Online Talk

Toric manifolds, that is, complete nonsingular toric varieties, are encoded by a PL sphere together with a characteristic map, which assigns a lattice vector to each vertex. Kleinschmidt and Batyrev classified those of Picard number at most 3, and all of them are projective. From Picard number 4 on, non-polytopal spheres appear, so polytopal methods no longer suffice. By a result of Choi and Park, every toric-colorable PL sphere of fixed Picard number is obtained from one of finitely many seeds by successive wedge operations. In joint work with Suyoung Choi and Hyeontae Jang, we enumerated all toric-colorable seeds of Picard number 4 and the toric manifolds over them. All toric manifolds of Picard number 4 then follow by the puzzle method of Choi and Park. I will present these results and the general framework we now use for higher Picard numbers and dimensions, based on recent and ongoing work with Suyoung Choi.