Lagrangian Torus Fibration Models of Fano Threefolds
Thomas Prince
Mirror Symmetry and Related Topics

14 December 2017

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 2017

1 / 34

Topological Mirror Symmetry
In this talk we take the point of view suggested by the
Strominger–Yau–Zaslow (SYZ) conjecture: that Mirror Symmetry can berealised by dualising (special) Lagrangian torus fibrations. Our particularinspiration is the following result of Gross from the paper ‘Topological
Mirror Symmetry’ (2001).

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 2017

2 / 34

Topological Mirror Symmetry
In this talk we take the point of view suggested by the
Strominger–Yau–Zaslow (SYZ) conjecture: that Mirror Symmetry can berealised by dualising (special) Lagrangian torus fibrations. Our particularinspiration is the following result of Gross from the paper ‘Topological
Mirror Symmetry’ (2001).

Topological Mirror Symmetry
The quintic threefold X has a ‘well-behaved’ torus fibration π : X → B.
Moreover there is a duality for well-behaved fibrations such that if
π̆ : X̆ → B, is the dual fibration in this case, X̆ is diffeomorphic to anon-singular minimal model of the mirror quintic.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 2017

2 / 34

Topological Mirror Symmetry
In this talk we take the point of view suggested by the
Strominger–Yau–Zaslow (SYZ) conjecture: that Mirror Symmetry can berealised by dualising (special) Lagrangian torus fibrations. Our particularinspiration is the following result of Gross from the paper ‘Topological
Mirror Symmetry’ (2001).

Topological Mirror Symmetry
The quintic threefold X has a ‘well-behaved’ torus fibration π : X → B.
Moreover there is a duality for well-behaved fibrations such that if
π̆ : X̆ → B, is the dual fibration in this case, X̆ is diffeomorphic to anon-singular minimal model of the mirror quintic.
Note that as a consequence of the dual fibration structures we have thatH ev (X , R) ∼= H 3 (X̆ , R) and H 3 (X , R) ∼= H ev (X̆ , R) as expected.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 2017

2 / 34

Quintic threefold
There are two important remarks to make about this torus fibration on thequintic.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 20173 / 34

Quintic threefold
There are two important remarks to make about this torus fibration on thequintic.

Lagrangian Fibrations
Castaño-Bernard–Matessi have shown that the topologial torus fibrationon the quintic can be adjusted slightly to make it a Lagrangian fibration
on a symplectic manifold (homeomorphic to the quintic).

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 20173 / 34

Quintic threefold
There are two important remarks to make about this torus fibration on thequintic.

Lagrangian Fibrations
Castaño-Bernard–Matessi have shown that the topologial torus fibrationon the quintic can be adjusted slightly to make it a Lagrangian fibration
on a symplectic manifold (homeomorphic to the quintic).

Integral Affine structure
The base manifold B ∼= S 3 , is equipped with an affine structure whichcontains certain singularities. In ‘simple’ situations (which will always holdin our constructions) there is a construction of π : X → B from B together
with its affine structure.
In general, we denote the total space of the torus fibration constructedfrom B as X̆ (B).
Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 20173 / 34

Torus fibrations on Fano manifolds
We will describe an extension of the construction of a topological torus
fibration to the Fano threefolds. In this context the base of the fibration we
obtain is a manifold with boundary, and we have the following statement.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 20174 / 34

Torus fibrations on Fano manifolds
We will describe an extension of the construction of a topological torus
fibration to the Fano threefolds. In this context the base of the fibration we
obtain is a manifold with boundary, and we have the following statement.

Main Statement
Given a Fano threefold X there is an integral affine manifold (with corners
and singularities) B such that the numerical invariants of the pair(X̆ (B), π −1 (∂B)) coincide with those of (X , D) for a divisor D ∈ | − KX |.
In the case that X has Picard rank one we show that X̆ (B) is
homeomorphic to X .

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 20174 / 34

Torus fibrations on Fano manifolds
We will describe an extension of the construction of a topological torus
fibration to the Fano threefolds. In this context the base of the fibration we
obtain is a manifold with boundary, and we have the following statement.

