Stably semiorthogonally indecomposable varieties

This paper introduces the concept of noncommutatively stably semiorthogonally indecomposable (NSSI) varieties, proves that varieties with finite Albanese morphisms and certain fibrations possess this property, and demonstrates that NSSI varieties ensure the indecomposability of their subvarieties' derived categories and the absence of phantom subcategories in specific products like $C \times \mathbb{P}^1$.

The Gist

Imagine you are a master architect designing a massive, complex building. In the world of algebraic geometry, these "buildings" are called varieties (shapes defined by equations), and the "blueprints" or "structural integrity" of these buildings are described by something called a Derived Category.

This category is like a giant library of all the possible ways you can build with the materials of that shape. Sometimes, this library is so huge and messy that you can split it into two completely separate, non-interacting wings. If you can do this, the building is "decomposable." If you cannot split it, it is "indecomposable."

Most mathematicians are happy knowing a building is indecomposable. But Dmitrii Pirozhkov in this paper asks a much tougher question: Is the building indecomposable even if you try to shake it, twist it, or attach it to other buildings?

He introduces a super-powerful version of indecomposability called NSSI (Noncommutatively Stably Semiorthogonally Indecomposable).

Here is the breakdown of his discovery using simple analogies:

1. The "Ghost" Problem (Phantom Subcategories)

Imagine you have a library of books (the derived category). Sometimes, you can find a secret section of books that are so "ghostly" that they don't leave a single trace in the catalog. You can't count them, you can't see them in the index, but they are technically there. These are called Phantom Subcategories.

Pirozhkov wants to prove that in certain special buildings, these ghosts cannot exist. If a building is "NSSI," it's so solid that no ghost sections can hide inside it.

2. The "Anchor" (Abelian Varieties)

How do you build a structure that is this solid? Pirozhkov finds a special type of foundation: Abelian Varieties.

• Analogy: Think of an Abelian Variety as a perfect, infinite, doughnut-shaped grid (a torus). It has a very rigid, symmetrical structure.
• The Discovery: If your building is built directly on top of this perfect doughnut grid (or can be mapped to it without stretching), it becomes NSSI. It becomes "stuck" to the grid so firmly that it cannot be split apart, and no ghosts can hide inside.

3. The "Lego" Rule (Fibrations and Products)

The paper also explains how to build new solid structures using old ones.

• The Rule: If you have a solid foundation (a NSSI base) and you build a tower on it where every single floor is also a solid NSSI structure, the entire tower becomes NSSI.
• Analogy: Imagine a skyscraper where the ground floor is made of unbreakable steel, and every single floor above it is also made of unbreakable steel. The whole building is unbreakable.
• Real-world example: The author uses this to show that a surface made by taking a curve (like a circle) and attaching a line (like a stick) to every point on it is solid. Even though it looks complex, it's "NSSI."

4. The "Rigid" Subcategories

Why is this important?
In normal buildings, you might be able to take a room out and move it to a different building without the whole thing collapsing. In an NSSI building, the rooms are "rigid." They are glued to the walls so tightly that you cannot move them or split them off.

Pirozhkov proves that if a building is NSSI:

1. No Ghosts: You can't have those invisible "phantom" sections.
2. No Splitting: You can't split the library of blueprints into two separate, non-interacting wings.
3. Stability: Even if you combine this building with a simple one (like a line or a sphere), the resulting complex building still refuses to split or hide ghosts.

The Big Picture

Before this paper, mathematicians knew some shapes were indecomposable. But they didn't know if those shapes were robust enough to stay indecomposable when mixed with other shapes.

Pirozhkov says: "If you start with a shape that is anchored to a perfect doughnut-grid (Abelian variety), or if you stack these shapes on top of each other, you get a 'Super-Indecomposable' shape."

This is a big deal because it solves a mystery about "phantom" parts in the math world. It tells us that in these specific, well-behaved geometric worlds, everything is visible, everything is connected, and nothing can hide in the shadows.

In short: The paper builds a fortress of mathematical certainty where nothing can be split apart and nothing can hide in the dark.

Technical Summary

Here is a detailed technical summary of the paper "Stably semiorthogonally indecomposable varieties" by Dmitrii Pirozhkov.

1. Problem Statement and Motivation

The derived category of coherent sheaves, $D^b_{\text{coh}}(X)$, is a fundamental invariant in algebraic geometry. A central question is determining when such a category is indecomposable, meaning it admits no non-trivial semiorthogonal decompositions (SODs). While many varieties (e.g., Calabi-Yau varieties, curves of positive genus) are known to have indecomposable derived categories, the standard notion of indecomposability is often insufficient for studying products or fibrations.

Specifically, the author addresses the following gaps:

• Stability under Products: If $Y$ has an indecomposable derived category, does $X \times Y$ inherit specific structural properties regarding its SODs?
• Phantom Subcategories: Do "phantom" subcategories (non-zero admissible subcategories with zero class in the Grothendieck group $K_0$) exist in products involving varieties with indecomposable derived categories?
• Rigidity: Can one define a stronger notion of indecomposability that forces any admissible subcategory in a "linear" setting to be closed under the action of the base scheme's perfect complexes?

2. Methodology and Definitions

The paper utilizes the formalism of stable $\infty$-categories (following Lurie and Perry) to handle linear structures rigorously.

