Counting real pseudo-holomorphic discs and spheres in dimension four and six

First, we provide another proof that the signed count of the real

J

-holomorphic spheres (or

J

-holomorphic discs) passing through a generic real configuration of

k

points is independent of the choice of the real configuration and the choice of

J

, if the dimension of the Lagrangian submanifold

L

(fixed points set of the involution) is two or three, and also if we assume

L

is orientable and relatively spin, and

M

is strongly semi-positive. This theorem was first proved by Welschinger in a more general setting, and we provide more natural approach using the degree of evaluation maps from the moduli spaces of

J

-holomorphic discs. Then, we define the invariant count of discs intersecting cycles of a symplectic manifold at fixed interior marked points, and intersecting real points at the boundary under certain assumptions. The last result is new and was not proved by Welshinger's method.