Main Statement
Given a Fano threefold X there is an integral affine manifold (with corners
and singularities) B such that the numerical invariants of the pair(X̆ (B), π −1 (∂B)) coincide with those of (X , D) for a divisor D ∈ | − KX |.
In the case that X has Picard rank one we show that X̆ (B) is
homeomorphic to X .
The numerical invariants of X we refer to are −KX3 , h2,1 , ρX and the index
of −KX . Note that our choice of codimension 2 submanifold
D := π −1 (∂B) is completely analogous to the toric situation.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 20174 / 34

Fano classification
We briefly recall the classification of the 105 classes of Fano threefolds
completed by Mori–Mukai, building on work of Fano and Iskovskikh. In
particular, we summarise the classification below according to properties
relevant later.
Picard rank

Total

Toric

−KX not very ample

1
2
3
4
5
≥6

17
36
31
13
3
5

1
4
7
4
2
0

2
3
0
0
0
2

Check: 105 total, 18 smooth toric, and 98 with very ample −KX .
Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 2017

5 / 34

Plan
Rather than identify a fibration on each Fano threefold in turn we aregoing to reconstruct them by (topologically) smoothing Fano toricvarieties. The method proceeds as follows:Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 2017

6 / 34

Plan
Rather than identify a fibration on each Fano threefold in turn we aregoing to reconstruct them by (topologically) smoothing Fano toricvarieties. The method proceeds as follows:1

Find a way of putting (non-trivial) affine structures on moment
polytopes of toric (Q-Gorenstein) Fano 3-folds.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 2017

6 / 34

Plan
Rather than identify a fibration on each Fano threefold in turn we aregoing to reconstruct them by (topologically) smoothing Fano toricvarieties. The method proceeds as follows:1

Find a way of putting (non-trivial) affine structures on moment
polytopes of toric (Q-Gorenstein) Fano 3-folds.

2

For each such affine manifold B, compute invariants of X̆ (B).

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 2017

6 / 34

Plan
Rather than identify a fibration on each Fano threefold in turn we aregoing to reconstruct them by (topologically) smoothing Fano toricvarieties. The method proceeds as follows:1

Find a way of putting (non-trivial) affine structures on moment
polytopes of toric (Q-Gorenstein) Fano 3-folds.

2

For each such affine manifold B, compute invariants of X̆ (B).

3

In the rank one case use topological classification results to proveX̆ (B) is homeomorphic to a Fano manifold.
XP/ X̆ (B)




/B

P◦

Idea: start on the top left, and finish on the top right going via thedeformation of affine structures.
Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 2017

6 / 34

Fano classification and Mirror Symmetry
Our approach dovetails with the computation of quantum periods of Fano
threefolds completed by Coates–Corti–Galkin–Kasprzyk. Starting from any
Fano 3-fold X the authors give Laurent polynomials f such thatZ
πf (t) :=3

1 ^ dxi
xi
T 3 1 − tf
i=1

coincides with the quantum period of X .

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 20177 / 34

Fano classification and Mirror Symmetry
Our approach dovetails with the computation of quantum periods of Fano
threefolds completed by Coates–Corti–Galkin–Kasprzyk. Starting from any
Fano 3-fold X the authors give Laurent polynomials f such thatZ
πf (t) :=3

1 ^ dxi
xi
T 3 1 − tf
i=1

coincides with the quantum period of X .

Compatibility with Mirror Symmetry
Every B we consider such that X̆ (B) models a Fano threefold X isconstructed by deforming the affine structure on a polytope P ◦ , such that
P supports a Laurent polynomial mirror to X .

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 20177 / 34

Affine manifolds with corners...
We first need to describe the objects B which appear in our constructions.

Integral Affine manifold
A manifold B with a maximal atlas A with charts in the group of integral
affine transformations GLn (Z) o Zn .

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 20178 / 34

Affine manifolds with corners...
We first need to describe the objects B which appear in our constructions.

