Holomorphic $p$-Contact and $s$-Symplectic Line Bundles

This paper generalizes scalar-valued holomorphic $p$-contact and $s$-symplectic structures to line bundle-valued analogues on compact complex manifolds, demonstrating that unlike their scalar counterparts, these new structures can admit projective examples, and investigates the positivity properties of their canonical bundles using recent regularisation results for $m$-psh functions.

The Gist

Imagine you are an architect trying to design a building with very specific, magical rules. In the world of mathematics, this "building" is a complex shape called a manifold, and the "rules" are about how you can wrap it in special fabrics (called forms and bundles) that don't tear or crumple.

This paper by Kyle Broder and Dan Popovici is about upgrading the blueprints for these buildings. They take old, strict rules and make them more flexible, discovering that some of these new buildings can actually be built in the real world (unlike the old ones, which were purely theoretical).

Here is the breakdown using everyday analogies:

1. The Old Rules: The "Scalar" Problem

For a long time, mathematicians studied two types of special shapes:

• Holomorphic Contact Shapes: Think of these as shapes with a specific "twist" or "knot" that must exist everywhere.
• Holomorphic Symplectic Shapes: Think of these as shapes with a perfect, balanced "dance" between two forces.

The Problem: In the old version, these shapes had to be made of a single, uniform material (called "scalar-valued"). It turned out that if you tried to build these using this single material, they could never be "Kähler" manifolds. In math-speak, a Kähler manifold is like a perfectly smooth, stable, and symmetrical crystal. The old rules said: "If you have this twist, you can't be a perfect crystal."

2. The New Innovation: The "Line Bundle" Upgrade

The authors asked: "What if we don't use a single uniform material? What if we wrap our shape in a special, flexible fabric (a holomorphic line bundle) that changes slightly as you move around the shape?"

This is the core of their paper. Instead of a rigid, single-material rule, they allow the "twist" or "dance" to be carried by this flexible fabric.

• The Analogy: Imagine trying to wrap a gift. The old rule said you must use one giant, unbreakable sheet of paper. If the gift has a weird shape, the paper tears. The new rule says: "Use a sheet of paper that can stretch and shrink locally, as long as the overall pattern holds together."

3. The Big Surprise: Projective Spaces

The most exciting discovery is that with this new "flexible fabric" rule, these special shapes can now be "Projective."

• What does this mean? In the old world, these shapes were like ghosts—they existed in theory but couldn't be built in the real world of standard geometry.
• The New Reality: The authors show that you can build these shapes using standard, well-known geometric objects, like Complex Projective Spaces (which are like the mathematical version of a sphere or a flat plane, but with complex numbers).
• The Result: They proved that a specific type of projective space (where the dimension is 3, 7, 11, etc.) can now be a "Contact" shape. It's like discovering that a standard Lego brick can actually be a magical, self-assembling robot if you just look at it through the right lens.

4. The "Spin" Connection

The paper also connects these shapes to something called Spin Structures.

• The Analogy: Think of a "Spin Structure" as a hidden "double-layer" identity. A shape is "Spin" if it can be covered by a double-layered blanket that fits perfectly without any twists or knots.
• The Finding: The authors show that any shape that follows their new "Contact" or "Symplectic" rules must be a "Spin" shape. It's like saying, "If your building has this specific magical twist, it automatically comes with a hidden double-layered foundation."

5. The "Curvature" Check (The Stability Test)

The authors also checked the "stability" of these new shapes. They asked: "Can these shapes be 'positive' or 'flat' (like a calm lake)?"

• The Answer: They found that these shapes cannot be perfectly flat or positively curved in certain ways. They must have some "negative" curvature (like a saddle shape or a Pringles chip).
• Why it matters: This tells us these shapes are inherently "Fano" manifolds. In simple terms, they are shapes that are "attracted" to themselves, often covered in rational curves (think of them as being made of straight lines that loop back on themselves). This makes them very different from the "calm lake" shapes we usually study.

