Deformations of Noncompact Calabi-Yau threefolds

TL;DR

This paper investigates the deformation theory of specific noncompact Calabi-Yau threefolds and their vector bundle moduli, revealing qualitative differences among cases and contrasting with simpler twofolds previously studied.

Contribution

It provides explicit calculations of deformations for $W_k$ threefolds and analyzes their vector bundle moduli, highlighting differences across cases and from simpler models.

Findings

Deformations vary qualitatively for each $k=1,2,3$.

Explicit $H^1(W_k, TW_k)$ calculations are provided.

Differences from simpler twofolds are discussed.

Abstract

We describe deformations of the noncompact Calabi-Yau threefolds $W_k = \mbox{Tot}(\mathcal{O}_{\mathbb{P}^1}(-k) \oplus \mathcal{O}_{\mathbb{P}^1}(k-2))$ for $k=1,2,3$, as well as their moduli of holomorphic vector bundles of rank $2$. Deformations are computed concretely by calculations of $H^1(W_k, TW_k)$. Information about the moduli of vector bundles is obtained by analysing bundles that are extensions of line bundles. We show that for each $k=1,2,3$ the associated structures are qualitatively different, and we also comment on their difference from the analogous structures for the simpler noncompact twofolds $\mbox{Tot}(\mathcal{O}_{\mathbb{P}^1}(-k))$ which had been studied previously by the authors. We describe deformations of the noncompact Calabi-Yau threefolds $W_k = \textrm{Tot}(\mathcal{O}_{\mathbb{P}^1}(-k) \oplus \mathcal{O}_{\mathbb{P}^1}(k-2))$ for $k=1,2,3$. We compute deformations concretely by calculations of $\textrm{H}^1(W_k, TW_k)$ via \v{C}ech cohomology. We show that for each $k=1,2,3$ the associated structures are qualitatively different, and we also comment on their difference from the analogous structures of simpler noncompact twofolds $\textrm{Tot}(\mathcal{O}_{\mathbb{P}^1}(-k))$.