Integral Affine manifold
A manifold B with a maximal atlas A with charts in the group of integral
affine transformations GLn (Z) o Zn .

Affine manifold with corners
A manifold with boundary B with a maximal atlas A with charts in thegroup of integral affine transformations GLn (Z) o Zn . Moreover we insist
that any point b ∈ ∂B has a chart making it the origin of a smooth cone(Rn−k × Rk≥0 , for some k ∈ {1, 2, 3}).

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 20178 / 34

Affine manifolds with corners...
We first need to describe the objects B which appear in our constructions.

Integral Affine manifold
A manifold B with a maximal atlas A with charts in the group of integral
affine transformations GLn (Z) o Zn .

Affine manifold with corners
A manifold with boundary B with a maximal atlas A with charts in thegroup of integral affine transformations GLn (Z) o Zn . Moreover we insist
that any point b ∈ ∂B has a chart making it the origin of a smooth cone(Rn−k × Rk≥0 , for some k ∈ {1, 2, 3}).
Note that for such a manifold X̆ (B) is easy to define:X̆ (B) := T ? B/Λ̆.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 20178 / 34

...and singularities.
Affine manifold with corners and singularities
A triple (B, A, ∆), where B is a topological manifold with boundary, A is
an atlas making B0 := B \ ∆ an affine manifold with corners and ∆ is aunion of submanifolds of codimension at least two.
In fact we only consider a very special class of singularities (simple) which
we can describe explicitly. All of these are derived from the focus-focus
singularity in dimension 2.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 2017

9 / 34

...and singularities.
Affine manifold with corners and singularities
A triple (B, A, ∆), where B is a topological manifold with boundary, A is
an atlas making B0 := B \ ∆ an affine manifold with corners and ∆ is aunion of submanifolds of codimension at least two.
In fact we only consider a very special class of singularities (simple) which
we can describe explicitly. All of these are derived from the focus-focus
singularity in dimension 2.

focus-focus singularity
This is an affine structure with singularities on R2 , with ∆ = {0}, andmonodromy of Λ equal to the monodromy defined on integral homology bya Dehn twist.
Indeed, X̆ (B0 ) is (topologically) exactly the usual local model in
Picard–Lefschetz theory.
Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 2017

9 / 34

...and singularities.
An affine manifold with simple singularities is a triple (B, A, ∆) such that
B0 is an affine manifold with corners and for any point b ∈ ∆ there isneighbourhood U of b ∈ B such that U is affine isomorphic to one of thefollowing possibilities.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201710 / 34

...and singularities.
An affine manifold with simple singularities is a triple (B, A, ∆) such that
B0 is an affine manifold with corners and for any point b ∈ ∆ there isneighbourhood U of b ∈ B such that U is affine isomorphic to one of thefollowing possibilities.
1

The product of a focus-focus singularity times an open interval.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201710 / 34

...and singularities.
An affine manifold with simple singularities is a triple (B, A, ∆) such that
B0 is an affine manifold with corners and for any point b ∈ ∆ there isneighbourhood U of b ∈ B such that U is affine isomorphic to one of thefollowing possibilities.
1

The product of a focus-focus singularity times an open interval.

2

A positive node. This is a trivalent point of ∆ with local monodromies



1 0 1
0 1 0 
0 0 1

Thomas Prince (Magdalen College, Oxford)




1 1 0
0 1 0 
0 0 1

Lagrangian fibrations
1 −1 −1
0 1
0 .
0 0
1

14 December 201710 / 34

...and singularities.
An affine manifold with simple singularities is a triple (B, A, ∆) such that
B0 is an affine manifold with corners and for any point b ∈ ∆ there isneighbourhood U of b ∈ B such that U is affine isomorphic to one of thefollowing possibilities.
1

The product of a focus-focus singularity times an open interval.

