The Connectedness Problem of Calabi--Yau Moduli Spaces
Date Issued
2015
Date
2015
Author(s)
Wang, Sz-Sheng
Abstract
We develop criteria for a Calabi--Yau 3-fold to be a conifold, i.e. to admit only ODPs as singularities, in the context of extremal transitions. There are birational contraction and smoothing involved in the process, and we give such a criterion in each aspect. More precisely, given a small projective resolution pi : widehat{X} rightarrow X of Calabi--Yau 3-fold X, we show that (1) If the fiber over a singular point P in X is irreducible then P is a cA_1 singular point, and an ODP if and only if there is a normal surface which is smooth in a neighborhood of the fiber. (2) If the natural closed immersion Def(widehat{X}) hookrightarrow Def(X) is an isomorphism then X has only ODPs as singularities. There are topological constraints associated to a smoothing widetilde{X} of X. It is well known that $e(widehat{X}) - e(widetilde{X}) = 2 | Sing(X) | if and only if X is a conifold. Based on this and a Bertini-type theorem for degeneracy loci of vector bundle morphisms, we supply a detailed proof of the result by P.S.~Green and T.~Hübsch that all complete intersection Calabi--Yau 3-folds in product of projective spaces are connected through projective conifold transitions (known as the standard web).
Subjects
Calabi-Yau threefold
conifold transition
small contraction
determinantal contraction
standard web
Type
thesis
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