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Toward the enumeration of Picard number 4 (Real)
Toric manifolds.
Mathieu ValléeENS Rennes, University of Rennes 1, Ajou University
Jointly with Suyoung Choi and Hyuntae Jang.

March 24, 2021

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

Backgrounds
The way to deal with the problem and store the obtained data
Known and new results

1Backgrounds
Important definitions
Theorems

2The way to deal with the problem and store the obtained data
How to enumerate every Toric manifolds
Database and website

3

Known and new results
History
STEP 1 of the process
New method

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

Backgrounds
The way to deal with the problem and store the obtained data
Known and new results

Important definitions
Theorems

Table of Contents

1Backgrounds
Important definitions
Theorems

2

The way to deal with the problem and store the obtained data
How to enumerate every Toric manifolds
Database and website

3

Known and new results
History
STEP 1 of the process
New method

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

Backgrounds
The way to deal with the problem and store the obtained data
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Important definitions
Theorems

Important definitions
Let K be a simplicial complex on [m].
Definition 1 (Characteristic map).
A (non-singular) Z2 -characteristic map over K is a map λ : [m] → Zn2 . It is
non-singular if it satisfies the so-called non-singularity condition:{i1 , . . . , is } ∈ K ⇔ λ(i1 ), . . . , λ(is ) are linearly independant.
Λ(K ) = {Characteristic maps over K}.
GLn (Z2 ) y Λ(K ), orbits are D-J classes DJ(K ).
λ = λ1 λ2 . . . λm .
We find λ̄ ∈ Mm,m−n (Z2 ) such that λλ̄ = 0.
Definition 2 (Dual characteristic map).
Such λ̄ defines a dual characteristic map λ̄ : [m] → Zm−nover K .
2A simplicial complex K is said colorizable if it supports a (dual) characteristicmap.
Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

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Proposition 3.
Let K be a simplicial complex on [m] of dimension n − 1, λ ∈ DJ(K ), and λ̄ its
dual. Let J be a subset of [m]. The following are equivalent:
1λ̄(J c ) is a basis of Zm−n;22λ(J) is a basis of Zn2 .

So dual characteristic maps and characteristic maps share equivalent data.
Definition 4 (Wedge operation).
Let K be a simplicial complex on V and p ∈ V being a vertex of K . The wedgeof K as p is the simplicial complex on V ∪ {p1 , p2 }\{p} defined as follows:
Wedp (K ) := (I ∗ LkK (p)) ∪ (∂I ∗ K \{p}),(1.1)

where I is the 1-simplex with vertices {p1 , p2 }, and K \F := {σ ∈ K : F 6⊂ σ},
for a face F ∈ K .

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

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Important definitions
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2The pentagon P5435112
Wed1 (P5 )Simplicial complexes which are not wedges are called seeds.
Wedge operation: commutative and associative. We can define a more generalwedge operation.
Definition 5 (Extended wedge operation,Bahri-Benderski-Cohen-Gtiler ).
Let K be a simplicial complex on [m], and J = (j1 , . . . , jm ) ∈ (N? )m . We define
the wedged simplicial complex K (J) as the simplicial complex obtained after
performing ji − 1 wedges on the vertex i for i = 1, . . . , m.

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

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Remark 6.
Any simplicial complex L which is not a seed can always be represented as awedged simplicial complex K (J) with K being a seed.
The combinatorial data of a simplicial complex L is a pair (K , J) with K a seedand J an m-tuple.
But in toric topology, we are working on pairs (K , λ).
Question 7.
Is there a constructive way of obtaining Λ(L) from Λ(K ) and J ?
(Λ(K ), J) −→ (Λ(K (J)).

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

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Definition 8 (projection).
We define the projection of λ over K with respect to a vertex p of K as follows:
projp (λ)(w ) := λ(w )/hλ(p)i.
The projection is a characteristic map on the link of K at the vertex p.
λ1 and λ2 are called p-adjacent if there exists a CM λ over Wedp (K ) such thatprojp1 (λ) = λ1 and projp2 (λ) = λ2 .
G (J): 1-skeleton of ∆J := ∆j1 −1 × . . . × ∆jm −1
its irreducible cycles are triangles and squares.
Definition 9 (Puzzle. Choi, Park, 2017).
A puzzle on a wedged simplicial complex K (J), with K on [m] andJ = (j1 , . . . , jm ) is a map π : V(G (J)) → DJ(K ).
A puzzle is called realizable if the image of the edges, resp. subsquares, of G (J)are p-adjacent, resp. realizable.
A realizable puzzle creates a unique D-J class over K (J).
Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

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Important theorems
K supports λ ⇔ Wed(K ) supports λ0 ;Pic(K ) = Pic(Wed(K ));λ̄ over a seed ⇒ λ̄ is injective (so finite number of seed for a fixed Picard
number). Namely, we have m ≤ 2Pic(K ) − 1 (Choi-Park, 2017);
Puzzle (Choi-Park, 2017):
1:1{Realizable puzzles} ←→ DJ(K ).
(the wedge operation is commutative and associative)

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

Backgrounds
The way to deal with the problem and store the obtained data
Known and new results

How to enumerate every Toric manifolds
Database and website

Table of Contents1Backgrounds
Important definitions
Theorems

2

The way to deal with the problem and store the obtained data
How to enumerate every Toric manifolds
Database and website

3

Known and new results
History
STEP 1 of the process
New method

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

Backgrounds
The way to deal with the problem and store the obtained data
Known and new results

How to enumerate every Toric manifolds
Database and website

Let p be a fixed Picard number.
The fundamental theorem for toric geometry: toric manifolds are classified bycomplete non-singular fans.
If a simplicial complex K supports a non-singular fan, then it always supports amod 2 characteristic map.
The strategy is then to restrict our case to K ’s which support a mod 2characteristic map.

