On the geometry and homology of certain simple stratified varieties

We study certain mild degenerations of algebraic varieties which appear in the analysis of a large class of supersymmetric theories, including superstring theory. We analyze Witten's σ-model [Nucl. Phys. B 403 (1993) 159] and find that the non-transversality of the superpotential induces additional singularities and a stratification of the ground state variety. This stratified variety admits certain homology groups such that ⊕ _qH^{2 q} satisfies the "Kähler package" of requirements [Ann. Math. Studies 102 (1982) 303]. Also, this ⊕ _qH^{2 q} extends the "flopped" pair of small resolutions [Nucl. Phys. B 416 (1994) 414; Nucl. Phys. B 330 (1990) 49; Commun. Math. Phys. 119 (1988) 431] to an "(exo)flopped" triple, and is compatible with both mirror symmetry [S.-T. Yau (Ed.), Mirror Manifolds, International Press, Hong Kong, 1990; B. Greene, S.-T. Yau (Eds.), Mirror Manifolds II, International Press, Hong Kong, 1996] and string theory [Mod. Phys. Lett. A 12 (1997) 521; Nucl. Phys. B 451 (1995) 96] results. Finally, we revisit the conifold transition [Nucl. Phys. B 330 (1990) 49] as it applies in our formalism.