By Dusa McDuff and Dietmar Salamon
J$-holomorphic curves revolutionized the learn of symplectic geometry whilst Gromov first brought them in 1985. via quantum cohomology, those curves are actually associated with some of the most enjoyable new rules in mathematical physics. This booklet provides the 1st coherent and entire account of the idea of $J$-holomorphic curves, the main points of that are almost immediately scattered in numerous examine papers. the 1st half the booklet is an expository account of the sphere, explaining the most technical facets. McDuff and Salamon supply entire proofs of Gromov's compactness theorem for spheres and of the life of the Gromov-Witten invariants. the second one half the ebook specializes in the definition of quantum cohomology. The authors identify that this multiplication exists, and provides a brand new facts of the Ruan-Tian outcome that's associative on acceptable manifolds. They then describe the Givental-Kim calculation of the quantum cohomology of flag manifolds, resulting in quantum Chern periods and Witten's calculation for Grassmannians, which pertains to the Verlinde algebra. The Dubrovin connection, Gromov-Witten strength on quantum cohomology, and curve counting formulation also are mentioned. The e-book closes with an overview of connections to Floer conception
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J$-holomorphic curves revolutionized the examine of symplectic geometry whilst Gromov first brought them in 1985. via quantum cohomology, those curves at the moment are associated with the various most enjoyable new rules in mathematical physics. This publication provides the 1st coherent and whole account of the speculation of $J$-holomorphic curves, the main points of that are almost immediately scattered in a number of examine papers.
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Extra resources for J-holomorphic curves and quantum cohomology
U) (with p ~ 2) we have d(df) = o. 13), e, /\ e: = 0, ej /\ e, -e, /\ ej, = and, by Schwarz's 82f _ 82f 8x .. 13 d! f* L 0 o d, that is, the following diagram n;(V) . d nr+l(u) LnrH(v) =p-l =p-l The expression given here for the exterior derivative resorts to the canonical basis for Ar E*. 14. Proposition. l;(U) and eo, ... , er are elements of E, we have, for any x E U: r da(x)(eo, ... , er) = ~)-I)"a'(x)(e.. )(eo, ... , e. ,... ,er), .. =0 where a'(x) denotes the derivative of a: U stands for (eo, ...
2. 5 becomes G t 0 G a = G Ha for every t, 8 E R, and it expresses the fact that t ...... G t is a homomorphism from the additive group R into the group of homeomorphisms of U. We say that the G t (t E R) form a one-parameter group. In general, for arbitrary 1)(1), the G t form a semigroup of local homeomorphisms of U. 40 1. 5 says that walking along a for tl seconds, then for t seconds, is the same as walking along a for t + tl seconds. 6. Theorem. If f is a CP vector field defined on an open set U, then (i) 1'(1) is open in R x U, and (ii) a E CP(D(I)j U).
31 1. 1. Generalities Let E be a real Banach space and 4> a map from an open subset of R into E. If 4> is differentiable, its derivative 4>'(t) at t is a linear map from R into E, and thus of the form ). 1-+ ). V for some vector VeE (namely V = 4>'(t)(1), 1 E R). 4). 1. Definition. Let U C R X E be open and f : U map. A solution of the first-order differential equation -+ E a continuous dx dt = f(x, t) is any map 4> : I -+ E, where Ie R is an interval, such that 4> is of class Cl and for every tEl we have (t,4>(t)) E U and 4>'(t) = f(t,4>(t)).
J-holomorphic curves and quantum cohomology by Dusa McDuff and Dietmar Salamon