Methoddescription

Direct Garrison-Scott computation
Use the Garrison-Scott algorithm on K (J) directly.

Puzzle method
Use the optimized versionof the Garrison-Scott algorithm on the seed K . And
use the Puzzle algorithm
for getting all the CM on
K (J)..

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

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How to enumerate every Toric manifolds
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Proposition 10 (Choi-Jang-V, 2021).
The puzzle algorithm is more efficient than the traditional Garisson-Scottalgorithm for finding characteristic maps over wedged simplicial complexes.
Thus the last proposition gives us the following methods for finding ”every” real
toric manifolds of Picard number p.
STEP 1
STEP 2
STEP 3

Find CS(p) = {Colorizable seeds K of Pic p} and DJ(K )Compute D(K ) (the characteristic map relation diagram for a puzzle)
for every K ∈ CS(p)Find the realizable puzzles π : V(G (J)) → DJ(K ).
Table: The steps of the process.

Remark 11.
There are infinitely many PL-spheres of Picard number p but any given one
can be calculated from this algorithm;
The finite set CS(p) can be stored in a database for any Picard number p.
Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

Backgrounds
The way to deal with the problem and store the obtained dataKnown and new results

How to enumerate every Toric manifolds
Database and website

See the Website.
The upcoming idea for the website is the following:
A toric topologist wants to know about a specific simplicial complex L;
She or he visits the website and inputs the maximal faces set of L;
The website finds L = K (J) and uses the puzzle algorithm to find the DJclasses over K (J).

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

Backgrounds
The way to deal with the problem and store the obtained data
Known and new results

History
STEP 1 of the process
New method

Table of Contents

1Backgrounds
Important definitions
Theorems

2

The way to deal with the problem and store the obtained data
How to enumerate every Toric manifolds
Database and website

3

Known and new results
History
STEP 1 of the process
New method

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

Backgrounds
The way to deal with the problem and store the obtained data
Known and new results

History
STEP 1 of the process
New method

Some historical results for (general) toric manifolds:
Picard number 1 is trivial;
Picard number 2 (1988,Kleinschmidt) using linear transformations andmatroids;
Picard number 3 (1991, Batyrev based on the work of Kleinschmidtand Sturmfels) ;Enumeration of CS(p) for small p:
CS(1) = {∂∆1 };CS(2) = {∂∆1 ∗ ∂∆1 };CS(3) = {∂∆1 ∗ ∂∆1 ∗ ∂∆1 , P5 , C47 }.
CS(4) =?...

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

Backgrounds
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STEP 1 of the process
New method

We focus on the STEP 1 of the process: finding all seed PL-spheres and theircharacteristic maps.
Classic way:1

Find all PL-spheres (Bistellar move, lexicographic ordering);

2

Select the seeds among them;3

Use the Garrison-Scott algorithm for finding every characteristic maps overthem.

Why is it difficult? Up to n = 11, with 15 vertices: number of such simplicial
15complexes: 2(11) = 21365 ...
Methods B-M or Lexico lowered this complexity but results obtained only up ton = 6 (3 months), n = 7 unreachable.
(n,m)
Colorizable seed PLS
PLSPolytopes(2, 6)
111(3, 7)455(4, 8)
2139
37(5, 9)142337322(6, 10)7336257?

Table: Data for the dimensions where the results have been obtained with the classic
methods.
Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

Backgrounds
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STEP 1 of the process
New method

Description of the new method:
1Restrict the number of IDCM (orbits of the permutation action on thecolumns);

2Fix λ̄, injective in an orbit;
Select the maximal faces compatible with λ̄:

3the set MF(λ̄) = {F1 , . . . , Fq } (Maximal faces), and∂ MF(λ̄) = {f1 , . . . , fp } (facets);
4

Use linear algebra (pseudo manifold condition = a facet fi should beincluded in exactly two maximal faces):
Matrix of the (highest dimensional) boundary operatoron(1 fi ⊂ Fj,MF(λ̄): M = mi,j ∈ Mp,q (Z2 ), with mi,j =0 otherwise
and Fi ∈ MF(λ̄) and fj ∈ ∂ MF(λ̄).
We denote by K(λ̄) the set of simplicial complexes supporting λ̄.
A simplicial complex K ∈ K(λ̄) is a vector in Zq2 .
K(λ̄) ⊂ kerZ2 (M).
Find a basis of the kernel of M (Gaussian elimination) → Finite number oflinear combinations.

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

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Known and new results(n,m)
Colorizable seed PLS
PLSPolytopes(2, 6)
111(3, 7)455

History
STEP 1 of the process
New method(4, 8)
2139
37(5, 9)
142
337
322

(6, 10)733
6257?

(7, 11)
1190?
?

Table: Data for the dimensions where the results have been obtained.

(n,m)
Colorizable seed PLS(8, 12)≥ 627(9, 13)≥ 155(10, 14)≥ 22(11, 15)≥3

Table: Partial results obtained for higher dimensions.

The inequality m ≤ 2Pic(K ) − 1 is optimal for Pic(K ) = 4.

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.

Backgrounds
The way to deal with the problem and store the obtained data
Known and new results

History
STEP 1 of the process
New method

Thank you for listening !

Mathieu Vallée ENS Rennes, University of Rennes 1, Ajou University Jointly with Suyoung Choi and Hyuntae Jang.