Symplectic Conifold Transitions

Abstract

We introduce a symplectic surgery in six dimensions which collapses Lagrangian three-spheres and replaces them by symplectic two-spheres. Under mirror symmetry it corresponds to an operation on complex 3-folds studied by Clemens, Friedman and Tian. We describe several examples which show that there are either many more Calabi-Yau manifolds (e.g. rigid ones) than previously thought or there exist "symplectic Calabi-Yaus" -non-Kähler symplectic 6-folds with c1 = 0. The analogous surgery in four dimensions, with a generalisation to ADE-trees of Lagrangians, implies that the canonical class of a minimal complex surface contains symplectic forms if and only if it has positive square.

Complex/Symplectic Mirrors

“…Also, the canonical class K is not modified by the transition because a small resolution does not create a new divisor, only a new curve. 3 Actually, the conjecture that all Calabi-Yau are connected was initially formulated by Reid [12] for all complex manifolds (and not only Calabi-Yaus) with K = 0, extending ideas by Hirzebruch [27].…”

“…Now we perform a small resolution to obtain a manifold M ′ and ask whether this new manifold is still Kähler; this question has been considered also by [31]. As we have seen, the complex property is kept, and the symplectic property is not (though the question in [3] regards more generally symplectic manifolds, disregarding the complex structure, and in particular being more interesting without such a path in complex structure moduli space).…”

“…For one thing, now symplectic moduli are given by H 2 (M, R) [28], so it does not seem promising to look for a point in moduli space where three-cycles shrink. But 2.1 in [3] shows that we can nevertheless shrink a three-cycle symplectically, and replace it by a two-cycle. Whether the resulting M ′ will also be symplectic is not automatic, however.…”

“…Suppose now we start (in the IIA theory) with a symplectic manifold W (whose moduli space is, as we said, modeled on H 2 (M, R)), and that for some value of the symplectic moduli some curves shrink. Then, it turns out that one can always replace the resulting singularities by some three-cycles, and still get a symplectic manifold (Theorem 2.7, [3]). The trick is that T * S 3 , the deformed conifold, is naturally symplectic, since it is a cotangent bundle.…”

“…Whether the resulting M ′ will also be symplectic is not automatic, however. This can be decided using Theorem 2.9 in [3]: the answer is yes precisely when there is at least one relation in homology among the three-cycles. 4 The case of interest in this paper is actually a blending of the two questions considered so far, whether complex or symplectic properties are preserved.…”