From Affine manifolds to complex manifolds - Instanton ... · Mirror symmetry and tropical geomety The Mumford construction Model rings Starting data Structures The algorithm Concluding

From Affinemanifolds

to complexmanifolds

Bernd Siebert

Mirrorsymmetry andtropicalgeomety

The Mumfordconstruction

Model rings

Starting data

Structures

The algorithm

Concludingremarks

From Affine manifoldsto complex manifolds

Instanton corrections from tropical disks I

Bernd Siebert

Mathematisches InstitutUniversitat Freiburg

Germany

Workshop on Homological Mirror Symmetryand Applications I

IAS 2007


to complexmanifolds

Bernd Siebert


Picture

Discretization

Fan structures

Tropicaldictionary

The MainTheorem


Model rings

Starting data

Structures

The algorithm

Concludingremarks

Mirror symmetry andtropical geomety


to complexmanifolds

Bernd Siebert


Picture

Discretization

Fan structures

Tropicaldictionary

The MainTheorem


Model rings

Starting data

Structures

The algorithm

Concludingremarks

Picture

The arena of mirror symmetry is a Legendre dual pair(B, B) of integral affine manifolds.

The affine structure has singularities along a polyhedralcomplex ∆ ⊆ B with codimR ∆ = 2.

B, B can be thought of as bases of dual SYZ-fibrations,but in general honest SYZ-fibrations seem to havethickened ∆.

On the complex side the affine manifold can be obtainedfrom maximal holomorphic degenerations (GS) or byrigid-analytic methods (Kontsevich/Soibelman; ∆ = ∅ orn = 2).

Mirror data are obtained directly from the affine geometry(e.g., hp,q), or by tropical geometry on B and B.

Appropriately modified should work for any version ofmirror symmetry.


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Bernd Siebert


Picture

Discretization

Fan structures

Tropicaldictionary

The MainTheorem


Model rings

Starting data

Structures

The algorithm

Concludingremarks

Discretization — Tropicalization of B

In our approach B and B are itself tropical — they arise from aset of integral polyhedra by facewise gluing and by specifyingfan structures at the vertices.

Tropical data: (B,P, ϕ)

P: complex of integral polyhedra (|P| = B).

For τ ∈P a fan structure Στ on the normal space.

Polarization: A (multivalued) integral PL-function ϕ on B.

Discrete Legendre transformation

(B,P, ϕ)←→ (B, P, ϕ).


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Bernd Siebert


Picture

Discretization

Fan structures

Tropicaldictionary

The MainTheorem


Model rings

Starting data

Structures

The algorithm

Concludingremarks

Fan Structures

Fan structures

A fan structure along τ ∈P is acontinuous map Sτ : Uτ → Rk with

S−1τ (0) = Int τ .

τ ≤ σ =⇒ Sτ |Int σ is anintegral affine map.

the collection of cones{R≥0Sτ (σ ∩ Uτ )

∣∣ σ ∈P}

defines a fan Στ in Rk .


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Bernd Siebert


Picture

Discretization

Fan structures

Tropicaldictionary

The MainTheorem


Model rings

Starting data

Structures

The algorithm

Concludingremarks

Tropical dictionary on the complex side

Complex data: a polarized toric degeneration

A degeneration π : X→ S , S a discrete valuationk-algebra with

X0 = π−1(0) is a union of toric varieties,π is toroidal at 0-d toric strata of X0.

L ↓ X0 an ample line bundle.

Tropical data: fan and cone picture

Fan picture Cone picture

P local models for π (X0, L)

(Στ , ϕ : |Στ | → R) (X0, L) local models for π


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Bernd Siebert


Picture

Discretization

Fan structures

Tropicaldictionary

The MainTheorem


Model rings

Starting data

Structures

The algorithm

Concludingremarks

The Main Theorem

Theorem (Gross/S. 2007)

Any (B,P, ϕ) fulfilling a maximal degeneracy condition1 arisesfrom a polarized toric degeneration(

π : X −→ Spec k[t], L ↓ X0

).

