Connecting moduli spaces of Calabi-Yau threefolds

Abstract

We demonstrate that many families of Calabi-Yau threefolds consist generically of small resolutions of nodal forms in other families and, in fact, that a large class of families is connected by this relation. Our result resonates with a conjecture of Reid that Calabi-Yau threefolds may have a universal moduli space even though they are of different homotopy types. Such ideas tie quite naturally to alluring prospects of unifying (super)string models.

Calabi-Yau manifolds with large Picard number

“…Much has been written recently on the role of Calabi-Yau manifolds in supersting theory (see for example [-4, 5, 18, 19, 39, 48]), and many specific examples have been studied in great detail [12,11,13,14,17,36,37]. Recall that a Calabi-Yau manifold is a smooth complex projective threefold with trivial canonical bundle and no global 1-forms or 2-forms -equivalently it is a projective manifold with S U(3) holonomy [2].…”

Geometric transitions

“…On the other hand, in physics, the same property provides a mathematical tool to connect topologically distinct compactifications to 4 dimensions of 10-dimensional type II super-string theory vacua. This fact was firstly observed by P. Candelas, A. M. Dale, P. S. Green, T. Hübsch, C. A. Lütken and R. Schimmirgk in [17], [31], [32], [18], [19]. The physical interpretation of a geometric transition connecting two topologically distinct string vacua was given later, in 1995, by A. Strominger [68], at least in the case of a conifold transition i.e.…”

“…This is the so called Calabi-Yau web conjecture described in many insightful papers starting from 1988 (see [17], [31], [32], [18], [19]). A more precise version of this conjecture will be given later following M. Gross (see 5.3).…”

“…Then the question is: can that graph be enlarged to a very bigger graph connecting deformation classes of all simply connected Calabi-Yau 3-folds? Evidences for such a conjecture were firstly given in [32], where the moduli spaces of some Calabi-Yau 3-folds, which are complete intersections in products of projective spaces, were connected each other. Most significative evidences are given in [20] where a general procedure for connecting up Calabi-Yau 3-folds which are complete intersections in some toric variety, is described.…”

A linear bound on the Euler number of threefolds of Calabi–Yau and of general type

“…For physical reasons (such as the compactification of the heterotic string) it became important to study the Hodge theory of Calabi-Yau threefolds (see for example [CHSW, COGP]) and in particular of their Euler numbers (and this carried all the consequences and beautiful predictions nowadays known as mirror symmetry conjectures). The similarity with K3 surfaces already suggested that some kind of finiteness result should hold for Calabi-Yau threefolds (see for example [R, H1, G2]) but the ideas of physics contributed both to clarify what the picture might be for the moduli space of Calabi-Yau threefolds ( [CGH,GH1,GH2,G1]) and, at least from a numerical point of view, what Euler numbers to expect (currently all the known examples satisfy −960 ≤ e(X) ≤ 960 [G1,HS]). …”