By Dusa McDuff and Dietmar Salamon
J$-holomorphic curves revolutionized the research of symplectic geometry while Gromov first brought them in 1985. via quantum cohomology, those curves are actually associated with the various most fun new rules in mathematical physics. This publication offers the 1st coherent and entire account of the idea of $J$-holomorphic curves, the main points of that are shortly scattered in numerous learn papers. the 1st half the ebook is an expository account of the sphere, explaining the most technical elements. McDuff and Salamon supply whole proofs of Gromov's compactness theorem for spheres and of the life of the Gromov-Witten invariants. the second one 1/2 the e-book makes a speciality of the definition of quantum cohomology. The authors identify that this multiplication exists, and provides a brand new evidence of the Ruan-Tian outcome that's associative on applicable 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 power on quantum cohomology, and curve counting formulation also are mentioned. The publication closes with an summary of connections to Floer idea
Read or Download J-holomorphic curves and quantum cohomology PDF
Best differential geometry books
Minimal surfaces and Teichmuller theory
The notes from a suite of lectures writer brought at nationwide Tsing-Hua collage in Hsinchu, Taiwan, within the spring of 1992. This notes is the a part of ebook "Thing Hua Lectures on Geometry and Analisys".
Complex, contact and symmetric manifolds: In honor of L. Vanhecke
This ebook is concentrated at the interrelations among the curvature and the geometry of Riemannian manifolds. It includes examine and survey articles in keeping with the most talks introduced on the overseas Congress
Differential Geometry and the Calculus of Variations
During this e-book, we examine theoretical and useful points of computing equipment for mathematical modelling of nonlinear structures. a couple of computing recommendations are thought of, comparable to tools of operator approximation with any given accuracy; operator interpolation concepts together with a non-Lagrange interpolation; tools of process illustration topic to constraints linked to strategies of causality, reminiscence and stationarity; equipment of procedure illustration with an accuracy that's the most sensible inside of a given type of versions; equipment of covariance matrix estimation;methods for low-rank matrix approximations; hybrid equipment in keeping with a mixture of iterative approaches and top operator approximation; andmethods for info compression and filtering below situation clear out version should still fulfill regulations linked to causality and kinds of reminiscence.
- An Introduction to Tensor Analysis
- Lectures on complex manifolds
- Diffeology
- Foliations: Geometry and Dynamics
Extra resources for J-holomorphic curves and quantum cohomology
Sample text
If f ∈ HomC(A) (X • , Y • ) is null-homotopic with homotopy {s i }, then {F (s i )} is a null-homotopy for F (f ). Thus F induces a functor F : K(A) → K(B) between the homotopy categories. As an example, let X and Y be topological spaces and f : X → Y a continuous map, and take A = Sh(X), B = Sh(Y ). Then direct image f∗ : K(X) → K(Y ) and inverse image f ∗ : K(Y ) → K(X) are well-defined functors on the homotopy category of complexes of sheaves. 4 Derived Categories Let A be an abelian category.
To do this, we need to specify a translation automorphism T and a collection of distinguished triangles. Let T = [1], the shift on complexes. 3 The Triangulation of the Homotopy Category C n (u) = X n+1 ⊕ Y n , n 0 dX[1] n = dC(u) n+1 u dYn 33 . We have the natural inclusion i : Y • → C • (u) and projection p : C • (u) → The triangle X • [1]. p X • −→ Y • −→ C • (u) −→ X • [1] u i is called a standard triangle. Let Δ be the collection of all triangles in K(A) isomorphic to standard triangles. 1 The homotopy category K(A) together with the shift functor [1] : K(A) → K(A) and Δ as the collection of distinguished triangles forms a triangulated category.
We claim there is no quasiisomorphism t : Z • → X • with f t null-homotopic. Suppose we had such a t = {t n }n∈Z . Pick a cycle z ∈ Z 1 such that the class of t 1 (z) is the generator of H 1 (X • ) = Z/2. This implies that t 1 (z) is an odd integer. Now t 1 (2z) = 2t 1 (z) is even, so 0 in H 1 (X • ). As t is in Qis, it follows that the class of 2z is 0 in H 1 (Z • ) and 2z = dZ (z ) for some z ∈ Z 0 . 4 Derived Categories Z −1 dZ Z0 t0 s0 0 Z1 t1 s1 Z Z 1 2 1 Z 0 49 Z/3 We have 2t 1 (z) = t 1 (dZ (z )) = 2t 0 (z ) = 2s 1 (dZ (z )) and hence t 1 (z) = s 1 (dZ (z )) = s 1 (2z) = 2s 1 (z), contradicting t 1 (z) odd.