A Small and Non-simple Geometric Transition

Abstract

Following notation introduced in the recent paper [41], this paper is aimed to present in detail an example of a small geometric transition which is not a simple one i.e. a deformation of a conifold transition. This is realized by means of a detailed analysis of the Kuranishi space of a Namikawa cuspidal fiber product, which in particular improves the conclusion of Y. Namikawa in Remark 2.8 and Example 1.11 of [29]. The physical interest of this example is presenting a geometric transition which can't be immediately explained as a massive black hole condensation to a massless one, as described by A. Strominger [44].

Elliptic threefolds with high Mordell-Weil rank

“…The second way to compute to H ch is via the localised deformations of the singular fibration X defined in Theorem 2.1. Rossi proves that each of the six singularities of X can be deformed to 3 nodes [22]. Each node contributes +1 to H ch , yelding H ch = 3 × 6 = 18 as well.…”

“…Namikawa and Rossi studied the deformation properties of a particular class of Schoens, "the Namikawa examples" [20,22]. We investigate the properties of these threefolds as elliptic varieties.…”

“…Of the proofs in [20] and [22] we report only the following Lemmas, which prove (2), (3), ( 4) and ( 5) of Theorem 2.1, because we will use some details of the proofs.…”