三維卡拉比-丘空間奇異點及模空間連結性研究

Abstract

本文探討在雙有理映射及形變理論的操作下，給出判別三維卡拉比-丘簇的奇異點是否為節點（即米爾諾數等於一）的條件。同時也對於P.S. Green和T. Hübsch教授的結果：在乘積射影空間裡的三維完全交集卡拉比—丘流形皆可由錐過渡變換連接，提供一個詳細的證明。

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).