Deformations of Noncompact Calabi-Yau threefolds

Abstract

We introduce a new notion of deformation of complex structure, which we use as an adaptation of Kodaira's theory of deformations, but that is better suited to the study of noncompact manifolds. We present several families of deformations illustrating this new approach. Our examples include toric Calabi-Yau threefolds, cotangent bundles of flag manifolds, and semisimple adjoint orbits, and we describe their Hodge theoretical invariants, depicting Hodge diamonds and KKP diamonds.

Classical deformations of noncompact surfaces and their moduli of instantons

“…Identifying B = C k−1 with its tangent space at 0 and using (5.5) we denote a fibre 18 shows that Z k (τ ) is an affine algebraic variety for each τ = 0, whence H 1 (Z k (τ ), T Z k (τ ) ) = 0. Thus Z k (τ ) admits no infinitesimal deformations.…”

“…We nevertheless chose to maintain our direct proofs, working only with the noncompact surfaces, because this approach also proved to be useful for other noncompact spaces, which may admit larger families of deformations than those obtained from a compactification. For instance, for the Calabi-Yau threefold W 2 := Tot(O P 1 (−2) ⊕ O P 1 ) ≃ Z 2 × C, the cohomology H 1 (W 2 , T W2 ) is infinite dimensional over C and [18,19] show that indeed infinitely many directions of this vector space produce inequivalent deformations.…”

Infinite dimensional families of Calabi–Yau threefolds and moduli of vector bundles

“…Every holomorphic vector bundle on W 1 is algebraic [K,Thm. 3.10], and W 1 is formally rigid [GKRS,Thm. 11].…”

“…In contrast, if k > 1, then W k has as infinite-dimensional family of deformations. In particular, a deformation family for W 2 can be given by GKRS,Thm. 13] and this family contains infinitely many distinct manifolds [BGS,Thm.…”