Remarks

Up to isomorphism (π, L) is determined uniquely afterspecifying gluing data that determine X0 out of (B,P, ϕ).

Any finite order deformation Xk → Spec k[t]/(tk+1) isconstructed by an explicit tropical algorithm on (B,P, ϕ).

1a primitivity requirement on the local monodromy polytopes


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Bernd Siebert



The Projconstruction

Discussion

Alternativeviewpoint

Primarydecomposition

Model rings

Starting data

Structures

The algorithm

Concludingremarks

TheMumford construction


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Bernd Siebert




Discussion



Model rings

Starting data

Structures

The algorithm

Concludingremarks

The Mumford Construction

As a motivating example we consider the case where B ⊆ Rd isan integral, convex polyhedron (=⇒ ∆ = ∅, ϕ : B → Rsingle-valued). In this case a toric construction produces a toricdegeneration:

The Proj construction (Mumford)

The upper convex hull of the graph of ϕ is an integral,unbounded polyhedron:

Ξ :={(m, h) ∈ B × R

∣∣ h ≥ ϕ(m)}.

Let X be the polarized toric variety associated to Ξ:

X = Proj(C[C (Ξ) ∩ Zd+2]

),

with C (Ξ) = cl(R≥0 · (Ξ× {1})

)⊆ Rd+2 the cone over Ξ.

Define π : X→ D by the action of ρ = (0, 1) on Ξ.


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Bernd Siebert




Discussion



Model rings

Starting data

Structures

The algorithm

Concludingremarks

Discussion

The associated tropical manifold

. . . equals (B,P, ϕ) (cone picture):

Toric strata of X0

= faces of Ξ= elements of P.

Near a zero-dimensional stratum(↔ vertex v ∈P) thedegeneration π is defined by thecone at the correspondingvertex of Ξ=⇒ the polarization equals ϕ.

(B,P)

Ξ

v


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Bernd Siebert




Discussion



Model rings

Starting data

Structures

The algorithm

Concludingremarks

Reminder: Tropical dictionary

Tropical data: fan and cone picture

Fan picture Cone picture

P local models for π (X0, L)

(Στ , ϕ : |Στ | → R) (X0, L) local models for π


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Bernd Siebert




Discussion



Model rings

Starting data

Structures

The algorithm

Concludingremarks

Alternative Viewpoint

Reduction to finite order

To find X→ D starting from X0, itsuffices to construct a compatiblesystem of k-th order thickenings Xk

of X0, for any k > 0:

Xk = Proj(C[C (Ξ) ∩ Zd+2]/(tk+1)

).

Note: Xk can be visualized as athickening of ∂Ξ ⊆ Ξ of thickness k:


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Bernd Siebert




Discussion



Model rings

Starting data

Structures

The algorithm

Concludingremarks

Decomposition of Xk

Primary decomposition of Xk

Once we have reduced to finite order, we can decompose intoirreducible components (primary decomposition). There are noembedded components, so a primary component is just athickening of an irreducible component of X0:

X0 = Z 1 ∪ . . . ∪ Z r =⇒ Xk = Z 1k ∪ . . . ∪ Z r

k .

The thickenings Zµk can be read off from ϕ!


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Model rings

The general case

Rings

Affine gluing

Example I

Example II

Starting data

Structures

The algorithm

Concludingremarks

Model rings


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Bernd Siebert



Model rings

The general case

Rings

Affine gluing

Example I

Example II

Starting data

Structures

The algorithm

Concludingremarks

The General Case

Plan

The plan is now to build Xk similarly from standard pieces,which are completely determined from the tropical geometry.We will soon run into troubles because of monodromy, and thewhole point of the construction is to make the necessaryadjustments order by order.

