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

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It would help if you described where you saw this term, and what you already know, so one can answer with some specificity. – S. Carnahan Nov 23 '10 at 7:09
@ Scott - Its everywhere in Mirror symmetry, in fact Mirror symmetry is a statement about the CYs near large complex structure limit point. e.g. its in page 10 of Morrison's What I am looking for is the simplest definition of LCL point and geometric notions associated with it. – J Verma Nov 23 '10 at 7:46

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 "Maximal Unipotent Monodromy for Complete Intersection CY Manifolds Authors: Bong H. Lian, Andrey Todorov, Shing-Tung 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.

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@ Misha - Thanks for you reply. the point which is not clear to me is that : why do we need to consider the large complex structure limit, is it just for simplification OR something from physics. – J Verma Nov 28 '10 at 20:42
It has many applications in both mathematics and physics. In physics, the canonical version of the Mirror Symmetry (equivalence of the counting function for rational curves and the hypergeometric function associated with the period mapping for a mirror dual variety) is in fact stated using the large complex structure limit (the hypergeometric function is computed around this limit). In mathematics, existence of large complex structure limit has important consequences stated by Todorov, but there are many contradicting claims, and I am not sure which is true. – Misha Verbitsky Nov 28 '10 at 21:19

There are several reasons to consider the large complex structure limit of Calabi-Yau 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 Strominger-Yau-Zaslow 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.

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In the book "Calabi-Yau 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.

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@ gernot - I am actually reading this book and I saw this definition, but like you I am also looking for a more conceptual reason and definition. I read somewhere that Morrison gave the first definition based on Classical Mirror symmetry. – J Verma Nov 23 '10 at 22:02

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