Given a Lagrangian fibration, we provide a natural construction of a mirror Landau-Ginzburg model consisting of a rigid analytic space, a superpotential function, and a dual fibration based on Fukaya’s family Floer theory. The mirror in the B-side is constructed by the counts of holomorphic disks in the A-side together with the non-archimedean analysis and the homological algebra of the A infinity structures. It fits well with the SYZ dual fibration picture and explains the quantum/instanton corrections and the wall crossing phenomenon. Instead of a special Lagrangian fibration, we only need to assume a weaker semipositive Lagrangian fibration to carry out the non-archimedean SYZ mirror reconstruction

In his Annals of Mathematics paper (2009), Berndtsson proves an important result on the curvature of holomorphic infinite-rank vector bundles whose fibers are Hilbert spaces consisting of holomorphic $L^2$ functions with respect to a family of weight functions $e^{-{\varphi}^(t,\cdot)}$, varying in $t$, over a pseudoconvex domain. Using H&‌ouml;rmander's classical theorem on $L^2$-estimates for the $\bar{\partial}$-operator, he shows that such bundles are positively curved in the sense of Nakano provided that the function $\varphi$ is plurisubharmonic with respect to both variables. This result is at the center of a long-standing project of Berndtsson aiming at formulating a complex analogue of Brunn-Minkowski theory, which first started with his result on the log-plurisubharmonicity of Bergman kernels over pseudoconvex domains (2006). In addition to extending the Brunn-Minkowski theorem and its generalization -the Pr&‌eacute;kopa-Leindler theorem- to the complex setting, Berndtsson's result has deep applications in complex analysis and geometry. For example, his result leads to alternative proofs of existence and uniqueness theorems for K&‌auml;hler-Einstein metrics, as well as optimal $L^2$-extension (or Ohsawa-Takegoshi type) theorems.

Berndtsson’s result can easily be extended to the geometric setting by taking the pseudoconvex domain to be a Stein manifold and the family of weights $e^{-{\varphi}^(t,\cdot)}$ to be a family of positively curved hermitian metrics for a line bundle over the manifold. Using a variant of H&‌ouml;rmander’s theorem due to Donnelly and Fefferman, we show that Berndtsson’s Nakano-positivity result holds under different (in fact, more general) curvature assumptions. This is of particular interest when the manifold admits a negative plurisubharmonic function, as these curvature assumptions then allow for some curvature negativity. We describe this setting as a "twisted" setting. In particular, we extend Berndtsson’s Nakano-positivity result to general trivial families of Stein manifolds. As immediate applications of this result, we prove $log$-plurisubharmonic variation theorems for Bergman kernels, as well as families of compactly supported measures and currents, over general trivial families of Stein manifolds. We then generalize these $log$-plurisubharmonic variation results to a certain class of non-trivial families of Stein manifolds. Finally, we also discuss Prékopa-Leindler type theorems showing, for instance, that the consequence of the Prékopa-Leindler theorem holds under weaker convexity assumptions.

For symplectic 4-manifolds with a real structure, Welschinger (2003) showed that counts of real rational pseudo-holomorphic curves, with appropriate signs, are well-defined invariants. They are called Welschinger invariants and are analogues of Gromov-Witten invariants in the real setting. In 2007, Solomon proposed two WDVV-type relations for them, which determine these numbers recursively in many good cases. They are real analogues of the usual WDVV relation.

We establish Solomon’s WDVV-type relations by implementing Georgieva’s suggestion to lift homology relations from the Deligne-Mumford moduli spaces of stable real curves. This is accomplished by lifting judiciously chosen cobordisms realizing these relations. Our topological approach provides a general framework for lifting relations via morphisms between not necessarily orientable spaces.

A complex projective variety is called rational if there is a Zariski-open subset on which it is isomorphic to a Zariski-open subset of projective space. There has been a huge amount of progress and activity in determining when varieties are rational. One the other hand, one can ask: given a projective variety whose nonrationality is known, can we measure how far it is from being rational?

Measures of irrationality provide an answer to the question above; they are birational invariants that offer an orthogonal viewpoint to questions concerning rationality. They have recently gained interest, in part due to work of Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery [BDELU] on hypersurfaces of large degree. In this dissertation, we make advances in the study of measures of irrationality on abelian surfaces and codimension two complete intersections, which answer a few questions posed in [BDELU].