The relation between minimal resolution and crepant resolution

The conclusion is Crepant resolution is necessarily minimal resolution, but minimal resolution is not necessarily crepant resolution!

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Something about SYZ conjecture

The main idea is to construct the Mirror Calabi-Yau Y from X intrinsically. The starting point is the moduli of special Lagrangian tori with a flat U(1) bundle on it in X, then one can put a metric on this moduli (possibly plus corrections coming from J-holomorphic disks) and expect that this moduli and the metric given on it is the mirror Calabi-Yau.
E.Zaslow’s viewpoint:
The SYZ conjecture can be viewed from different perspectives: geometric, symplectic and topological.

(1) Geometric: this is the most difficult and vague, mathematically, since the geometric counterpart of even a conformal field theory is approximate in nature. For example, a SUSY sigma model with target a compact complex manifold X is believed to lie in the universality class of a conformal field theory when X is CY, but the CY metric does not give a conformal field theory on the nose — only to one loop. Likewise, the arguments about creating a boundary conformal field theory using minimal (CFT) + Lagrangian (SUSY) are only valid to one loop, as well. To understand how the corrections are organized, we should compare to (closed) GW theory, where “corrections” to the classical cohomology ring come from worldsheet instantons — holomorphic maps contributing to the computation by a weighting equal to the exponentiated action (symplectic area). The “count” of such maps is equivalent by supersymmetry to an algebraic problem. No known quantity (either spacetime metric or Kahler potential or aspect of the complex structure) is so protected in the open case, with boundary. That’s why the precise form of the instanton corrections is unknown, and why traction in the geometric lines has been made in cases “without corrections” (see the work of Leung, e.g.). Nevertheless, the corrections should take the form of some instanton sum, with known weights. The sums seem to correspond to flow trees of Kontsevich-Soibelman/ Moore-Nietzke-Gaiotto/Gross-Siebert.

(2) Topological: Mark Gross has proven that the dual torus fibration compactifies to produce the mirror manifold.

(3) Symplectic: Wei Dong Ruan has several preprints which address dual Lagrangian torus fibrations, which come to the same conclusion as Gross (above).

The followings are some papers on SYZ worth reading:

(1) Hitchin’s “The moduli space of special Lagrangian submanifolds” arXiv:dg-ga/9711002

(2) M. Gross’s survey

Hitchin’s paper was written shorly after Mirror Symmetry is T-duality and it is a matematical explanation of the paper. Essentially Maclean proved that the moduli space of SL submanifolds is unobstructed and its tangent space is the space of harmonic 1-forms on the SL submanifold. A natural metric which you can put on the moduli space is the L2 metric on harmonic forms. When the SL submanifold is a torus, the moduli space also has an “affine structure”. This was already known from integrable systems, they are called action coordinates. They are affine because they are defined up to affine transformations (with linear part having integral coefficients). Hitchin shows that with respect to these coordinates the metric can be expressed as the Hessian of a function. Hitchin also shows that the moduli space has two affine strutures (this is because of the “special” condition). The two affine structures are related by Legendre transform using the Hessian (i.e.the metric). So one could say that mirror symmetry is “Legendre transform”.

Things have developed a lot since Hitchin’s paper, and M. Gross surveys these developements.
How to do “quantum corrections” to the metric is a major open problem, there are many approaches.
(1) Auroux deals with it in the paper mentioned above.
(2) Fukaya wants to do it with Floer homology for families.
(3) Kontsevich and Soibelmann use rigid analytic geometry.
(4) Gross-Siebert’s program.
It seems that quantum corrections could be understood in terms of “tropical geometry” in the moduli space of SL tori (an “affine manifold with singularities”). In a recent paper of M. Gross (“Mirror symmetry for P^2 and tropical geometry”), he explains how “period calculations” can be understood in terms of tropical geometry (at least for P^2).

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