TOWARD THE CLASSIFICATION OF (REAL) TORIC MANIFOLDS OFPICARD NUMBER 4
MATHIEU VALLEE

Toric varieties (resp., real toric varieties) are classified by fans (resp., mod 2 fans). Moregenerally, (real) toric spaces can be classified by pairs (K, λ) consisting of a simplicial complex K
and a (mod 2) characteristic map λ over K. It is known that the space obtained from a pair (K, λ)is a smooth manifold if the simplicial complex K is a PL-sphere and λ is non-singular.
The classification of smooth (real) toric varieties for Picard number smaller than 4 has been
entirely achieved (see [1, 2]), so the next step is Picard number 4. The Picard number of a toricvariety over an (n − 1)-dimensional star-shaped PL-sphere K having m vertices is m − n. Thus,
it is natural to define the Picard number of K as Pic(K) = m − n. The wedge operation on avertex of a simplicial complex preserves both the PL-sphereness and the Picard number. Simplicial
complexes which cannot be described as the wedge of a lower dimensional simplicial complex are
called seeds. In [3], Choi and Park have described a way to construct the characteristic maps overany wedged simplicial complex by using a puzzle starting from the characteristic maps over its
seed. In addition, they also showed there are at most finite seeds supporting a characteristic map.
More precisely, in this case when Pic K = 4, we have n ≤ 11. Hence, toward the classification of(real) toric manifolds of Picard number 4, we try to enumerate every seed PL-spheres of Picard
number 4 supporting a mod 2 characteristic map as the very first step.
As for the enumeration of seed PL-spheres supporting mod 2 characteristic maps, the classicalway was to find all PL-spheres and check the seedness and the existence of mod 2 characteristic
maps using the famous Garrison-Scott algorithm. This method leads us to get the complete list
up to n = 6.
In this work, we additionally develop the linear algebraic method using the condition of supporting mod 2 characteristic functions. This method allowed us to obtain the result for n = 7 andto show that the inequality n ≤ 11 is optimal as presented in the following table.
(n,m)(2, 6)(3, 7)(4, 8)(5, 9)(6, 10)(7, 11)
Number of Seed PLS
14211427331190
supporting chr. ftns
Number of PLS
15393376257?
Number of Polytopes
1537322?
?
(n,m)(8, 12)(9, 13)(10, 14)(11, 15)(n, n + 4) with n > 11
Number of Seed PLS>0>0>0≥20
supporting chr. ftns
Table 1. Data for the dimensions where the results have been obtained.
If time permits, I will present a website associated with a database containing all the known
results concerning the topological data of every known PL-spheres.
This is a joint work with Suyoung Choi and Hyuntae Jang.
References
1. Victor V. Batyrev, On the classification of smooth projective toric varieties, Tohoku Math. J. (2) 43 (1991),
no. 4, 569–585. MR 1133869
2. Suyoung Choi and Hanchul Park, Wedge operations and torus symmetries, Tohoku Math. J. (2) 68 (2016), no. 1,91–138. MR 3476138, Wedge Operations and Torus Symmetries II, Canad. J. Math. 69 (2017), no. 4, 767–789. MR 3679694
3.
Department of mathematics, Ajou University, 206, World cup-ro, Yeongtong-gu, Suwon 16499, Republic of Korea
Email address: mathieu.vallee@ens-rennes.fr1