Key Definitions

• Perf(Y)-linear Category: A $k$-linear stable $\infty$-category $\mathcal{D}$ equipped with an action functor $\mathcal{D} \otimes_k \text{Perf}(Y) \to \mathcal{D}$.
• NSSI (Noncommutatively Stably Semiorthogonally Indecomposable): A scheme $Y$ is NSSI if for any $\text{Perf}(Y)$-linear category $\mathcal{D}$ (proper over $Y$ with a classical generator) and any left admissible subcategory $\mathcal{A} \subset \mathcal{D}$, the subcategory $\mathcal{A}$ is closed under the action of $\text{Perf}(Y)$.
  • Implication: This is strictly stronger than standard indecomposability. It implies that the derived category of any connected closed subscheme is also indecomposable.
• Phantom Subcategory: A non-zero admissible subcategory $\mathcal{A} \subset \mathcal{T}$ such that the class $[A] = 0$ in $K_0(\mathcal{T})$ for all $A \in \mathcal{A}$.

Technical Tools

• Rigidity of Admissible Subcategories: The author proves that in suitable settings, the condition of an object lying in a fixed admissible subcategory is an open condition in the base scheme (Theorem 3.1).
• Fourier-Mukai Transforms: The paper leverages the action of the Picard scheme $\text{Pic}^0(Y)$ via Fourier-Mukai transforms to show that admissible subcategories are invariant under the action of line bundles in the connected component of the identity.
• Base Change and Fibrations: The paper establishes how the NSSI property behaves under flat proper morphisms and base changes, utilizing the existence of mapping objects in linear categories.

3. Key Contributions and Results

A. Characterization of NSSI Schemes

The paper provides two primary theorems establishing large classes of NSSI varieties:

1. Affine Morphisms to Abelian Varieties (Theorem 1.4):
   If a scheme $Y$ admits an affine morphism to an abelian variety $A$, then $Y$ is NSSI.

   • Proof Strategy: The author first proves that abelian varieties are NSSI using the Fourier-Mukai equivalence with their duals and the rigidity of admissible subcategories under the action of $\text{Pic}^0(A)$. The result is then extended to $Y$ via the affine morphism property (Lemma 2.14).

2. Fibrations over NSSI Bases (Theorem 1.5):
   If $\pi: Y \to B$ is a flat proper morphism where the base $B$ is NSSI and every fiber $Y_b$ is NSSI, then the total space $Y$ is NSSI.

   • Significance: This allows the construction of NSSI varieties that do not admit affine maps to abelian varieties.
   • Example: Bielliptic surfaces are proven to be NSSI. Their Albanese map is a fibration over an elliptic curve (NSSI) with elliptic fibers (NSSI), yet the Albanese map is not finite (so they don't fit the first theorem).

B. Structural Consequences for Products

Lemma 5.3 establishes a crucial structural result: If $Y$ is NSSI and $X$ is any smooth projective variety, then any admissible subcategory $\mathcal{A} \subset D^b_{\text{coh}}(X \times Y)$ is induced from $X$.

• Specifically, $\mathcal{A} \cong \mathcal{A}_X \boxtimes D^b_{\text{coh}}(Y)$ for some admissible $\mathcal{A}_X \subset D^b_{\text{coh}}(X)$.
• This implies that the "complexity" of the derived category of the product is entirely determined by the factor $X$; the NSSI factor $Y$ cannot support "new" non-trivial decompositions.

C. Non-existence of Phantom Subcategories

The paper applies the NSSI property to rule out phantom subcategories in specific geometric settings (Proposition 1.6 / Proposition 5.4):

• Case 1: If $Y$ is NSSI and $X$ is $\mathbb{P}^1$ or a del Pezzo surface, then $D^b_{\text{coh}}(X \times Y)$ has no phantom subcategories.
• Case 2: If $\pi: X \to Y$ is an étale-locally trivial fibration with fiber $\mathbb{P}^1$ or $\mathbb{P}^2$ and $Y$ is NSSI, then $D^b_{\text{coh}}(X)$ has no phantom subcategories.

Proof Logic: Since $Y$ is NSSI, any admissible subcategory in the total space is determined by its restriction to a fiber. Since the fibers ($\mathbb{P}^1, \mathbb{P}^2$) are known to have no phantom subcategories, and the pushforward map on $K$-theory is injective in these specific fibration cases, the total space cannot contain phantoms.

4. Significance and Impact

• Strengthening Indecomposability: The paper introduces "NSSI" as a robust property that is preserved under products and fibrations, unlike standard indecomposability. This provides a new tool for classifying varieties based on their derived categorical structure.
• Resolution of Phantom Questions: It provides a definitive negative answer to the existence of phantom subcategories in a wide range of surfaces and threefolds (e.g., $C \times \mathbb{P}^1$ where $C$ is a curve of genus $g > 0$), a problem that was previously open for these specific cases.
• Geometric Examples: The identification of bielliptic surfaces as NSSI varieties demonstrates that the property is not limited to varieties with finite Albanese maps, broadening the scope of "rigid" derived categories.
• Methodological Advance: The use of $\infty$-categorical linearity and the rigidity of admissible subcategories via the Picard scheme offers a powerful framework for future investigations into semiorthogonal decompositions in families.

In summary, Pirozhkov establishes that varieties "close" to abelian varieties (via affine maps or fibrations) possess a rigid derived structure that prevents non-trivial decompositions and phantom phenomena, significantly advancing the understanding of the stability of derived categories under geometric operations.