries 

Corollary (Odd Betti numbers are even) 

Let X be a compact Kähler manifold. Then b2k+1 ∈ 2Z. 

Proof. 

The Hodge Diamond ⇒ br (X ) = 

⇒ b2k+1 = 

p+q=2k+1 

h p,q = 2 

p+q=r 

h p,q 

k 

h p,2k+1−p . 

p=0 


Example 1: Smooth Elliptic Curve 

h 1,1 = 1 

h 1,0 = 1 h 0,1 = 1 

h 0,0 = 1 


Example 2: Hodge Diamond of a K3 Surface 

1 

0 0 

1 20 1 

0 0 

1 


Example 3: Hodge Diamond of a CY threefold 

X = Fermat quintic X 

1 

0 0 

1 1 1 

1 101 101 1 

1 1 1 

0 0 

1 

ˆ X = X /(Z5) 3 

1 

0 0 

1 101 1 

1 1 1 1 

1 101 1 

0 0 

1 


Hodge Diamonds of Mirror Pairs 

h n,n 

h n,n−1 h n−1,n 

h n,n−2 h n−1,n−1 h n−2,n 

hn,0 · · · 

. .. 

. 

. .. 

 

Serre · · · h0,n ↕ H 

. 

h 2,0 h 1,1 h 0,2 

h 1,0 h 0,1 

h 0,0 

←→ 



Hodge Diamonds of Mirror Pairs 

h n,n 

h n,n−1 h n−1,n 

h n,n−2 h n−1,n−1 h n−2,n 

. 

. .. 

hn,0 · · · hm,m · · · h0,n ↕ H 

. .. 

. 

h2,0 h1,1 h0,2 h 1,0 h 0,1 

h 0,0 

←→ 



Compact Calabi Yaus are few... 

◮ 1D: Only tori, hence elliptic curves in the algebraic case. 

◮ 2D: K3 surfaces, possibly a couple more examples depending 

on your definition. 

◮ 3D: Quintic on P 4 , and unknown classification, open problem, 

conjecturally there are finitely many. 


Mirror Symmetry is a duality 

For our CYs: 

The duality is: 

Calabi–Yau = trivial canonical bundle 

trivial cotangent bundle ⇔ parallelizable manifold 

vector space ⇔ dual vector space 

tangent bundle ⇔ cotangent bundle 

complex geometry ⇔ symplectic geometry 


Kontsevich’s Homological Mirror Symmetry conjecture 

New version of Mirror Symmetry: 

DCoh(X ) ∼ = DFuk( ˆX , W ) 

These are equivalences of derived categories. By Bondal–Orlov 

reconstruction theorem, in the case of case of Fano or general type 

the derived categories determine the varieties. 

(Detailed explanations in Maranhão before Carnaval 2013) 


Known cases of the HMS conjecture 

The HMS conjecture has been proved in very few cases. 

The mirror of an elliptic curve is another elliptic curve 

(Zaslow–Polishschuk). 

Various partial results were obtained for the case of toric varieties 

(Hori–Vafa, P. Clarke). 

However, full computations of the Fukaya–Seidel category were 

completed in just a couple of cases. 


AKO Mirror of a Projective Plane 

Theorem (Auroux-Katzarkov–Orlov) 

The Mirror of CP 2 is the Landau–Ginzburg model which is the 

elliptic fibration with 3 singular fibers, determined by the fiberwise 

compactification of the superpotential W0 : (C ∗ ) 2 → C given by 

W0 = x + y + 1 

xy 

This superpotential has a natural compactification to an elliptic 

fibration 

W0 : M → CP 1 

in which the fiber over infinity consists of 9 rational components. 


AKO Mirror of a Del Pezzo Surface 

Theorem (Auroux–Katzarkov–Orlov) 

Given any Del Pezzo surface obtained by blowing up CP 2 at k 

points, there exists a complexified symplectic form B + iω on M 

for which 

D b (Coh(Xk)) ∼ = D b (Lag(Wk)) 

where Wk : M → CP 1 is a deformation of Wo such that k of the 

critical points in the fiber over infinity have been displaced towards 

finite values of the superpotential. 


This is NOT where our vanishing cycle lives 


Vanishing cycle - mathematician’s picture 


Vanishing cycle - physicist’s picture 


Fukaya category 

A category whose objects are vanishing cycles or Lagrangian 

thimbles. One considers a symplectic fibration 

W : (M, ω) → C 

with isolated non-degenerate critical points, at distinct critical 

values λ0, . . . , λr of W . 

On picks a regular value λ∗ of W and draws arcs γ0, . . . γr joining 

λ∗ to the various critical values. By parallel transport along γi one 

obtains a Lagrangian thimbles associated to the vanishing cycles. 


System of arcs 


Fukaya–Seidel category 

Definition 

The category of vanishing cycles Lag(W , γi) is an A∞-category 

which objects L0, . . . , Lr corresponding to the vanishing cycles. 

The morphisms between the objects are given by 

⎧ 

⎨ 

Hom(Li, Lj) = 

⎩ 

CF ∗ (Li, Lj; R) = R [Li ∩Lj ] if i < j 

R · id if i = j 

0 if i > j 

The differential m1, composition m2 and higher order products Mk 

are defined in terms of Lagrangian Floer homology. 


Topological <strong>Lefschetz</strong> fibration - definition 

Let X be a compact complex variety of dimension n together with 

a surjective map f : X → P 1 . We say that f is a topological 

<strong>Lefschetz</strong> fibration (TLF) if: 

1. The differential df is surjective outside a finite set of points 

{Q1, . . . , Qµ} ⊂ X . 

2. For any p ∈ S 2 \ {f (Q1), . . . , f (Qµ)} the fibre f −1 (p) is a 

smooth complex variety of dimension n. 

3. The images qi := f (Qi) are distinct for different Qi. 

4. Let Xi be the fibre f containing Qi. Then for each i there 

exist small discs qi ∈ Ui and Qi ⊂ UQi ⊂ X such that 

f : UQi → Ui is a holomorphic Morse function and in 

coordinates (x0, . . . , xn) ∈ UQi e z ∈ Ui, 

z = f (x0, . . . , xn) = x 2 0 + · · · + x 2 n. 

5. f |X − {Xi} is locally trivial. 


Monodromy representation 

Since X \ Xi is a fibration it has the unique path lifting property. 

Thus, the choice of a system of arcs determines the monodromy 

representation 

π1(S 2 \ {q1, . . . , qµ}, o) → Aut(π1(X0)) 

ci ↦→ mon(ci) 

where X0 is the regular fibre. 

Remark 

If f : X → S 2 is a TLF in dimnesion 4 and of genus g ≥ 2, then 

the monodromy representation determines the diffeomorphism type 

of f . A characterization for higher dimensions is an open question. 

Elizabeth Gasparim <strong>Lefschetz</strong> <strong>fibrations</strong> on adjoint orbits 

,

Monodromy 


Symplectic <strong>Lefschetz</strong> <strong>fibrations</strong> 

Let (X , ω) be a symplectic manifold together with a TLF 

f : X → S 2 . We say that f is a symplectic Letschetz fibration if for 

every p ∈ S 2 the symplectic form ω is nondegenerate on the fibre 

Xp over p in the sense that: 

◮ Xp is a symplectic sub manifold of X , and 

◮ for each i the symplectic form ωQi is non degenerate over the 

tangent cone of Xi at Qi. 


Examples from algebraic geometry 

Modify a <strong>Lefschetz</strong> pencil by blowing up the base locus 

transforming it to a TLF. 

Figure : Pencil to fibration. 


Symplectic structure on the blow-up 

The pull-back of the symplectic form becomes degenerate on the 

blow-up, hence does not give the TLF the structure of a symplectic 

fibration. 

Seidel constructs a symplectic form over the <strong>Lefschetz</strong> fibration by 

modifying the symplectic form on a tubular neighborhood of the 

exceptional divisor using partitions of unity. 

But now we loose track of vanishing cycles and thimbles. 


A flag 


A flag 


<strong>Lefschetz</strong> <strong>fibrations</strong> via Lie theory 

Notation/definition: 

Let g be a complex semisimple Lie algebra and G a connected Lie 

group with Lie algebra g. 

The Cartan–Killing form of g, 

〈X , Y 〉 = tr (ad (X ) ad (Y )) ∈ C 

is symmetric and nondegenerate. 

Fix a Cartan subalgebra h ⊂ g. A root of h is a linear functional 

α : h → C, α = 0. Denote the set of all roots by Π. 

An element H ∈ h is regular if α (H) = 0 for all α ∈ Π. 


Adjoint orbit 

The adjoint representation of G in g is denoted by Ad (g) X , 

g ∈ G and X ∈ g. An adjoint orbit is given by 

O (X ) = Ad(G) · X = {Ad(g) · X ∈ g : g ∈ G}. 


Main result 

Given H0 ∈ h and H ∈ hR with H a regular element. Then, the 

“height function ” fH : O (H0) → C defined by 

fH (x) = 〈H, x〉 x ∈ O (H0) 

has a finite number of isolated singularities and defines a <strong>Lefschetz</strong> 

fibration, that is, 


<strong>Lefschetz</strong> <strong>fibrations</strong> from Lie theory 

the following properties hold: 

1. The singularities are nondegenerate (Hessian non degenerate). 

2. If c1, c2 ∈ C are regular values then the level manifolds 

f −1 

H (c1) and f −1 

H (c2) are diffeomorphic. 

3. There exists a symplectic form Ω in O (H0) such that if c ∈ C 

is a regular value then the level manifold f −1 

H (c) is 

symplectic, that is, the restriction of Ω to f −1 

H (c) is a 

symplectic (nondegenerate) form. 

4. If c ∈ C is a singular value, then f −1 

H (c) contains affine 

subspaces (contained in O (H0)). These subspaces are 

symplectic with respect to the form Ω from the previous 

item. 


Most interesting part of the proof 

Lemma 

x is a singular point for fH if and only if x ∈ O (H0) ∩ h = W · H0, 

where W is the Weyl group. 


What I think we proved 

Theorem 

The Fukaya category for the Landau–Ginzburg model (O(H0), fH) 

has at least as many generators as the Weyl group of G. 

Proof. 

By the fundamental lemma of Picard–<strong>Lefschetz</strong> theory, to every 

singularity of the <strong>Lefschetz</strong> fibration there corresponds at least one 

vanishing cycle. 

for me details, and applications to HMS... 


Third Latin Congress on Symmetries in Geometry and 

Physics - São Luís, Maranhão, February 1 –10, 2013

Lefschetz fibrations on adjoint orbits - PMA

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