Derived Moduli Spaces and Applications

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Working with his colleagues, the investigator will develop the formalism of Derived Deformation Theory. This theory aims to resolve in a systematic fashion many of the difficulties arising from the fact that moduli spaces in algebraic geometry relatedto higher dimensional varieties are typically singular. The basic idea is that the correct object to consider for a moduli problem is some ``derived moduli space'' which should be manifestly smooth in an appropriate sense and should carry a differential-graded structure sheaf. The usual moduli space is obtained from the derived version as the degree-zero truncation, and this explains its singularnature. In recent work, Ciocan-Fontanine and Kapranov defined andstudied the ``right derived category of schemes'', whose objects are differential-graded schemes, and they have constructed the derived version of Grothendieck's Quot scheme. The investigator will extend the formalism to a larger category of differential-graded stacks, and use it to construct derived versions of some other important moduli spaces in algebraic geometry (such as moduli of vector bundles, stable maps, etc.). As a first application of the theory, it is proposed to give a simpler and more general construction of the virtual fundamentalclasses of Behrend-Fantechi and Li-Tian. This new construction is expected to be better suited to investigate the properties of virtual fundamental classes in the case of moduli of stable maps of higher genus to a hypersurface in a Fano manifold and to understand mathematicallymirror symmetry at higher genus. The project has also a part dealingwith some explicit calculations of genus zero Gromov-Witten invariants of flag manifolds, and applications to mirror symmetry.This is research in the field of algebraic geometry, which is one of the oldest branches of modern mathematics. In recent years, the methods and ideas of algebraic geometry, especially the study of moduli spaces, have been employed in string theory, a very active part of theoretical physics. Developments in string theory have sparked a fruitful interaction between the two communities of researchers and have led to the discovery and study of many striking new phenomena. Mirror symmetry is one example. It is expected that a better understanding of moduli spaces in algebraic geometry will lead to more applications to string theory.