Stability Conditions, Wall-crossing and weighted Gromov-Witten Invariants

Details Stability Conditions, Wall-crossing and weighted Gromov-Witten Invariants Stability Conditions, Wall-crossing and weighted Gromov-Witten Invariants

2006-07-23

arxiv.org/abs/math/0607580v2

Mosc. Math. J., 2009, 9, 3-32

Mathematics

Algebraic Geometry

28 pages; v2: references added and updated, addressed referee comments; to appear in Moscow Math Journal

Scientific paper

We extend B. Hassett's theory of weighted stable pointed curves ([Has03]) to weighted stable maps. The space of stability conditions is described explicitly, and the wall-crossing phenomenon studied. This can be considered as a non-linear analog of the theory of stability conditions in abelian and triangulated categories. We introduce virtual fundamental classes and thus obtain weighted Gromov-Witten invariants. We show that by including gravitational descendants, one obtains an $\LL$-algebra as introduced in [LM04] as a generalization of a cohomological field theory.

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