Vanishing Cycles of Hypersurfaces

• Vanishing cycles of hypersurfaces are key invariants that capture the topological discrepancies between singular fibers and their smooth deformations.
• They are computed using advanced techniques involving functorial frameworks, perverse sheaves, and mixed Hodge modules for precise monodromy and homology estimates.
• These cycles provide essential insights for bounding Betti numbers and understanding the degeneracy phenomena in complex algebraic varieties.

Vanishing cycles of hypersurfaces constitute a core structural element in both the topology and Hodge theory of complex algebraic varieties. They measure the discrepancy between the cohomology of singular hypersurfaces and their smooth deformations, encode monodromy actions in degenerating families, and furnish explicit invariants governing the topology of degenerations and the behavior of singularities. Modern theory operationalizes vanishing cycles via sophisticated functorial frameworks, perverse sheaves, and mixed Hodge modules, with deep implications for monodromy, Betti number bounds, and degeneracy phenomena.

1. Foundational Concepts: Vanishing Cycles, Homology, and Functorial Constructs

The theory begins with a holomorphic map $f: Y \to \mathbb{C}$, central fiber $X = f^{-1}(0)$, and smooth nearby fibers $X_t$ for small $t \neq 0$. The discrepancy between the topological invariants of $X$ and $X_t$ is captured via the nearby and vanishing cycle functors: $$R\Psi_f(\mathcal{F}) := i^* Rj_* (\mathcal{F}|_{Y^*}), \qquad R\Phi_f(\mathcal{F}) := \operatorname{Cone}(i^*\mathcal{F} \to R\Psi_f(\mathcal{F}))[-1]$$ where $i: X \hookrightarrow Y$, $j: Y^* := Y \setminus X \hookrightarrow Y$, and $\mathcal{F}$ is a constructible complex of sheaves (Maxim et al., 2020). The resulting vanishing cohomology groups

$$H^k_v(X) := H^k(\mathbb{P}^n, R\Phi_f(\mathbb{Q}_X)) \cong H^k(X, R\Phi_f(\mathbb{Q}_X))$$

quantify the cohomological differences induced by singularities, and their homological analogs are defined via the relative homology of the total space and nearby smooth fiber, leading to the vanishing homology groups

$$H^{\curlyvee}_k(V) := H_k(\mathcal{V}, V_t;\mathbb{Z}),$$

with $(\mathcal{V}, V_t)$ the smoothing pencil associated to a projective hypersurface (Siersma et al., 2014).

2. Homological Concentration and Stratified Singular Loci

A principal outcome for projective hypersurfaces with a singular locus $\Sigma$ of dimension $s$ is that the vanishing homology (or cohomology) is concentrated in consecutive degrees $n+1, n+2, \ldots, n+s+1$ (Siersma et al., 2014, Maxim et al., 2020). In the special case $s=1$, Siersma–Tibăr prove: $$H^{\curlyvee}_k(V) = 0 \quad \text{for}~k \notin \{n+1, n+2\}$$ Explicit ranks for the nontrivial groups are then described in terms of monodromy data around irreducible components $\Sigma_i$ of $\Sigma$, their genera, intersection numbers with the pencil axis, and special points where singular type jumps.

Table: Degrees Supporting Vanishing Homology (Maxim et al., 2020, Siersma et al., 2014)

$\dim_\mathbb{C} \Sigma$	Nontrivial Vanishing Homology Degrees	Key Paper
0 (isolated)	$n+1$	(Siersma et al., 2014)
1	$n+1, n+2$	(Siersma et al., 2014)
$s$	$n+1, \ldots, n+s+1$	(Maxim et al., 2020)

This framework directly generalizes the classical Picard–Lefschetz theory, recovering it for $\Sigma = \emptyset$ (smooth case) and extending to higher-dimensional singular loci.

3. Monodromy, Local Systems, and Rank Formulas

Vanishing cycles are deeply intertwined with monodromy phenomena. The rank of the highest vanishing homology group is given as the intersection of kernels of local and global monodromy operators acting on the generic transversal Milnor fiber homology: $$H^{\curlyvee}_{n+2}(V) \cong \left\{ w = (w_i) \in \bigoplus_i H_{n-1}(F'_i) ~\bigg|~ w_i \in \bigcap_{s, q} \operatorname{Ker}(A_{s,i} - I) \cap \bigcap_{j \in G_i} \operatorname{Ker}(A_{j,i} - I)\right\}$$ where $A_{s,i}$ and $A_{j,i}$ encode local and global monodromy along $\Sigma_i$ (Siersma et al., 2014). If monodromy acts without eigenvalue $1$ along some loop, the highest vanishing homology group vanishes altogether.

Table: Monodromy-Invariant Contributions (Siersma et al., 2014)

Monodromy Type	Effect on $H^{\curlyvee}_{n+2}(V)$	Condition
Eigenvalue $1$	Kernel intersection contributes nontrivially	Generic case
No eigenvalue $1$	Forces vanishing of top group	Rigid monodromy

Lower degrees are determined via an Euler–characteristic formula involving the ranks of the higher group and contributions from Milnor fibers at special points and isolated singularities.

4. Perverse Sheaf Formulation and Stratified Computation

Vanishing cycles admit a full perverse-sheaf package, where the stalk cohomology of the vanishing cycles sheaf $\phi_f[-1]\mathbb{Z}_U[n+1]$ at a point $p$ computes reduced Milnor fiber cohomology: $$H^k(\phi_f[-1]\mathbb{Z}_U[n+1])_p \cong \widetilde{H}^{n+k}(F_{f,p};\mathbb{Z})$$ Successive hyperplane slices and applications of nearby and further vanishing cycles yield the Lê cycles and Lê numbers, which enumerate cell attachments in the topological model of the Milnor fiber and encode the local topological complexity of non-isolated singularities (Massey, 2014, Maxim et al., 2020).

For singular loci of dimension $s>0$, perverse cosupport arguments show that reduced cohomology survives only for $k$ in $[n-s, n]$, with lower degrees determined by restriction to higher-dimensional strata and explicit monodromy-invariant submodules. The Lê–Tibăr slicing mechanism provides an inductive approach to reduce dimension and recursively compute vanishing cohomology, leveraging alternations of nearby and vanishing cycle functors (Maxim et al., 2020).

5. Betti Number Bounds, Lefschetz Supplements, and Degeneracy

Vanishing cohomology yields robust upper bounds for the Betti numbers $b_k(X)$ of singular hypersurfaces: $$b_k(X) \leq b_k(X_t) + \dim H^k_v(X) + \dim H^{k-1}_v(X)$$ for $k$ in $[n+1, n+s+1]$, where $X_t$ is a smooth neighbor; the top new group satisfies

$b_{n+s+1}(X) \leq 1 + \sum_i \mu_i$

with summation over $s$-dimensional strata and contributions from transversal Milnor numbers (Maxim et al., 2020). Artin vanishing for perverse sheaves on affine open complements underpins a supplement to the Lefschetz hyperplane theorem: $$H^k(X, X_H; \mathbb{Z}) = 0 \quad \text{for}~k < n~\text{and}~n+s+1 < k < 2n$$ where $X_H = X \cap H$ is a hyperplane section.

These methods yield new proofs of results, including Kato’s theorem on high-degree cohomology of projective hypersurfaces with positive-dimensional singular loci.

6. Explicit Computations and Examples

Concrete models illuminate the structure of vanishing cycles:

• Cone over a smooth curve ($V = \text{Cone}(C) \subset \mathbb{P}^{n+1}$): Singular locus $\Sigma \cong \mathbb{P}^1$, topological reduction gives $H^{\curlyvee}_{n+2}(V) = 0$, and $b^{\curlyvee}_{n+1}(V) = 2g - 1$ where $g$ is genus of $C$ (Siersma et al., 2014).
• Suspension of a nodal curve ($V$ in $\mathbb{P}^3$): Monodromy data and Milnor numbers at nodes provide computations of $b^{\curlyvee}_4(V)$, $b^{\curlyvee}_3(V)$.
• Degenerations with two-strata singular sets: Perverse sheaf vanishing collapses most Milnor fiber cohomology, with tight support only in degrees $n-s$, $n-1$, $n$ under specific connectivity hypotheses (Tráng et al., 2011).

7. Open Problems and Research Directions

The current theory establishes full control for hypersurfaces with low-dimensional singular loci. For higher-dimensional $\Sigma$ ($\dim \Sigma \geq 2$), the distribution of vanishing cycles across successive degrees becomes intricate, challenging systematic description (Siersma et al., 2014, Maxim et al., 2020). Mixed Hodge structures on vanishing homology, involvement of limiting Hodge structures, and relations to monodromy representations remain active research areas. Applications extend to bounding the topology of degenerations, computing Alexander invariants, and analyzing global monodromy and vanishing lattice phenomena.

Summary: The vanishing cycles of hypersurfaces encode the topological and Hodge-theoretic complexity introduced by singularities. Precise rank formulas and perverse sheaf machinery enable effective computation and control, undergirding major advances in singularity theory, topology of algebraic varieties, and degeneracy phenomena (Siersma et al., 2014, Maxim et al., 2020).