Lecture 2. Gromov-Witten Invariants
Structural constants of the quantum cup-product on H* (X) are defined by(aob,c) := L:qt' ... q~r r evi(a)/\ev2(b)/\evj(c),
d J[x3,d]where ( d 1 , ... , dr) is the coordinate expression of the degree d in a basis of the
lattice H2(X, Z) (which we assume isomorphic to zr), the integral means evaluation
of a cohomology class on the virtual fundamental cycle, and a, b, c are arbitrary
co homology classes of X.
Exercises. (a) Show that symplectic area of a holomorphic curve in (almost)
Kahler manifold is positive. Deduce that the semigroup L C H2(X, Z) of degrees
of compact holomorphic curves fits some integer simplicial cone in the lattice at
least in the case of Kahler manifolds with H^2 (X) spanned by Kahler classes. (In
fact the same is true for generic almost Kahler structures and therefore - for any
almost Kahler X if L means the semigroup spanned by those degrees which actually
contribute to the structural constants.) Conclude from this that the structural con-
stants are (at worst) formal power series in q 1 , ... , qr with respect to an appropriate
basis in the lattice H2(X).
(b) Make precise sense of the statement that QH(X) is a q-deformation of
H(X).
( c) Prove that the quantum cup-product o respects the following grading on
H*(X, Q[[q]]): cohomology classes of X are assigned their usual degrees divided by 2
since we want to count dimensions of cycles in "complex" units, and the parameters
qi are assigned the degrees in accordance with the rule deg qd = ( c 1 (Tx), d).
One can define more general Gromov-Witten invariants(a1, ... , ak)d := r evi(ai) !\ ... !\ evk(ak)
J[xk,d]which have the meaning of
the number of degree d holomorphic spheres in X passing through generic cycles
Poincare-dual to the classes a 1 , ... , ak.
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