Open-closed decomposition

For our construction it also does not suffice to decompose intoclosed components. Rather we first need to go over to standardopen pieces, which are then further decomposed.


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Bernd Siebert



Model rings

The general case

Rings

Affine gluing

Example I

Example II

Starting data

Structures

The algorithm

Concludingremarks

Standard pieces

The rings

For each k ∈ N and inclusion ω ⊆ τ we now define a ring Rkω,τ :

If ω = v is a vertex ϕ is given by a single-valued ϕv on|Σv | = Rd . The upper convex hull of ϕv defines an affinetoric variety Spec k[Pv ] as before. Take the k-th orderneighbourhood of the stratum given by τ to define Rk

v ,τ .

For each pair ω ⊆ τ we can similarly define the k-th orderneighbourhood of the τ -stratum wrt. ϕω : |Σω| → R.Removes all toric strata not containing the ω-stratum.

τ : selects stratum, ω: selects open set.

Remarks

Rkω,τ neglects all monomials that vanish to order ≥ k on

any maximal cell σ containing τ .

Monomials have directions: Pv → Λv ⊆ TvB, m 7→ m.


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Bernd Siebert



Model rings

The general case

Rings

Affine gluing

Example I

Example II

Starting data

Structures

The algorithm

Concludingremarks

Gluing of standard pieces

R3v ,τ1×R3

v,vR3

v ,τ2

v

τ1

τ2

v τ2

R3v ,τ2

v

R3v ,v

v

τ1

R3v ,τ1


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Bernd Siebert



Model rings

The general case

Rings

Affine gluing

Example I

Example II

Starting data

Structures

The algorithm

Concludingremarks

Local inconsistencies

Near the discriminant locus the result of gluing depends on thechoice of affine chart!

Example

Take ϕ = 0 on left triangle and with slope 1 on right triangle(ϕ(1, 0) = 1).

Result 1: C[x , y ,w ,w−1, t]/(xy − t).

Result 2: C[x , y ,w ,w−1, t]/(xyw−1 − t)

= C[x , y ,w ,w−1, t]/(xy − wt).

wx w−1y

xw

y


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Bernd Siebert



Model rings

The general case

Rings

Affine gluing

Example I

Example II

Starting data

Structures

The algorithm

Concludingremarks

Local correction

The inconsistency can be cured by introduction of a factoremanating from the singularity:

Example

Computation 1:C[x , y ,w ,w−1, t]/(xy(1 + w)−1 − t)

= C[x , y ,w ,w−1, t]/(xy − (1 + w)t).

Computation 2:C[x , y ,w ,w−1, t]/

(xyw−1(1 + w−1)−1 − t)

= C[x , y ,w ,w−1, t]/(xy(w + 1)−1 − t)

= C[x , y ,w ,w−1, t]/(xy − (1 + w)t).

(1 + w)−1

(1 + w−1)−1

(To do these computations we should also localize at 1 + w .)


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Bernd Siebert



Model rings

Starting data

Structures

The algorithm

Concludingremarks

Starting data


to complexmanifolds

Bernd Siebert



Model rings

Starting data

Structures

The algorithm

Concludingremarks

Local models for π and log structure on X0

Situation along codimension 1 stratum Xρ ⊆ X0:

π : X→ Spec k[t] locally described by

xy − fρ(w1, . . . ,wn−1)tb = 0.

fρ = fρ,v ∈ Γ(U,O∗Xρ) is determined uniquely if we choose

xi = zmi ∈ k[Pv ] with m0 = −m1 ∈ Λv ⊆ TvB.

Note: fρ generalizes 1 + w in the previous example.

Relation to log structures

A compatible choice of fρ,v defines a log structure on X0 with alog-smooth morphism to the standard log point. If ∆ ∩ ρ 6= ∅the fρ,v vanish along the singular locus of the log structureZ ⊆ X0. (codim Z = 2.)

