RT info:eu-repo/semantics/article
T1 Real logarithmic models for real analytic foliations in the plane
A1 Corral, Nuria
A1 Sanz, Fernando
K1 Singular holomorphic foliation · Logarithmic foliations · Generalized curves · Center-focus plane vector fields
AB Let S be a germ of a holomorphic curve at (C2, 0) with finitely manybranches S1, . . . , Sr and let I = (I1, . . . , Ir ) ∈ Cr . We show that there exists a nondicriticalholomorphic foliation of logarithmic type at 0 ∈ C2 whose set of separatricesis S and having index Ii along Si in the sense of Lins Neto (Lecture Notes in Math.1345, 192–232, 1988) if the following (necessary) condition holds: after a reductionof singularities π : M →(C2, 0) of S, the vector I gives rise, by the usual rules oftransformation of indices by blowing-ups, to systems of indices along components ofthe total transform ¯S of S at points of the divisor E = π−1(0) satisfying: (a) at anysingular point of ¯S the two indices along the branches of ¯S do not belong to Q≥0 andthey are mutually inverse; (b) the sum of the indices along a component D of E forall points in D is equal to the self-intersection of D in M. This construction is usedto show the existence of logarithmic models of real analytic foliations which are realgeneralized curves. Applications to real center-focus foliations are considered.
PB Springer
SN 1139-1138
YR 2011
FD 2011
LK https://uvadoc.uva.es/handle/10324/68187
UL https://uvadoc.uva.es/handle/10324/68187
LA eng
NO Rev Mat Complut (2012) 25:109–124
DS UVaDOC
RD 11-jul-2024