On Milnor and Tjurina numbers of Pairs of Holomorphic Functions Germs

Technical Summary: Milnor and Tjurina Numbers of Pairs of Holomorphic Function Germs

Problem Statement
The study of numerical invariants for holomorphic function germs with isolated singularities is a central topic in Singularity Theory, primarily focusing on the Milnor number $\mu(f)$ and the Tjurina number $\tau(f)$. However, numerical invariants associated with meromorphic function germs $f/g$ have received less attention. Recently, A. Szawlowski introduced the Milnor number of a pair $(f, g)$, denoted $\mu(f, g)$, which can be interpreted as the Milnor number of the meromorphic function $f/g$. This invariant is derived from the singular holomorphic foliation $\mathcal{F}_{\omega_{f/g}}$ defined by the 1-form $\omega_{f/g} = gdf - fdg$.

The primary problem addressed in this paper is the absence of a corresponding Tjurina number for pairs of holomorphic functions. While $\mu(f, g)$ is established, a definition and characterization of the Tjurina number of a pair, $\tau(f, g)$, are needed to provide a complete analytic invariant framework for meromorphic germs and their associated pencils of curves.

Methodology
The authors employ the theory of singular holomorphic foliations to define and analyze these invariants. The core methodology involves:

1. Foliation Association: Associating the pair $(f, g)$ with the foliation $\mathcal{F}_{\omega_{f/g}}$ generated by the pencil of curves $\mathcal{P}_{f,g} = \{ \alpha f + \beta g = 0 \}$.
2. Relative Tjurina Numbers: Utilizing the concept of the Tjurina number of a foliation $\mathcal{F}$ with respect to an invariant reduced curve $C$, defined as $\tau(\mathcal{F}, C) = \dim_{\mathbb{C}} \mathcal{O}_2 / (P, Q, h)$, where $\omega = Pdx + Qdy$ and $C = \{h=0\}$.
3. Balanced Divisors of Separatrices: Applying the theory of balanced divisors of separatrices (introduced by Genzmer) for the foliation $\mathcal{F}_{\omega_{f/g}}$. These divisors are constructed using bifurcation fibers (singular members of the pencil) and generic fibers.
4. Analytic Invariance: Proving that the proposed invariants are independent of the specific choice of balanced divisor, thereby establishing them as well-defined analytic invariants.
5. Intersection Theory and Formulas: Deriving explicit formulas by relating the invariants of the pair to the intersection multiplicity $I_0(f, g)$, the Tjurina numbers of individual fibers, and the GSV-index (Gómez-Mont, Seade, Verjovsky).

Key Contributions and Results

• Definition of $\tau(f, g)$: The paper introduces the Tjurina number of a pair $(f, g)$, denoted $\tau(f, g)$, defined as the Tjurina number of the foliation $\mathcal{F}_{\omega_{f/g}}$ with respect to any primitive balanced divisor of separatrices adapted to the curve $fg=0$. The authors prove this definition is independent of the choice of the balanced divisor.

• Explicit Formulas:

  • Fiber-wise Relation: For a specific fiber $C_{[\alpha:\beta]}$ of the pencil, the paper establishes $\tau_{[\alpha:\beta]}(f, g) = I_0(f, g) + \tau(C_{[\alpha:\beta]})$.
  • General Formula: An explicit formula for $\tau(f, g)$ is derived in terms of the Tjurina numbers of the zero and pole divisors of a balanced divisor, the number of bifurcation separatrices $r$, and the intersection multiplicity $I_0(f, g)$. Specifically, $\tau(f, g) = \tau(B_0) - \tau(B_\infty) + r(r-2)I_0(f, g)$.
  • Bifurcation Formula: A bifurcation formula for $\tau(f, g)$ is established, analogous to the known formula for $\mu(f, g)$:
    $$\tau(f, g) = \tau(fg) + \sum_{j=1}^r (\tau(h_j) - \tau_{\text{gen}}) + r^2 I_0(f, g) + \Theta_{B_0} + \Theta_{h_1 \dots h_r} - \Theta_{B_\infty}$$
    where $h_j$ are bifurcation fibers and $\tau_{\text{gen}}$ is the generic Tjurina number.

• Topological vs. Analytic Nature: The paper demonstrates that while the generic Milnor number of a pair is a topological invariant, the generic Tjurina number of a pair is not a topological invariant. This is shown via an example where topologically equivalent pairs yield different generic Tjurina numbers.

• Characterization of Semitame Germs: The authors provide a new characterization of semitame meromorphic function germs (those whose only bifurcation fibers are $f$ and $g$). A pair $(f, g)$ is semitame if and only if $\tau(f, g) = \tau(fg)$. This condition is shown to be equivalent to $\mu(f, g) = \mu(fg)$.

• Teissier's Lemma for Pairs: A version of Teissier's Lemma is established for pairs, relating the polar intersection number of the pair along a balanced divisor to the Milnor number of the pair:
  $$i_0((f, g), B) = \mu(f, g) + \nu(f) + \nu(g) - 1$$
  where $\nu$ denotes algebraic multiplicity.

• Upper Bounds and Applications:

  • An upper bound for $\mu(f, g)$ is derived: $\mu(f, g) \leq 2\tau(fgh_1 \dots h_r) - 2r(r+2)I_0(f, g)$.
  • The results are applied to generic pencils of algebraic curves in $\mathbb{P}^2$. The paper provides an alternative proof for Dimca's formula regarding the sum of global Milnor numbers of singular members in a generic pencil, showing $\sum \mu(C^s_j) = 3(k-1)^2$.
  • A corresponding inequality for the sum of global Tjurina numbers is derived: $\frac{9}{4}(k-1)^2 < \sum \tau(C^s_j) \leq 3(k-1)^2$, with equality in the upper bound holding if and only if all singularities are quasi-homogeneous.

Significance
The paper claims significance by extending the classical theory of singularities from single holomorphic functions to pairs of functions (meromorphic germs). By introducing $\tau(f, g)$, the authors complete the parallel between Milnor and Tjurina numbers in the context of pencils of curves. The work provides explicit computational tools (formulas) for these invariants and clarifies their topological and analytic properties. Specifically, the characterization of semitame germs via the Tjurina number offers a new perspective on the structure of meromorphic functions. Furthermore, the application to generic pencils of algebraic curves connects local singularity theory with global geometric properties, offering an alternative proof to existing results by Dimca and establishing new bounds for Tjurina numbers in this context.