Existence: GS’03


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Bernd Siebert



Model rings

Starting data

Structures

Slabs

Walls

Structures

Gluing

Consistency

The algorithm

Concludingremarks

Structures


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Bernd Siebert



Model rings

Starting data

Structures

Slabs

Walls

Structures

Gluing

Consistency

The algorithm

Concludingremarks

Slabs

Change of vertex for fρ,v

If ρ ∈Pn−1, v , v ′ ∈ ρ vertices then2

fρ,v ′ = zmρ

v′v · fρ,v .

mρv ′v ∈ Λρ: Monodromy for path around ρ ∩∆ through v , v ′.

Definition

A slab b is a convex polyhedral subset of some ρ ∈Pn−1

x ∈ b \∆ with

fb,v(x ′) = zmρ

v(x′)v(x) · fb,v(x) via parallel transport, where fory ∈ ρ \∆, v(y) ∈ ρ denotes a vertex in the sameconnected component.

fb,x ≡ fρ,v(x) mod t.

2We omit factors arising from non-standard gluing data for X0.


to complexmanifolds

Bernd Siebert



Model rings

Starting data

Structures

Slabs

Walls

Structures

Gluing

Consistency

The algorithm

Concludingremarks

Walls

Slabs yield perturbations of closed gluings, but these are notenough: We need to also apply automorphisms to affine opensubsets. These are implemented consistently by walls.

Definition

A wall consists of an(n − 1)-dimensional polyhedralsubset p of some σ ∈Pmax ofthe form

p = (q− R≥0 ·mp) ∩ σ,

with p 6⊆ ∂σ, together withmp ∈ Pσ, ord(mp) > 0, andcp ∈ k.

Function analogous to fb,x of slabs: fp := 1 + cpzmp .


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Bernd Siebert



Model rings

Starting data

Structures

Slabs

Walls

Structures

Gluing

Consistency

The algorithm

Concludingremarks

Walls

Slabs yield perturbations of closed gluings, but these are notenough: We need to also apply automorphisms to affine opensubsets. These are implemented consistently by walls.

Definition

A wall consists of an(n − 1)-dimensional polyhedralsubset p of some σ ∈Pmax ofthe form

p = (q− R≥0 ·mp) ∩ σ,

with p 6⊆ ∂σ, together withmp ∈ Pσ, ord(mp) > 0, andcp ∈ k.

Function analogous to fb,x of slabs: fp := 1 + cpzmp .


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Bernd Siebert



Model rings

Starting data

Structures

Slabs

Walls

Structures

Gluing

Consistency

The algorithm

Concludingremarks

Structures

Definition

A structure S is a locally finite set of slabs and walls, suchthat

the slabs define a polyhedral decomposition of∣∣P≤n−1

∣∣.any σ ∈Pmax is subdivided by {p ∈ S | p ⊆ σ} intoconvex polyhedral subsets, called Chambers.

System of rings

For each ω ⊆ τ and chamber u with ω ∩ u 6= ∅ andτ ⊆ σu ∈Pmax obtain a ring Rk

ω,τ (u).


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Bernd Siebert



Model rings

Starting data

Structures

Slabs

Walls

Structures

Gluing

Consistency

The algorithm

Concludingremarks

Gluing morphisms

Change of strata

For ω ⊆ ω′ ⊆ τ ′ ⊆ τ have natural morphisms

Rkω,τ (u)

localiz.−→ Rkω′,τ (u)

quot.−→ Rkω′,τ ′(u).

Crossing a wall or slab

Crossing p or b (p, b ⊆ u ∩ u′) gives the log isomorphism

θ : Rkω,τ (u) ∈ zm 7−→ f −〈m,n〉 · zm ∈ Rk

ω,τ (u′),

with f = 1 + cpzmp or f = fb,x , x ∈ b \∆.


