Maximal unipotent monodromy for complete intersection CY manifolds. (English) Zbl 1067.14035
The authors construct one parameter families of \(n\)-dimensional Calabi-Yau manifolds, which are complete intersections of toric varieties and which have a monodromy operator \(T\) such that \((T^N-\text{id})^{n+1}=0\) but \((T^N-\text{id})^n\neq 0\) for some \(N\in {\mathbb N}\). (The presence of \(N\) is explained by the well-known property of the operators of geometric monodromy their eigenvalues to be roots of unity; the two conditions imply that \(T\) has a Jordan block of size \(n+1\) which is the maximal possible.) In such a case one says that \(T\) is maximal unipotent.
The results are inspired by string theory in the B model which requires the existence of degenerations of Calabi-Yau manifolds with maximum unipotent monodromy. In string theory such a point in the moduli space is called a large radius limit (or large complex structure limit). The authors base their method of proof on the Clemens theory of monodromy and on the theory of mixed Hodge structures.
The results are inspired by string theory in the B model which requires the existence of degenerations of Calabi-Yau manifolds with maximum unipotent monodromy. In string theory such a point in the moduli space is called a large radius limit (or large complex structure limit). The authors base their method of proof on the Clemens theory of monodromy and on the theory of mixed Hodge structures.
Reviewer: Vladimir P. Kostov (Nice)
MSC:
14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
14B05 | Singularities in algebraic geometry |
32Q25 | Calabi-Yau theory (complex-analytic aspects) |
14M10 | Complete intersections |
14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |