From a polynomial with complex coefficients in 2 variables (algebraic data), one may associate a
very interesting geometric object of a considerable mathematical complexity: the partition of C2 = R4 obtained by its level surfaces:
Ct := {f (z1 , z2 ) = t} t∈C .
For t general, this is a differentiable surface, and for a finite number of values t0 , it will be a singular surface.
If p ∈ C2 is a non-singular point (i.e. one of the partial derivatives ∂f/∂zj(p)is non-zero), the
Implicit Function Theorem guarantees that Ct is a smooth surface in a neighborhood of p.
If ∂f/∂zj(p) for j = 1, 2 do not have a common factor, then Ct0 will be an isolated singular point at points where
∂f/∂z1(p0)=∂f/∂z2(p0) =0 f (p0 ) = t0
in the sense that C(t0) − {p0 } is a smooth surface in a small ball B with centre p0 .
The intersection C’(t0) := Ct ∩ B, t ∈ ∆ε − {t0} , is called the Milnor fibre of f at p0 .
It is a finite surface, with a finite number of boundary components and finite genus. The number of boundary components is determined by the number of irreducible factors of f in the ring of germs of holomorphic functions O(C2,p0) and the dimension of its first homology group may be algebraically computed
dimC H1 (C’(t0) 0 , C) = dimC O(C2,p0)/ (∂f/∂z1(p0),∂f/∂z2(p0))
where the denominator is the ideal generated by its terms in the ring O(C2,p0) . This homology group is called the vanishing homology of f at p0 .
The local study of the isolated singularity at p0 consists in understanding what happens in B, i.e.
how the different surfaces C(t0) assemble themselves to form the ball B. It is not difficult to realize that
f | B−C(t_0) has the structure of a locally trivial fibre bundle over the punctured disk ∆ − {t0} ⊂ C. The
fundamental group of ∆ − {t0 } has as generator a closed loop γ0 around t0 , and the structure of a
locally trivial fibation is codified by the map g : C’(t) → C’( t ), called the monodromy map, obtained by trivializing the fibre bundle over γ0 .
Its action g∗ on the vanishing homology H1 (C’(t) , C) is called the algebraic monodromy.
We will see that the monodromy is a quasi-periodic map, in the sense that on a large open set C’’(t)
of C’(t) it is periodic, but the complement C’(t) − C’’(t) is a disjoint union of tubes (annuli) where the monodromy is a Dehn twist (a twist by an integer, being the identity on the boundary of the tube).
We will show how to construct a symmetric bilinear form on the vanishing homology using the
intersection product of 1 − cycles on Ct and the algebraic monodromy, that codifies the twists along
the different tubes. The Theorem we have is that this bilinear form is positive definite, once we
have cancelled the annihilator.
This is a joint work with L. Alanis, E. Artal, Ch. Bonatti, M. González-Villa and P. Portilla, which may be downloaded from Arxiv.