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Bernd Siebert



Model rings

Starting data

Structures

Slabs

Walls

Structures

Gluing

Consistency

The algorithm

Concludingremarks

Consistency and k-th order deformation

Consistency

A structure S is consistent to order k if changing chamberscyclically around a codimension two stratum j of S (a joint)gives the identity in Rk

ω,τ (u):

θj := θr ◦ . . . ◦ θ1 = 1.

Theorem

Assume S is consistent to order k. Then there exists alog-smooth deformation

πk : Xk −→ Spec k[t]/(tk+1)

of X0 of the required sort.

Note: The log structure enters via the order 0-terms in fb.


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Model rings

Starting data

Structures

The algorithm

Step O: Theinitial structure

Logautomorphisms

Step I

Steps II and III

Concludingremarks

The algorithm


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Bernd Siebert



Model rings

Starting data

Structures

The algorithm


Logautomorphisms

Step I

Steps II and III

Concludingremarks

Step O: The initial structure

Construct Sk , consistent to order k, inductively by modifyingSk−1 by order k stuff.

S0

Has only slabs b = ρ ∈Pn−1, fb,x = fρ,v(x).


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Model rings

Starting data

Structures

The algorithm


Logautomorphisms

Step I

Steps II and III

Concludingremarks

Log automorphisms

Log automorphisms

For a joint j ⊆ u consider algebraic group Hj of logautomorphisms of Rk

ω,τ (u) with Lie algebra

hj =⊕

m 6∈Λj, 1≤ord(m)≤k

zm(k⊗Z (Λ⊥j ∩m⊥)

),

[zm∂n, zm′

∂n′ ] = zm+m′∂〈m′,n〉n′−〈m,n′〉n.

(Kontsevich/Soibelman).

Properties

hj preserves holomorphic log volume form.

exp(zm∂n)(zm′

) = exp(zm)〈m′,n〉 · zm′

.

hkj =

⊕ord(m)=k zm

(k⊗ (Λ⊥j ∩m⊥)

)⊆ hj is abelian.


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Bernd Siebert



Model rings

Starting data

Structures

The algorithm


Logautomorphisms

Step I

Steps II and III

Concludingremarks

Step I: Insertion of new walls

Step I

For each joint j ∈ Sk−1 expand θj in Hkj :

θj = exp(∑

mcmzm∂n

).

For any m insert wall((j− R≥0m) ∩ σ, cm,m

).

Remarks

Point: The order of m increases along −m.

If j ⊆ |P≤n−1| we can not literally work in Hkj , because

fb,x contains monomials of order 0. Under the maximaldegeneracy condition it still works, but if j ⊆ |P≤n−2| wehave to modify fb,x for slabs b ⊇ j (hard!).


to complexmanifolds

Bernd Siebert



Model rings

Starting data

Structures

The algorithm


Logautomorphisms

Step I

Steps II and III

Concludingremarks

Steps II and III: Adjustment of slabs andnormalization

Step II: Adjustment of slabs

The adjustment of slabs from the codimension 2 scatteringcauses new inconsistencies. These can be dealt with byspreading the changes along slabs contained in any ρ ⊆Pn−1

by a homological argument.

Step III: Normalization

The remaining inconsistencies are undirectional, i.e. of the formctk∂n. These can be removed by requiring the Taylor series oflog fb,x at any vertex v ∈ ρ = ρb not to contain any puret-terms.


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Bernd Siebert



Model rings

Starting data

Structures

The algorithm

Concludingremarks

Concluding remarks

Local mirror symmetry computations show that t shouldbe the canonical coordinate. Depends crucially on thenormalization procedure!

Structures contain all information about the complexdegeneration.

Remaining task on complex side: Extract period data andA∞-category from structures.

Structures on fan side ←→ tropical disks(to be explained in Mark’s talk).

Documents

From Affine manifolds to complex manifolds - Instanton ... · Mirror symmetry and tropical geomety The Mumford construction Model rings Starting data Structures The algorithm Concluding