Double solids

Geometric transitions

“…The present section will be essentially devoted to explain the basic argument given by H. Clemens in [21]. As a consequence we get that locally a conifold transition is described by a suitable surgery.…”

“…between elliptic curves). (3) The transition process was firstly (locally) observed by H. Clemens in the study of double solids V admitting at worst nodal singularities [21]: in his Lemma 1.11 he pointed out "the relation of the resolution of the singularities of V to the standard…”

“…We will refer to the matrix A as the Clemens' matrix : in fact it is the same matrix appearing in formula (1.18) of [21]. For any [y] ∈ P 1 C , one can easily check that A[y] ∈ SO(4) and moreover it is antisymmetric i.e.…”

“…Analogously geometric transitions may be the right way to give a notion of "connectedness" to the "moduli space" of Calabi-Yau 3-folds. This is essentially the famous Reid's fantasy [62] founded on deep speculations due to H. Clemens [21], R. Friedman [27], F. Hirzebruch [38] and J. Werner [76]. 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.…”

“…In particular the fundamental example of a non-trivial conifold transition involving a quintic 3-fold in P 4 is developed in detail. Section 2 is a revised version of some of the "topological considerations" given by H. Clemens in [21], which allow to locally think a conifold transition as a surgery in topology (Proposition 2.10). In section 3 the global change in topology induced by a conifold transition is carefully studied, relying each other homological invariants of all of the three poles of a conifold transition (theorem 3.2).…”