2

A positive node. This is a trivalent point of ∆ with local monodromies



1 0 1
0 1 0 
0 0 1
3




1 1 0
0 1 0 
0 0 1




1 −1 −1
0 1
0 .
0 0
1

A negative node. This is a trivalent point of ∆ with local
monodromies given by the transpose of the matrices given for the
positive node.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201710 / 34

...and singularities.
An affine manifold with simple singularities is a triple (B, A, ∆) such that
B0 is an affine manifold with corners and for any point b ∈ ∆ there isneighbourhood U of b ∈ B such that U is affine isomorphic to one of thefollowing possibilities.
1

The product of a focus-focus singularity times an open interval.

2

A positive node. This is a trivalent point of ∆ with local monodromies



1 0 1
0 1 0 
0 0 1




1 1 0
0 1 0 
0 0 1




1 −1 −1
0 1
0 .
0 0
1

3

A negative node. This is a trivalent point of ∆ with local
monodromies given by the transpose of the matrices given for the
positive node.

4

The product of a focus-focus singularity times a half-open interval.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201710 / 34

Compactifications

We have a very straightforward definition of X̆ (B0 ) → B0 , but how should
we extend this to π : X̆ (B) → B? The problem is a local one, and solution
to this is given in detail in D-Branes and Mirror Symmetry, Chapter 6. For
each type of point in ∆ we describe the topology of the fibre we insert.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201711 / 34

Compactifications

We have a very straightforward definition of X̆ (B0 ) → B0 , but how should
we extend this to π : X̆ (B) → B? The problem is a local one, and solution
to this is given in detail in D-Branes and Mirror Symmetry, Chapter 6. For
each type of point in ∆ we describe the topology of the fibre we insert.
1

Over generic singular points insert a pinched T 2 times S 1 .

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201711 / 34

Compactifications

We have a very straightforward definition of X̆ (B0 ) → B0 , but how should
we extend this to π : X̆ (B) → B? The problem is a local one, and solution
to this is given in detail in D-Branes and Mirror Symmetry, Chapter 6. For
each type of point in ∆ we describe the topology of the fibre we insert.
1

Over generic singular points insert a pinched T 2 times S 1 .

2Over a positive node insert a T 3 with a collapsed T 2 .

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201711 / 34

Compactifications

We have a very straightforward definition of X̆ (B0 ) → B0 , but how should
we extend this to π : X̆ (B) → B? The problem is a local one, and solution
to this is given in detail in D-Branes and Mirror Symmetry, Chapter 6. For
each type of point in ∆ we describe the topology of the fibre we insert.
1

Over generic singular points insert a pinched T 2 times S 1 .

2Over a positive node insert a T 3 with a collapsed T 2 .

3Over a negative node insert a T 3 with a pair of solid tori attached.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201711 / 34

Compactifications

We have a very straightforward definition of X̆ (B0 ) → B0 , but how should
we extend this to π : X̆ (B) → B? The problem is a local one, and solution
to this is given in detail in D-Branes and Mirror Symmetry, Chapter 6. For
each type of point in ∆ we describe the topology of the fibre we insert.
1

Over generic singular points insert a pinched T 2 times S 1 .

2Over a positive node insert a T 3 with a collapsed T 2 .

3Over a negative node insert a T 3 with a pair of solid tori attached.

4Over a singular point in ∂B insert a pinched T 2 .

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201711 / 34

Polytopes to affine manifolds
Starting from a Fano polytope P, how can we construct B? We will
encode the data used to define B in degeneration data, which depends on
two auxiliary combinatorial notions. We fix a Fano polytope P ⊂ NR ∼= R3
and a rational fan Σ in MR .

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201712 / 34

Polytopes to affine manifolds
Starting from a Fano polytope P, how can we construct B? We will
encode the data used to define B in degeneration data, which depends on
two auxiliary combinatorial notions. We fix a Fano polytope P ⊂ NR ∼= R3
and a rational fan Σ in MR .

Edge data
Edge data is a one-dimensional torus invariant cycle C on the toric variety
XP . Moreover we demand that C is supported on the collection of thosetorus invariant curves of XP whose images under the moment map
XP → P ◦ are contained in a two-dimensional cone of Σ.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201712 / 34

Polytopes to affine manifolds
Starting from a Fano polytope P, how can we construct B? We will
encode the data used to define B in degeneration data, which depends on
two auxiliary combinatorial notions. We fix a Fano polytope P ⊂ NR ∼= R3
and a rational fan Σ in MR .

