Perverse Schober Structures for Conifold Degenerations

Abstract: We study a one parameter degeneration of Calabi Yau threefolds whose central fiber contains a single ordinary double point. Using the nearby and vanishing cycle formalism, we construct a canonical perverse object on the singular fiber from the variation morphism between vanishing and nearby cycles. We show that this object restricts to the constant perverse sheaf on the smooth locus and differs from the intersection complex by a single rank one contribution supported at the node. Thus the object isolates the vanishing cycle contribution associated with the conifold degeneration in a canonical sheaf theoretic form. We also explain how this construction aligns with the rank-one Picard Lefschetz phenomenon that appears categorically through spherical monodromy, making it a natural comparison object for the decategorified effect of spherical twists in the ordinary double point case.

• The paper establishes a canonical perverse sheaf by decategorifying categorical monodromy through vanishing cycles at ordinary double point singularities.
• It employs nearby and vanishing cycle functors along with distinguished triangles to precisely capture the topological, Hodge theoretic, and categorical corrections in Calabi–Yau degenerations.
• The method extends to multi-node settings and lays groundwork for future studies in mixed Hodge theory, wall-crossing, and explicit schober constructions.

Canonical Perverse Schober Structures for Conifold Degenerations

Introduction and Context

The paper provides a systematic investigation of perverse sheaf and schober phenomena associated with conifold degenerations—degenerations of Calabi–Yau threefolds where the central fiber acquires a single ordinary double point (ODP) singularity. Conifold transitions serve as a basic model for topological change—specifically, the appearance and vanishing of 3-spheres in the smoothing—and are foundational in both the geometric theory of Calabi–Yau manifolds and their physical realizations in string theory (notably in the context of vanishing cycles and monodromy acting on middle cohomology). The conifold case also offers a particularly transparent interplay between the topology (via vanishing cycles), Hodge theory (limiting mixed Hodge structures), and categorical structures (notably, spherical objects and their twists in derived categories).

A key technical framework is the analysis of the nearby and vanishing cycles functors, their corresponding distinguished triangles in the derived category of constructible sheaves, and the resulting canonical morphisms. The connection with categorical Picard–Lefschetz theory and the theory of perverse schobers provides a natural backdrop, especially in light of the Kapranov-Schechtman program for categorifying perverse sheaves.

Perverse Sheaf Construction from Vanishing Cycles

Let $\pi: \mathcal{X} \to \Delta$ be a one-parameter smoothing degeneration of Calabi–Yau threefolds, with $X_0 = \pi^{-1}(0)$ having a single ODP $p$. The vanishing cycle contribution is governed by the Milnor fiber, which in this case is $S^3$, yielding one-dimensional vanishing cohomology in degree $3$. At the sheaf level, the key functors are $\psi_\pi$ (nearby cycles) and $\phi_\pi$ (vanishing cycles); their interplay encodes the local modification of the sheaf-theoretic and topological structure.

The main construction is the canonical perverse sheaf

$$\mathcal{P} = \mathrm{Cone}(var_F)[-1], \qquad F := \mathbb{Q}_{\mathcal{X}}[3],$$

where $var_F: \phi_\pi(F) \to \psi_\pi(F)$ is the variation morphism.

Structural Properties

• Local Behavior: On the smooth locus $U = X_0 \setminus \{p\}$, $X_0 = \pi^{-1}(0)$0 restricts to the shifted constant sheaf, $X_0 = \pi^{-1}(0)$1.
• Relation to Intersection Complex: There is a short exact sequence of perverse sheaves:

$X_0 = \pi^{-1}(0)$2

indicating $X_0 = \pi^{-1}(0)$3 differs from $X_0 = \pi^{-1}(0)$4 by a single, point-supported rank-one contribution at $X_0 = \pi^{-1}(0)$5.

• Uniqueness of Local Correction: The perverse stalk $X_0 = \pi^{-1}(0)$6 at $X_0 = \pi^{-1}(0)$7 is one-dimensional; this isolates precisely the topological vanishing cycle contribution of the ODP.

These results show that $X_0 = \pi^{-1}(0)$8 provides a canonical, functorial extension of the constant sheaf through the singularity, encoding the specific local correction corresponding to the vanishing cycle, as opposed to the broader intersection-space programs that are designed to model duality-satisfying cohomology.

Categorical and Schober Interpretations

In the context of homological mirror symmetry and categorical Picard–Lefschetz theory, the vanishing sphere at the ODP is represented by a spherical object $X_0 = \pi^{-1}(0)$9 in $p$0. The associated spherical twist $p$1 acts on the derived category, categorifying the classical Picard–Lefschetz monodromy:

$p$2

on $p$3, intertwining the categorical and cohomological pictures.

Kapranov and Schechtman's perverse schober formalism upgrades perverse sheaf data to categorical data (stable $p$4-categories and spherical functors). In the present geometric situation, the local degeneration data induces a perverse schober over the disk, where the monodromy is generated precisely by the rank-one spherical twist associated with the singularity.

Decategorification yields the perverse sheaf $p$5 as the sheaf-theoretic shadow of this rank-one phenomenon: $p$6 captures all the extra information encoded by the categorical spherical monodromy that is visible at the level of perverse sheaves.

Extension to Multiple Nodes and Stratified Degenerations

The methodology extends directly to the case of several isolated nodes. If $p$7 contains $p$8 ODPs, then

$p$9

and the corresponding perverse object $S^3$0 sits in

$S^3$1

providing a global extension by direct sum of the local vanishing corrections. For more general stratified singular loci, the vanishing cycle contributions aggregate into a perverse sheaf supported on the singular stratification.

Implications and Future Directions

Practical and Theoretical Relevance

The canonical perverse object constructed here provides a precise target for decategorification in contexts governed by vanishing cycles and Picard–Lefschetz monodromy, relevant to both the topology of degenerations and the representation of categorical autoequivalences in Fukaya and derived categories. It serves as a natural comparison for intersection-space complexes and clarifies the effect of singularities on the extended perverse sheaf structure.

This approach further suggests an organizing principle for understanding categorical monodromy in degenerations: the critical features (and their sheaf-theoretic analogs) are entirely controlled by the data in the nearby/vanishing cycle triangle, and their perverse extensions.

Outlook

Potential generalizations include:

• Mixing with Hodge Theory: Lifting the construction to mixed Hodge modules could give a more refined object incorporating Hodge-theoretic filtrations.
• Categorical Wall-Crossing: Spherical twists corresponding to vanishing cycles underlie wall-crossing phenomena in spaces of stability conditions; understanding the perverse sheaf avatar can inform computable invariants in enumerative geometry.
• Quiver Structures: In multi-node settings, the extension structure may be organized via quiver-like data determined by intersection pairings of vanishing cycles.
• Explicit Schober Constructions: While this work focuses on functorial perverse sheaf objects as decategorified shadows, explicit schober realizations matching these invariants can be constructed and compared in the context of mirror symmetry and singularity categories.

Conclusion

The paper establishes a canonical, functorial perverse sheaf $S^3$2 associated with conifold degenerations of Calabi–Yau threefolds, characterized as the shifted cone on the vanishing cycle variation morphism. $S^3$3 aligns precisely with both the local topological vanishing cycle data and the categorical monodromy induced by the spherical twist, differing from the intersection complex by a single, rank-one contribution supported at the node. This construction situates the perverse sheaf invariants as natural decategorifications of categorical monodromy and schober data, with extension to more general singularities and Hodge-theoretic enhancement as promising future directions.