6. How to Build Them (The Recipe)

The paper doesn't just say these things exist; it shows how to build them:

• Mixing and Matching: You can take a "Symplectic" shape (the dance partner) and a "Contact" shape (the twist) and glue them together. The result is a bigger, more complex "Contact" shape.
• The Projective Space Example: They explicitly built a "Contact" shape out of a standard projective space (like a 3D or 7D version of a sphere) by carefully choosing the "flexible fabric" (the line bundle) to wrap around it.

Summary

In a nutshell:
Mathematicians used to think certain "twisted" shapes were too weird to exist in the real, stable world of geometry. Broder and Popovici realized that if you allow these shapes to wear a special, flexible "coat" (a line bundle) instead of being made of a single rigid material, the rules change. Suddenly, these shapes can exist in the real world, they are related to deep "spin" properties, and they have a very specific, interesting geometry that prevents them from being too "flat."

It's like realizing that a puzzle piece you thought was broken actually fits perfectly, as long as you rotate it and look at it from a slightly different angle.

Technical Summary

1. Problem Statement and Context

The paper addresses the generalization of classical notions of holomorphic contact and holomorphic symplectic structures on compact complex manifolds.

• Background: Recently, scalar-valued (complex-valued) generalizations known as holomorphic $p$-contact and $s$-symplectic structures were introduced (by Kasuya, Popovici, and Ugarte). In the scalar case, these structures impose strict topological constraints: a manifold carrying a scalar $p$-contact structure is never Kähler.
• The Gap: The authors seek to extend these definitions to line bundle-valued differential forms. The central question is whether allowing the forms to take values in a non-trivial holomorphic line bundle $L$ relaxes the rigidity of the scalar case, potentially allowing for Kähler or even projective examples.
• Specific Goals:
  1. Define $L$-valued holomorphic $p$-contact and $s$-symplectic structures.
  2. Investigate the topological implications (specifically regarding spin structures).
  3. Analyze the curvature properties of the canonical bundle $K_X$ associated with these structures, particularly in the presence of singular Hermitian metrics.
  4. Determine if such manifolds can be projective and provide concrete examples.

2. Methodology

The authors employ a blend of complex differential geometry, sheaf cohomology, and advanced techniques in pluripotential theory.

• Generalized Definitions: They define $L$-valued forms $\Gamma$ (for $p$-contact) and $\Omega$ (for $s$-symplectic) satisfying $\bar{\partial}\Gamma = 0$ (or $\bar{\partial}\Omega=0$) and a non-degeneracy condition ($\Gamma \wedge \partial \Gamma \neq 0$ or $\Omega \wedge \Omega \neq 0$).
  • Technical Challenge: Since $\partial$ is not a connection on a general line bundle, $\partial \Gamma$ is not globally defined. The authors resolve this by showing that $\Gamma \wedge \partial \Gamma$ (or $\Omega \wedge \Omega$) is a well-defined global section of $L^2 \otimes K_X^{-1}$ (or similar), effectively making the non-degeneracy condition well-posed.
• Bochner-Kodaira-Nakano (BKN) Identities: The core analytical tool is the application of the BKN identity to the line bundles involved. They utilize the identity to relate the curvature of the bundle to the existence of global holomorphic sections.
• Singular Metrics and Regularization: A key methodological innovation is the handling of singular Hermitian fibre metrics. The authors use a recent regularization theorem (by Dinew and Popovici) to approximate singular metrics with smooth ones while preserving $m$-positivity properties. This allows them to apply standard BKN identities to singular settings.
• Cohomological Vanishing: They utilize exact sequences (normal bundles, tensor products) and vanishing theorems (Akizuki-Nakano, Kodaira) to prove non-existence results for specific cohomology groups on hypersurfaces.

3. Key Contributions and Definitions

A. New Definitions

The paper introduces locally holomorphic $p$-contact and $s$-symplectic manifolds:

• $p$-Contact: A compact complex manifold $X$ ($n=2p+1$, $p$ odd) with a line bundle $L$ and a form $\Gamma \in H^0(X, \Omega^p \otimes L)$ such that $\bar{\partial}\Gamma=0$ and $\Gamma \wedge \partial \Gamma \neq 0$.
• $s$-Symplectic: A compact complex manifold $X$ ($n=2s$, $s$ even) with a line bundle $L$ and a form $\Omega \in H^0(X, \Omega^s \otimes L)$ such that $\bar{\partial}\Omega=0$ and $\Omega \wedge \Omega \neq 0$.

