We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D∈|-KY| is a smooth divisor with D^2=d , then X=Y\D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y=P^2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type I_d fiber appearing as a singular fiber in a rational elliptic surface π: Y →P^1.

We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles ≤ π (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles ≤ π, possibly with boundary consisting of totally geodesic hyperbolic turnovers.To that end we first generalize the local rigidity result contained in [Wei] to the setting of hyperbolic cone-3-manifolds of finite volume as above. We then use the techniques developed in [BLP] to deform the cone-manifold structure to a complete non-singular or a geometric orbifold structure, where global rigidity holds due to Mostow-Prasad rigidity, cf. [Mos], [Pra], in the hyperbolic case,resp. [deR], cf. also [Rot], in the spherical case. This strategy has already been implemented successfully by [Koj] in the compact hyperbolic case if the singular locus is a link using HodgsonKerckhoff local rigidity, cf. [HK].

In this paper we obtain a stability theorem of generalized K¨ahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of K¨ahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact K¨ahler manifolds which admits deformations to generalized complex manifolds and obtain non-trivial generalized K¨ahler structures on Fano surfaces and toric K¨ahler manifolds. In particular, we show that every nonzero holomorphic Poisson structure on a K¨ahler manifold induces deformations of non-trivial generalized K¨ahler structures.

Getzler-Jones-Petrack introduced $A_\infty$ structures on the equivariant complex for manifold $M$ with smooth $\mathbb{S}^1$ action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation we obtained a new set of $A_\infty$ structures. We extend and prove Fukaya's conjecture relating this Witten's deformed equivariant de Rham complexes, to a new Morse theoretical $A_\infty$ complexes defined by counting gradient trees with jumping which are closely related to the $\mathbb{S}^1$ equivariant symplectic cohomology proposed by Siedel.

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