Generic vector fields on isolated complex hypersurface germs

Technical Summary: Generic Vector Fields on Isolated Complex Hypersurface Germs

Problem Statement
The paper investigates the space of holomorphic vector fields on isolated hypersurface singularities $(V, 0) \subset (\mathbb{C}^{n+1}, 0)$. While it is established that the space of germs of holomorphic vector fields on such a singularity is infinite-dimensional, and the subset of those with an isolated singularity at the origin is open and dense, the specific properties of "generic" elements within this space remain to be characterized. Specifically, the authors seek to determine the minimal possible value of the GSV-index (Gómez-Mont, Seade, Verjovsky index) for these vector fields and to understand the geometric conditions under which this minimum is attained. Furthermore, the work aims to extend these local findings to derive global obstructions for the existence of holomorphic vector fields on compact singular varieties.

Methodology
The authors employ a combination of local singularity theory and global topological methods:

1. Local Analysis via GSV-Index: The study utilizes the GSV-index, defined as the Poincaré-Hopf index of a continuous extension of a vector field to a local Milnor fiber. The authors relate this index to algebraic invariants of the singularity, specifically the Tjurina-Greuel number $\tau_0(V)$ and the Milnor number $\mu_0(V)$.
2. Algebraic Characterization: The core of the local theory involves analyzing the ideal generated by the defining equation of the hypersurface $f$ and the components of a holomorphic extension of the vector field $\tilde{v} = (v_1, \dots, v_{n+1})$. The authors use dimension formulas over the local ring of holomorphic germs $\mathcal{O}_{n+1,0}$ and properties of regular sequences to establish lower bounds for the index.
3. Extension and Non-degeneracy: A central technique is examining whether a vector field on the hypersurface can be extended to the ambient space $\mathbb{C}^{n+1}$ with a non-degenerate isolated singularity at the origin.
4. Global Obstructions: For compact varieties, the authors apply the local results by summing the minimal indices at singular points. They utilize the Baum-Fulton-MacPherson Riemann-Roch theorem and Chern class calculations for virtual tangent bundles to relate these local sums to global topological invariants (such as the Euler characteristic and geometric genus).

Key Contributions and Results

• Characterization of Generic Vector Fields (Theorem 1):
  The authors prove that for a hypersurface germ $(V, 0)$ with an isolated singularity, the GSV-index of any holomorphic vector field $v$ with an isolated singularity is bounded below by:
  $$\text{Ind}_{GSV}(v, (V, 0)) \geq 1 + (-1)^n \tau_0(V)$$
  where $\tau_0(V)$ is the Tjurina-Greuel number.
  Crucially, the paper establishes that equality holds if and only if the vector field admits a holomorphic extension to the ambient space $\mathbb{C}^{n+1}$ with a non-degenerate singularity at the origin. Furthermore, the set of such vector fields (those achieving the minimal index) forms an open and dense subset of the space of all vector fields with isolated singularities.

• Characterization of Weighted Homogeneous Germs (Theorem 2):
  The paper provides a partial resolution to a conjecture regarding the equivalence of weighted homogeneity and the existence of specific vector fields:

  • For even $n$, the existence of a vector field extending to the ambient space with a non-degenerate singularity implies $(V, 0)$ is weighted homogeneous.
  • For odd $n$, the existence of a vector field everywhere transversal to the link $L_V$ implies $(V, 0)$ is weighted homogeneous.
    The authors note that proving the full equivalence in all dimensions remains an open task, but their results confirm the implication in these specific cases.

• Global Obstructions for Compact Varieties (Section 5 & 6):
  By summing the local lower bounds, the authors derive constraints on compact singular varieties admitting global holomorphic vector fields:

  • Projective Hypersurfaces: They derive an inequality relating the degree of the hypersurface, the number of singular points, and the sum of local indices (Theorem 3).
  • Singular Curves (Theorem 4): An irreducible compact singular complex curve admitting a non-trivial holomorphic vector field with zeros must be rational (geometric genus $g=0$) and possess at most two singular points.
  • Singular Surfaces in Kähler 3-folds (Theorem 5): If a singular surface $V$ in a compact Kähler 3-fold $X$ admits a non-trivial holomorphic vector field with zeros, and the adjoint line bundle $\mathcal{O}_X(K_X + [V])$ is nef, then the geometric genus is greater than or equal to the irregularity ($g(V) \geq q(V)$).

Significance and Claims
The paper claims to provide a precise description of the "generic" behavior of holomorphic vector fields on isolated hypersurface singularities, identifying the minimal index in terms of the Tjurina number. This characterization serves as a bridge between local algebraic geometry and global topology.

The authors emphasize that their results yield new obstructions for the existence of global vector fields on singular spaces, a topic where standard methods for smooth manifolds (relying on the isomorphism between the complex of regular forms and the Koszul complex) fail. Specifically, the paper demonstrates that the existence of such vector fields imposes strict topological constraints:

1. Singular curves carrying such fields are necessarily rational with very few singularities.
2. Singular surfaces in specific Kähler environments cannot have "excess" irregularity relative to their geometric genus.

The work also clarifies the relationship between the existence of transversal vector fields (or non-degenerate extensions) and the weighted homogeneous nature of the singularity, partially answering a conjecture in the field. The authors present these findings as a foundational step in understanding the interplay between singularity type and the global geometry of singular varieties.