B. Topological Implications

• Spin Structures: The existence of these structures implies that $L^2 \cong -K_X$ (the anti-canonical bundle). Consequently, $K_X$ has a square root, meaning $X$ must be a spin manifold.
• Hierarchy: The authors establish the implication chain:
  $$\text{Holomorphic Contact} \implies \text{Holomorphic } p\text{-Contact} \implies \text{Spin Manifold}$$
  This places $p$-contact manifolds as an intermediate class between classical contact manifolds and general spin manifolds.

C. Curvature and Positivity Results

The paper investigates the curvature of the canonical bundle $K_X$.

• Theorem 3.5 & 3.6: If $X$ admits an $L$-valued $p$-contact (or contact) structure, the canonical bundle $K_X$ cannot possess a Hermitian metric (even singular) with certain partial positivity properties (specifically $i\Theta \geq n-p, \omega 0$ or $i\Theta \geq n-1, \omega 0$).
• Scalar Curvature Vanishing (Theorem 3.7): If $X$ is a spin manifold with a Kähler metric $\omega$ and a metric on $-K_X$ such that the scalar curvature is non-negative everywhere and strictly positive somewhere, then $H^0(X, K_X) = 0$.
• Fano Property: These results imply that such manifolds can be Fano (where $-K_X$ is ample). This contrasts sharply with the scalar case, where $p$-contact manifolds are never Kähler.

4. Main Results

1. Existence of Projective Examples: Unlike scalar $p$-contact manifolds, locally holomorphic $p$-contact manifolds can be projective.

   • Complex Projective Space $\mathbb{P}^n$: The authors prove that $\mathbb{P}^n$ is a locally holomorphic $p$-contact manifold if and only if $n \equiv 3 \pmod 4$. They explicitly construct the required form $\Gamma$ with values in $\mathcal{O}(p+1)$.
   • Hypersurfaces: For smooth hypersurfaces $X \subset \mathbb{P}^{n+1}$, a holomorphic $p$-contact structure exists if and only if $X$ is isomorphic to $\mathbb{P}^n$ (i.e., degree $d=1$). For degrees $d \geq 3$, the relevant cohomology groups vanish, preventing the existence of such structures.

2. Geometric Consequences:

   • If $X$ is a projective manifold with an $L$-valued $p$-contact structure, its Kodaira dimension is $-\infty$.
   • Such manifolds are covered by rational curves and are not Kobayashi hyperbolic.

3. Symplectic Analogue:

   • For $s$-symplectic structures, the authors show that the sheaf of vector fields contracting the form to zero has rank zero.
   • They provide a construction for new examples via surjective holomorphic submersions: If $Y$ is $s$-symplectic and the fibers admit a $p$-contact structure, the total space $X$ admits an $(s+p)$-contact structure.

5. Significance

• Breaking Rigidity: The paper demonstrates that the "non-Kähler" rigidity of scalar holomorphic contact structures is an artifact of the triviality of the line bundle. By introducing line bundle values, the authors unlock a rich class of examples that include standard projective spaces and Fano manifolds.
• Unification: It unifies the theory of contact structures, symplectic structures, and spin geometry, showing that $p$-contact structures are a natural generalization that sits comfortably within the realm of spin manifolds.
• Analytical Tools: The application of singular metric regularization to prove vanishing theorems for $p$-contact structures extends the toolkit of complex geometry, offering a template for studying other geometric structures with singular metrics.
• Classification: The results provide a sharp classification for projective spaces and hypersurfaces, identifying $\mathbb{P}^n$ ($n \equiv 3 \mod 4$) as the primary example of a projective $p$-contact manifold.

In summary, Broder and Popovici successfully generalize the theory of holomorphic contact and symplectic manifolds to the bundle-valued setting, proving that these structures are compatible with Kähler and projective geometry, and providing a detailed classification of their existence on fundamental complex manifolds.