Residue formulas for logarithmic foliations and applications

Abstract

We prove a residual formula in terms of the logarithmic indices for one-dimensional holomorphic foliations, with isolated singularities, and logarithmic along normally crossing divisors. As application we provide a formula for the number of singularities in the complement of the invariant divisor on projective spaces. We also give a formula for the total sum of the logarithmic indices if the the singular set of the foliation is contained in the invariant divisor. Finally, we obtain a a Poincaré-Hopf type formula for singular normal projective varieties. iv

Polynomial bounds for automorphisms groups of foliations

“…The section C 0 being in the support of ∆ means that F is regular at a general point p ∈ C 0 and the leaf that passes through p is transverse Consequently, a = m, b = (a + 1)e, ∆ = aC 0 and there exist s = b + 2 − 2g invariant fibers. This proves case (5). If F does not fit in any of the previous cases, then it has a singularity on a generically transverse fiber.…”

“…In [5] the authors proved more general results for an one-dimensional foliation F on a compact complex manifold X, of dimension n, with isolated singularities and an F -invariant hypersurface S. They proved under mild hypotheses on the singularities that lie on S that X c n (T X (− log S) ⊗ K F ) = p∈Sing(F )∩{X\S}…”

Analytic varieties invariant by foliations and Pfaff systems

“…In the context of a hypersurface V invariant by a one-dimensional foliation F , we have the following Baum-Bott type theorem [34,33]: if F has isolated singularities and V is a normal crossing divisor, then…”

Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues

“…He showed that the image sheaf of the logarithmic residues coincides with the sheaf of regular meromorphic differential forms introduced by D. Barlet [5] and M. Kersken [15,16]. We refer the reader to [4,8,9,10,12,29,30] for more recent results on logarithmic residues.…”