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
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
Picture
Discretization
Fan structures
Tropicaldictionary
The MainTheorem
The Mumfordconstruction
Model rings
Starting data
Structures
The algorithm
Concludingremarks
Mirror symmetry andtropical geomety
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
Picture
Discretization
Fan structures
Tropicaldictionary
The MainTheorem
The Mumfordconstruction
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.
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
Picture
Discretization
Fan structures
Tropicaldictionary
The MainTheorem
The Mumfordconstruction
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).
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
Picture
Discretization
Fan structures
Tropicaldictionary
The MainTheorem
The Mumfordconstruction
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 .
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
Picture
Discretization
Fan structures
Tropicaldictionary
The MainTheorem
The Mumfordconstruction
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
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
Picture
Discretization
Fan structures
Tropicaldictionary
The MainTheorem
The Mumfordconstruction
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
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
The Projconstruction
Discussion
Alternativeviewpoint
Primarydecomposition
Model rings
Starting data
Structures
The algorithm
Concludingremarks
TheMumford construction
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
The Projconstruction
Discussion
Alternativeviewpoint
Primarydecomposition
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.
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
The Projconstruction
Discussion
Alternativeviewpoint
Primarydecomposition
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
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
The Projconstruction
Discussion
Alternativeviewpoint
Primarydecomposition
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
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
The Projconstruction
Discussion
Alternativeviewpoint
Primarydecomposition
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:
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
The Projconstruction
Discussion
Alternativeviewpoint
Primarydecomposition
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!
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
Model rings
The general case
Rings
Affine gluing
Example I
Example II
Starting data
Structures
The algorithm
Concludingremarks
Model rings
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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.
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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.
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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 .)
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
Model rings
Starting data
Structures
The algorithm
Concludingremarks
Starting data
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
Model rings
Starting data
Structures
Slabs
Walls
Structures
Gluing
Consistency
The algorithm
Concludingremarks
Structures
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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.
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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 .
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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 .
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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).
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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 \∆.
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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.
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
Model rings
Starting data
Structures
The algorithm
Step O: Theinitial structure
Logautomorphisms
Step I
Steps II and III
Concludingremarks
The algorithm
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
Model rings
Starting data
Structures
The algorithm
Step O: Theinitial structure
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).
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
Model rings
Starting data
Structures
The algorithm
Step O: Theinitial structure
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.
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
Model rings
Starting data
Structures
The algorithm
Step O: Theinitial structure
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!).
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
Model rings
Starting data
Structures
The algorithm
Step O: Theinitial structure
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.
From Affinemanifolds
to complexmanifolds
Bernd Siebert
Mirrorsymmetry andtropicalgeomety
The Mumfordconstruction
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).