Invariants

We construct our invariants, and study their transitions. For simplicity, the ground field is assumed to be the complex number field C in this chapter.

Let H^*(A) and H_*(A) denote the singular cohomology and homology groups of a topological space A with Q-coefficient. They are naturally Z/2Z-graded.

Let X be a smooth connected complex projective surface with a base point, and let D be a smooth hypersurface of X. We denote the Picard variety of X by Pic. We often use the natural identification H⁴(X) = Q. The intersection product of a, b ∈ H²(X) is denoted by a · b.

In Section 7.1, we explain a way of evaluation for our argument. In Section 7.2, we show transition formulas in simpler cases. By using them, we construct our invariants in Section 7.3. They are also sufficiently useful for the transition problems in the rank 2 case, which we study in Section 7.4.

In Section 7.5,we give transition formulas for the case p_g = dim H²(X,O_X)! >0. They are formally the same as those in the simpler case.

In Section 7.6, we study transition formulas for the case p_g = 0. By using it, we obtain a weak wall crossing formula in Section 7.7. We write down the weak wall crossing formula and a weak intersection rounding formula for the rank 3 case in Subsection 7.7.3–7.7.4. We also give a transition formula for variation of parabolic weights in Subsection 7.7.5. In Section 7.8, we derive weak intersection rounding formulas from weak wall crossing formulas.