臺灣大學數學系演講
博士生論文口試
 
 

演講者：王賜聖
講  題：The Connectedness Problem of Calabi-Yau Moduli Spaces
時  間：2015年06月03日 (星期三) 13:00
地  點：臺灣大學天數館430室
摘  要：
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 $\pi : \widehat{X} \to X$ 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) Suppose that for any $\pi$-flop $\widehat{X}' \to X$ and any factorization $\widehat{X}' \to X' \to X$ with $X' \ne \widehat{X}'$ the 3-fold $X'$ admits no smoothing, 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. Hubsch that all complete intersection Calabi-Yau 3-folds in product of projective spaces are connected through \emph{projective} conifold transitions.