Edge data
Edge data is a one-dimensional torus invariant cycle C on the toric variety
XP . Moreover we demand that C is supported on the collection of thosetorus invariant curves of XP whose images under the moment map
XP → P ◦ are contained in a two-dimensional cone of Σ.

Ray data
Ray data is a set J := {J(ρ) : ρ ∈ Σ(1)} of collections of nef divisors oneach torus invariant hypersurface Xρ of XΣ .

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201712 / 34

Degeneration data

This is our central combinatorial notion. We say the triple (Σ, C , J) forms
degeneration data for P if:
1

J is smooth: all polyhedra of sections are standard simplices.

2

C and J are compatible: the numbers C assigns to edges
neighbouring a vertex v of P ◦ contained in a ray ρ ∈ Σ(1) equal theedge lengths of the sum of the polyhedra in J(ρ).

3

C is smooth and convex. If P is reflexive this means that C assigns anumber to an edge E of P ◦ which is either equal to `(E ? ), the lengthof E ? , or to `(E ? ) − 1.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201713 / 34

Example
To show how this data is used to describe an affine structure we consider
the prototypical example of P3 . Let P be the convex hull of the standard
basis and the point (−1, −1, −1).

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201714 / 34

Example
To show how this data is used to describe an affine structure we consider
the prototypical example of P3 . Let P be the convex hull of the standard
basis and the point (−1, −1, −1).

Degeneration dataLet Σ be the normal fan of P, let C be the sum of the torus invariant
curves of P3 (assigning 1 to every edge of P ◦ ), and let J be the trivial
Minkowski decomposition of each facet of P into itself.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201714 / 34

Example
To show how this data is used to describe an affine structure we consider
the prototypical example of P3 . Let P be the convex hull of the standard
basis and the point (−1, −1, −1).

Degeneration dataLet Σ be the normal fan of P, let C be the sum of the torus invariant
curves of P3 (assigning 1 to every edge of P ◦ ), and let J be the trivial
Minkowski decomposition of each facet of P into itself.

Affine structure
To describe an affine structure we first identify ∆. Note that eachσ ∈ Σ(2) defines a polyhedral subset c of P ◦ and C defines a divisor on
the toric variety TV (c) (we call these pairs (c, D) slabs).

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201714 / 34

Example
Affine structure
Each toric variety TV (c) is isomorphic to P(1, 1, 4) and the divisor
described by C is in the class O(4) (corresponding to the base of thetriangle c). Let Γc be the dual graph to a triangulation of the polyhedronof sections of O(4). We embed these graphs into c as shown below. Glue
these curves together at points on the rays (positive nodes), trivalentpoints of each Γc become negative nodes.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201715 / 34

Polytopes to affine manifolds
In fact we rely on a number of methods for constructing degeneration
data, we describe the most important one here.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201716 / 34

Polytopes to affine manifolds
In fact we rely on a number of methods for constructing degeneration
data, we describe the most important one here.

Smooth Minkowski decompositions
Given a reflexive polytope P and Minkowski decompositions of all of its
facets into standard simplices, we can assign degeneration data as follows.
1

Let Σ be the normal fan of P.

2

Let C be the cycle E 7→ `(E ? ).

3

Let J be the divisors corresponding to the chosen Minkowski
decompositions.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201716 / 34

Polytopes to affine manifolds
In fact we rely on a number of methods for constructing degeneration
data, we describe the most important one here.

Smooth Minkowski decompositions
Given a reflexive polytope P and Minkowski decompositions of all of its
facets into standard simplices, we can assign degeneration data as follows.
1

Let Σ be the normal fan of P.

2

Let C be the cycle E 7→ `(E ? ).

3

Let J be the divisors corresponding to the chosen Minkowski
decompositions.

