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Title:
Moduli, calibrations, and singularities
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Abstract:
A very effective way to study singularities of Calabi-Yau threefolds is to express the singular space as a limit of smooth spaces. If the complex structure is held fixed while the Kähler class is varied, there are algebraic cycles (calibrated by a power of the Kähler form) which approach zero volume in the limit. If the Kähler class is held fixed while the complex structure is varied, there are ``vanishing cycles'' whose volumes approach zero in the limit. These vanishing cycles are conjectured to have special Lagrangian representatives under appropriate conditions, i.e., to be calibrated by the real part of a holomorphic 3-form. Both kinds of limit can be studied with techniques from algebraic geometry.
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