What is the Large Complex Structure limit(LCL) of complex moduli space of a CalabiYau 3fold and why do we need to consider LCL in Mirror symmetry.

The "large complex structure limit" is a family of CY manifolds over a punctured disk having the maximal possible unipotent (or, sometimes, quasiunipotent) monodromy. It seems that its existence is proven in this paper http://arxiv.org/abs/math/0008061 "Maximal Unipotent Monodromy for Complete Intersection CY Manifolds Authors: Bong H. Lian, Andrey Todorov, ShingTung Yau" for complete intersections. There are many claims of existence of such family in other papers of Todorov, I am not sure how much of them are correct. For simple examples (such as a K3) it's not hard to find. You can also complete this family adding a fiber in the center and call this fiber "a large complex structure limit", but this is (apparently) not as useful. 


There are several reasons to consider the large complex structure limit of CalabiYau families. Historically the first one was that the mirror of this limit corresponds to the large Kaehler structure limit, which leads to an important simplification because there are no quantum corrections, hence classical intersection theory suffices. A second reason derives from the StromingerYauZaslow conjecture and the work done in this direction by several people, among them Kontsevich, Soibelman, Gross, Wilson, and others. These authors conjectured that the large complex structure limit determines the base of the conjectured torus fibration of the CY variety. This conjecture has been proven in the K3 case by Gross and Wilson and further work has been done by Gross and Siebert. 


In the book "CalabiYau Manifolds and related geometries" D. Gross gives a definition of a large complex structure limit point. The basic intuition, as I understand it, is that the fiber over this point is a calabi yau manifold with a special kind of degeneration. (In one dimension, the elliptic curve over this point would have a pinched point, approaching the lcl point, there would be a vanishing cycle.) In the case that the base space is one dimensional, the conditions given in the book are satisfied, if the point has maximal unipotent mondodromy. In my opinion the definition is not really satisfactory and I am not sure if it is state of the art, although it works well enough to construct the mirror map for the quintic. For your second question you might take a look at what follows the definition in the above book. Basically the properties given suffice (the monodromy weight filtration gives an extra structure on your Cohomology) to find nice enough coordinates locally, that allow one to give an explicit description of the mirror map. That being said, I would love to see a more conceptional reason and definition. 