We observe that the condition on P is very restrictive. In fact 89 of the
105 Fano threefolds are constructed this way. Note that the previous
example is a special case of this construction, and that the slabs arealways wps, with divisor class O(`(E )`(E ? )).
Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201716 / 34

Two dimensional situation

Our constructions directly generalise a two-dimensional one, in which anyof the 10 del Pezzo surfaces may be obtained by making nodal trades froma polygon. We summarise the two-dimensional affine manifolds importantin our construction below.
BX̆ (B)

Affine structure

Polygonpolarised toric variety

B is the image of the moment map

S2

K3 surface

24 focus-focus singularities

Disc

Del Pezzo surface dPd

12 − d focus-focus singularities

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201717 / 34

Product constructions
In fact the two-dimensional cases automatically give us 5 examples bytaking products of non-toric del Pezzo surfaces with P1 . The images
below shows the base of a torus fibration on dP4 × P1 . We see that the
fan on which ∆ is supported is not the normal fan of P.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201718 / 34

Example: MM3–5

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201719 / 34

Example: MM4–2

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201720 / 34

Euler numbers
We can easily compute the Euler number of the manifold X̆ (B) by taking
the sum of the Euler numbers of the fibres which do not contain a circle
factor.
Special fibre
Positive node
Negative node
Point in ∆ ∩ ∂B
Vertex of B

Euler number
1
−1
1
1

Formula for Euler number
In the case that B has smooth boundary, and is constructed using our first
method, the Euler number obeys the following formula:Xe(X̆ (B)) = 24 + T −`(E )2 `(E ? ),E ∈edges(P)Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201721 / 34

Degree
The next invariant we need to consider is the anti-canonical degree, which
we replace in our topological model with the integer [π −1 (∂B)]3 .

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201722 / 34

Degree
The next invariant we need to consider is the anti-canonical degree, which
we replace in our topological model with the integer [π −1 (∂B)]3 .

Degree calculation
The self-intersection number D 3 , where D := [π −1 (∂B)], is equal to theanti-canonical degree of the toric variety XP .
This is expected from the deformation of XP to X̆ (B).

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201722 / 34

Degree
The next invariant we need to consider is the anti-canonical degree, which
we replace in our topological model with the integer [π −1 (∂B)]3 .

Degree calculation
The self-intersection number D 3 , where D := [π −1 (∂B)], is equal to theanti-canonical degree of the toric variety XP .
This is expected from the deformation of XP to X̆ (B). We can give anentirely topological proof:
1

Construct a (specific) push-off D 0 of D so that C := D 0 ∩ D is asurface in D.

2

Find the genus of C by taking the image under C and computing the
genus of the tropical curve.

3

Use Lefschetz theorem on (1, 1)-classes and genus formula tocompute C 2 = 2g (C ) − 2.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201722 / 34

Contraction map
We compute the Betti numbers of X̆ (B) via the contraction map
ξ : X̆ (B) → X̆0 (B), the union of toric varieties defined by decomposing P ◦along Σ. We tabulate the possible fibres over points in the interior of B.
codimension of π0 (p)˜p∈∆ξ −1 (p)

0

no

point

1

no

S1

1

yes

point

2

no

T2

2

yes

point or S 1

3

no

T3

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201723 / 34

A Leray spectral sequence
Our aim is to compute H 2 (X̆ (B), Q). If this is achieved all other Betti
numbers will follow from the Euler number formula and simply
connectedness of X̆ (B).

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201724 / 34

A Leray spectral sequence
Our aim is to compute H 2 (X̆ (B), Q). If this is achieved all other Betti
numbers will follow from the Euler number formula and simply
connectedness of X̆ (B).

Leray spectral sequence
The spectral sequence H p (X̆0 (B), R q ξ? Q) ⇒ H p+q (X̆ (B), Q) has thefollowing truncated E2 page.
Q
0?
0 QR−1 ?
Q
0
Q 0

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201724 / 34

A Leray spectral sequence
Our aim is to compute H 2 (X̆ (B), Q). If this is achieved all other Betti
numbers will follow from the Euler number formula and simply
connectedness of X̆ (B).

Leray spectral sequence
The spectral sequence H p (X̆0 (B), R q ξ? Q) ⇒ H p+q (X̆ (B), Q) has thefollowing truncated E2 page.
Q
0?
0 QR−1 ?
Q0
Q 0
We often make use of the maps ik from the disjoint union of thecodimension k toric non-boundary strata into X̆0 (B).
Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201724 / 34

A Leray spectral sequence
For example, consider H 0 (X̆0 (B), R 2 ξ? Q). We have an injection,
R 2 ξ? Q → i2? i2 ? R 2 ξ? Q,

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201725 / 34

A Leray spectral sequence
For example, consider H 0 (X̆0 (B), R 2 ξ? Q). We have an injection,
R 2 ξ? Q → i2? i2 ? R 2 ξ? Q,
By left exactness of global sections we have an inclusion
H 0 (R 2 ξ? Q) ,→ H 0 (i2? i2? R 2 ξ? Q).

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201725 / 34

A Leray spectral sequence
For example, consider H 0 (X̆0 (B), R 2 ξ? Q). We have an injection,
R 2 ξ? Q → i2? i2 ? R 2 ξ? Q,
By left exactness of global sections we have an inclusion
H 0 (R 2 ξ? Q) ,→ H 0 (i2? i2? R 2 ξ? Q).
In fact i2? i2? R 2 ξ? Q is the direct sum of its restrictions to the one
dimensional strata of the decomposition of P ◦ induced by Σ. Computing
this following the argument of Gross we have that,dim H 0 (X̆0 (B), R 2 ξ? Q) = dim H 0 (X̆0 (B), i2? i2 ? R 2 ξ? Q) = 0.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201725 / 34

Computing H 2
We describe a general procedure for computing b2 (X̆ (B)) in the case B is
obtained from smooth Minkowski decompositions of the facets of P.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201726 / 34

Computing H 2
We describe a general procedure for computing b2 (X̆ (B)) in the case B is
obtained from smooth Minkowski decompositions of the facets of P.

The space Γ(Σ, C )Fix an orientation of every 1-dimensional space MQ /hσi for σ ∈ Σ(2).
Then Γ(Σ, C ) ⊂ QΣ(2) is (almost!) defined toPbe the subspace of elements
γ = (γ1 , . . . , γn ) where n = |Σ(2)| such that i∈I γi = 0 whenever theelements of Σ(2) corresponding to I appear as the support of singularlocus near a point of ∆ ∩ Σ(1).

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201726 / 34

Computing H 2
We describe a general procedure for computing b2 (X̆ (B)) in the case B is
obtained from smooth Minkowski decompositions of the facets of P.

The space Γ(Σ, C )Fix an orientation of every 1-dimensional space MQ /hσi for σ ∈ Σ(2).
Then Γ(Σ, C ) ⊂ QΣ(2) is (almost!) defined toPbe the subspace of elements
γ = (γ1 , . . . , γn ) where n = |Σ(2)| such that i∈I γi = 0 whenever theelements of Σ(2) corresponding to I appear as the support of singularlocus near a point of ∆ ∩ Σ(1).

Computation of b2 (X̆ (B))
In this context
b2 (X̆ (B)) = dim Γ(Σ, C ) − 2

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201726 / 34

Example computation (V22 )Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201727 / 34

Topological Classification

To prove that the rank one Fano varieties are homeomorphic to the
manifolds X̆ (B) we construct we use the classification of 6-manifolds(Wall, Jupp). This says that 1-connected topological manifolds withtorsion-free homology which admit a smooth structure are classified by thefollowing invariants:
1b3 (X ), the third Betti number,2

H 2 (X , Z), the second integral cohomology group,3

w2 (X ) ∈ H 2 (X , Z2 ), the second Stiefel–Whitney class,

4

FX , the cubic form on H 2 (X , Z),

5

p1 (X ) ∈ H 4 (X , Z), the first Pontryagin class.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201728 / 34

Topological Classification
The Betti numbers have already been computed, for the form on H 2 wehave:

Cubic form
The form FX is determined in the case b2 (X̆ (B)) = 1 by the value of[π −1 (∂B)]3 as well as the index of [π −1 (∂B)] ∈ H 2 (X̆ (B), Z).
The characteristic classes are computed in similar ways, using the
restriction

Second Stiefel–Whitney classw2 (X̆ (B)) = PD[π −1 (∂B)] mod 2. We compute the cap product with D
using the projection formula. Let θ : D ,→ X̆ (B) be the inclusion, then,θ? θ? w2 (X̆ (B)) _ [D] = w2 (X̆ (B)) _ θ? [D]Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201729 / 34

Topological Classification
Computing w2 (X̆ (B))
We can computeθ? w2 (X̆ (B)) = w2 (T X̆ (B)|D )= w2 (TD ⊕ ν(D))= w2 (TD) + w2 (ν(D))
Using the Whitney product formula and naturality. Observing that
w2 (TD) = 0 and w2 (ν(D)) = e(ν(D)) mod 2 we see that the cap
product of w2 (X̆ (B)) with D is exactly the cup of D with itself.
Since b2 (X̆ (B)) = 1 this is sufficient to prove the result, so long as theindex of D is odd. In general we can use the same argument with an
integral lift of the second Stiefel Whitney class.
Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201730 / 34

Topological Classification
Lemma
Given a Fano threefold X , p1 (X )c1 (X ) = −KX3 − 48. This follows
immediately from HRR and p1 (X ) = −2c2 (X ) + c1 (X )2 .

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201731 / 34

Topological Classification
Lemma
Given a Fano threefold X , p1 (X )c1 (X ) = −KX3 − 48. This follows
immediately from HRR and p1 (X ) = −2c2 (X ) + c1 (X )2 .

Computing p1 (X̆ (B))
Noting that we can canonically identify H 2 (X , Z) and H 2 (X̆ (B), Z) andcubic forms so that [π −1 (∂B)] is identified with c1 (X ), we use theprojection formula
p1 (X̆ (B)) _ θ? [D] = θ? θ? (p1 (X̆ (B))) _ [D] ,and we have that θ? p1 (X̆ (B)) = p1 (D) + p1 (ν(D)). However, D isdiffeomorphic to a K3 surface so,
p1 (D) = −2c2 (D) + c1 (D)2 = −2c2 (D) = −48. Moreover p1 (ν(D)) is theEuler class of ν(D) ⊕ ν(D), which is precisely [D]3 .
Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201731 / 34

Example: B3 and its blow-up

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201732 / 34

Where next?
There are several things to be doing in this direction.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201733 / 34

Where next?
There are several things to be doing in this direction.

Fano 3-folds
Find a general procedure for more general facet decompositions. Find ageneralised formula for Euler number, understand the non-reflexive casebetter.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201733 / 34

Where next?
There are several things to be doing in this direction.

Fano 3-folds
Find a general procedure for more general facet decompositions. Find ageneralised formula for Euler number, understand the non-reflexive casebetter.

Calabi–Yau 3-folds and Fano 4-folds
Give analogous constructions for a class of reflexive 4-topes. Compute
topological invariants in this case. Smooth boundaries are more interesting
here, corresponding to CY hypersurfaces in general Fano 4-folds.

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201733 / 34

Where next?
There are several things to be doing in this direction.

Fano 3-folds
Find a general procedure for more general facet decompositions. Find ageneralised formula for Euler number, understand the non-reflexive casebetter.

Calabi–Yau 3-folds and Fano 4-folds
Give analogous constructions for a class of reflexive 4-topes. Compute
topological invariants in this case. Smooth boundaries are more interesting
here, corresponding to CY hypersurfaces in general Fano 4-folds.

Mirror Symmetry
Tropical superpotential calculation in some of these examples. Relate SYZ
mirror symmetry and Minkowski polynomials of
Coates–Corti–Galkin–Golyshev–Kasprzyk.
Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201733 / 34

The End

Thomas Prince (Magdalen College, Oxford)

Lagrangian fibrations

14 December 201734 / 34