We study how Gromov-Witten invariants of projective 3-folds transform under a standard flop and a small extremal transition in the algebro-geometric setting from the recent development of algebraic relative Gromov-Witten theory and its applications. This gives an algebro-geometric account of Witten's wall-crossing formula for correlation functions of the descendant nonlinear sigma model in adjacent geometric phases of a gauge linear sigma model and of the symplectic approach in an earlier work of An-Min Li and Yongbin Ruan on the same problem. A terse account from the stringy and the symplectic viewpoint is given in the appendix to complement and compare to the discussion in the main text.

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In this paper, it is proved that the Hodge metric completion of the moduli space of polarized and marked Calabi-Yau manifolds, ie the Torelli space, is a complex affine manifold. As applications we prove that the period map from the Torelli space and the extended period map from its completion space, both are injective into the period domain, and that the completion space is a bounded domain of holomorphy with a complete Khler-Einstein metric. As a corollary we show that the period map from the moduli space of polarized Calabi-Yau manifolds with level m structure is also injective.

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Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of Q-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincar´e transform, which has several consequences. In many concrete geometric contexts these functions provide useful invariants. We illustrate this with two different applications, the first one to GVsubschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties.