Speaker
Description
Let $G$ be a finite abelian subgroup of $\mathsf{SL}(n, \mathbb{C})$, and suppose there exists a toric crepant resolution $ \phi: X \longrightarrow \mathbb{C}^n / G$ of the quotient variety $\mathbb{C}^n / G$. Let $\mathsf{Exc}(\phi) = E_1 \cup \dots \cup E_s$ be the decomposition of the exceptional set of $\phi$ into irreducible components. In this seminar, I will show that for every $i$ there exists an open subset $U_i$ of $X$ such that $E_i \subset U_i$, and $U_i$ is isomorphic to the total space of the canonical bundle $\omega_{E_i}$ of $E_i$. Furthermore, $X = U_1 \cup \dots \cup U_s$. This contributes to the collection of results aimed at solving a classical problem, i.e., to determine which submanifolds of a complex manifold have a neighborhood isomorphic to a neighborhood of the zero section of their normal bundle.