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SUMMARY:On a Quadratic Form on the Homology of the Milnor Fibre of a Curve
  with an Isolated Singularity in C^2
DTSTART:20230629T143000Z
DTEND:20230629T153000Z
DTSTAMP:20260423T145300Z
UID:indico-event-9995@indico.math.cnrs.fr
DESCRIPTION:Speakers: Xavier Gomez-Mont (CIMAT Guanajuato\, Mexique)\n\nFr
 om a polynomial with complex coefficients in 2 variables (algebraic data)\
 , one may associate avery interesting geometric object of a considerable m
 athematical complexity: the partition of C2 = R4 obtained by its level sur
 faces:Ct := {f (z1 \, z2 ) = t} t∈C .For t general\, this is a different
 iable surface\, and for a finite number of values t0 \, it will be a singu
 lar surface. If p ∈ C2 is a non-singular point (i.e. one of the partial
  derivatives ∂f/∂zj(p)is non-zero)\, theImplicit Function Theorem guar
 antees 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 isolate
 d singular point at points where∂f/∂z1(p0)=∂f/∂z2(p0) =0 f (p0 ) =
  t0in 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\, w
 ith 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 dimensi
 on of its first homology group may be algebraically computeddimC 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) . Th
 is homology group is called the vanishing homology of f at p0 . The local
  study of the isolated singularity at p0 consists in understanding what ha
 ppens in B\, i.e.how the different surfaces C(t0) assemble themselves to f
 orm the ball B. It is not difficult to realize thatf | B−C(t_0) has the 
 structure of a locally trivial fibre bundle over the punctured disk ∆ 
 − {t0} ⊂ C. Thefundamental group of ∆ − {t0 } has as generator a c
 losed loop γ0 around t0 \, and the structure of alocally trivial fibation
  is codified by the map g : C’(t) → C’( t )\, called the monodromy m
 ap\, obtained by trivializing the fibre bundle over γ0 .Its action g∗ o
 n the vanishing homology H1 (C’(t) \, C) is called the algebraic monodro
 my. 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 co
 mplement C’(t) − C’’(t) is a disjoint union of tubes (annuli) wher
 e the monodromy is a Dehn twist (a twist by an integer\, being the identit
 y on the boundary of the tube). We will show how to construct a symmetric
  bilinear form on the vanishing homology using theintersection product of 
 1 − cycles on Ct and the algebraic monodromy\, that codifies the twists 
 alongthe different tubes. The Theorem we have is that this bilinear form i
 s positive definite\, once wehave cancelled the annihilator.This is a join
 t work with L. Alanis\, E. Artal\, Ch. Bonatti\, M. González-Villa and P.
  Portilla\, which may be downloaded from Arxiv.\n\nhttps://indico.math.cnr
 s.fr/event/9995/
LOCATION:René Baire (Dijon)
URL:https://indico.math.cnrs.fr/event/9995/
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