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arxiv: 2606.26193 · v1 · pith:AGJYNOXBnew · submitted 2026-06-24 · 🧮 math.AG

Inducing t-structures on semiorthogonal components

Alexander Kuznetsov , Shengxuan Liu , Alexander Perry This is my paper

Pith reviewed 2026-06-26 01:15 UTC · model grok-4.3

classification 🧮 math.AG
keywords semiorthogonal decompositionst-structuresperverse t-structurestriangulated categoriesderived categoriesphantom categoriesFano varietiesEnriques surfaces
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The pith

A method induces t-structures on semiorthogonal components of a triangulated category by first building an associated perverse t-structure on the ambient category.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a general procedure that takes a triangulated category equipped with a t-structure and produces t-structures on each piece of a semiorthogonal decomposition of that category. The procedure works by constructing a perverse t-structure on the whole category and then restricting its data to the components. When the construction succeeds, it supplies bounded t-structures on many categories that previously lacked them, including phantom and quasiphantom categories, the orthogonal complement to the structure sheaf on a Fano variety, the residual component of an Enriques surface, and the categorical resolution of a nodal cubic curve. A reader would care because t-structures turn the derived category into an abelian heart whose objects can be studied with homological algebra tools.

Core claim

Given a triangulated category with a t-structure, one constructs an associated perverse t-structure on the same category; this perverse t-structure then induces t-structures on the individual semiorthogonal components.

What carries the argument

The perverse t-structure associated to a given t-structure on the ambient triangulated category, which supplies the truncation functors and heart needed to define the induced t-structures on each semiorthogonal summand.

If this is right

  • Bounded t-structures exist on almost all known phantom and quasiphantom categories.
  • The semiorthogonal complement of the structure sheaf on a Fano variety admits a bounded t-structure.
  • The residual component of an Enriques surface admits a bounded t-structure.
  • The categorical resolution of a nodal cubic curve that appears in a counterexample to the Jordan-Hölder property admits a bounded t-structure.
  • Brill-Noether modifications of the derived category of a curve admit bounded t-structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same induction could be tested on semiorthogonal decompositions that arise from birational maps or flops.
  • The induced t-structures may be compatible with mutations, allowing one to move t-structures between different decompositions of the same category.
  • Existence of these t-structures supplies new hearts that could be used to define stability conditions on the components.

Load-bearing premise

The ambient triangulated category admits a t-structure whose associated perverse t-structure interacts compatibly with the given semiorthogonal decomposition.

What would settle it

An explicit triangulated category equipped with a t-structure and a semiorthogonal decomposition in which the proposed truncation triangles on one component fail to exist or fail to satisfy the required orthogonality conditions.

read the original abstract

Given a triangulated category with a t-structure, we introduce a method for inducing t-structures on its semiorthogonal components, based on the construction of an associated perverse t-structure on the ambient category. As applications, we construct bounded t-structures in many new examples, including: almost all known phantom and quasiphantom categories; the semiorthogonal complement of the structure sheaf on a Fano variety; the residual component of an Enriques surface; the categorical resolution of a nodal cubic curve appearing in an early counterexample to the Jordan-H\"{o}lder property for semiorthogonal decompositions; and Brill-Noether modifications of the derived category of a curve.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The paper introduces a method to induce t-structures on semiorthogonal components of a triangulated category equipped with a t-structure, by constructing an associated perverse t-structure on the ambient category whose truncation functors restrict to the components via orthogonality. Applications construct bounded t-structures on almost all known phantom and quasiphantom categories, the semiorthogonal complement of the structure sheaf on a Fano variety, the residual component of an Enriques surface, the categorical resolution of a nodal cubic curve, and Brill-Noether modifications of the derived category of a curve.

Significance. If the central construction holds, the result is significant: it supplies a uniform, direct verification procedure that equips many previously t-structure-less semiorthogonal components with bounded t-structures, resolving existence questions in several concrete geometric settings. The paper's strength lies in the explicit compatibility checks for each listed application rather than abstract generality alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, careful reading of the manuscript, and recommendation to accept. We are pleased that the uniform construction and its applications to phantom categories, Fano complements, Enriques residuals, and other examples were viewed as significant.

Circularity Check

0 steps flagged

No significant circularity; direct axiom verification on components

full rationale

The paper introduces an explicit construction of an associated perverse t-structure on the ambient triangulated category and verifies that its truncation functors restrict to the semiorthogonal components by direct checking of the t-structure axioms using the given orthogonality relations. No equations reduce a claimed prediction or result to a fitted input by construction, no load-bearing uniqueness theorem is imported from self-citation, and applications consist of case-by-case compatibility checks with standard ambient t-structures rather than self-referential definitions. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the standard axioms of triangulated categories and the existence of a t-structure on the ambient category; no free parameters or new entities are mentioned.

axioms (1)
  • domain assumption Triangulated categories admit t-structures that can be used to define perverse variants.
    Invoked as the starting point for the induction method.

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discussion (0)

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Reference graph

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