Measuring rationality of Schwede--Takagi pairs

1. Problem Statement and Context

The paper addresses the study of Schwede–Takagi pairs $(Y, \mathcal{I}^c)$, where $Y$ is a normal variety over a field of characteristic zero, $\mathcal{I}$ is a coherent ideal sheaf, and $c \in \mathbb{R}_{\geq 0}$. These objects arise naturally in the Minimal Model Program (MMP).

• Rational Singularities for Pairs: A pair $(Y, \mathcal{I}^c)$ is said to have rational singularities if, for a log resolution $f: \tilde{Y} \to Y$ (where $\mathcal{I}\cdot\mathcal{O}_{\tilde{Y}} = \mathcal{O}_{\tilde{Y}}(-G)$), the natural map $\mathcal{O}_Y \to Rf_*\mathcal{O}_{\tilde{Y}}(\lfloor c \cdot G \rfloor)$ is an isomorphism.
• The Gap: While derived categorical characterizations exist for singularities of schemes (e.g., locally complete intersections, rational singularities of varieties), extending these to pairs presents a challenge. Specifically, the natural map for pairs does not necessarily arise as the unit of an adjoint functor pair, rendering standard "generation" techniques inapplicable directly.
• The Goal: The authors aim to:
  1. Provide a derived categorical characterization of rational singularities for pairs.
  2. Define a numerical invariant (using "level") to measure the failure of rationality for pairs, particularly on affine varieties that are locally complete intersections (LCIs).

2. Methodology

The authors employ techniques from triangulated category theory, specifically the concept of generation and level.

• Level of an Object: For objects $E, G$ in a triangulated category $\mathcal{T}$, $\text{level}_G(E)$ is the smallest integer $n$ such that $E$ can be constructed from $G$ using shifts, direct summands, finite direct sums, and at most $n$ cones. If no such $n$ exists, the level is $\infty$.
• Derived Characterization: The paper seeks to characterize rationality by determining if $\mathcal{O}_Y$ belongs to the subcategory $\langle Rf_*\mathcal{O}_{\tilde{Y}}(\lfloor c \cdot G \rfloor) \rangle_1$ (i.e., level 1).
• Technical Strategy:
  • Local Algebra & Completion: Since the standard adjunction unit strategy fails for pairs, the authors pass to the completions of the local rings ($\widehat{\mathcal{O}}_{Y,p}$). They utilize the fact that for normal local rings, the completion is a normal domain.
  • Krull–Schmidt Categories: They leverage the property that the derived category of coherent sheaves over a complete local ring is a Krull–Schmidt category. This allows them to decompose objects and use indecomposability arguments (specifically that $\mathcal{O}_Y$ is indecomposable).
  • Duality: The proof utilizes Grothendieck duality, relating the pushforward of the structure sheaf to the dualizing complex $\omega^\bullet_Y$ and the multiplier submodule.
  • LCI Homotopy: For the measurement of failure, they utilize results by Pollitz regarding the generation of the derived category of affine LCI schemes.

3. Key Contributions and Results

A. Derived Characterization of Rational Pairs (Theorem 1.1)

The authors prove that a pair $(Y, \mathcal{I}^c)$ with $Y$ normal has rational singularities if and only if:
$$\mathcal{O}_Y \in \langle Rf_*\mathcal{O}_{\tilde{Y}}(\lfloor c \cdot G \rfloor) \rangle_1$$
This means $\mathcal{O}_Y$ is a direct summand of $Rf_*\mathcal{O}_{\tilde{Y}}(\lfloor c \cdot G \rfloor)$ (up to shift).

• Significance: This extends previous work by Lank–Venkatesh (which characterized rational singularities for varieties) to the setting of pairs. The proof strategy differs significantly from prior work because the map $\mathcal{O}_Y \to Rf_*\mathcal{O}_{\tilde{Y}}(\lfloor c \cdot G \rfloor)$ is not a unit of an adjunction.

B. Measuring the Failure of Rationality

The paper investigates whether the "level" can quantify how far a pair is from being rational.

• The Challenge: In general, $\text{level}_{Rf_*\mathcal{O}_{\tilde{Y}}(\lfloor c \cdot G \rfloor)}(\mathcal{O}_Y)$ might be infinite.
• The Solution (LCI Case): For normal, quasi-affine varieties that are Locally Complete Intersections (LCIs), the authors prove that this level is always finite.
• Connection to Multiplier Submodules: When $Y$ is Cohen–Macaulay (which is true for LCIs), the level of $\mathcal{O}_Y$ with respect to the pushforward sheaf coincides with the level of the multiplier submodule $\mathcal{J}(\omega_Y, \mathcal{I}^c)$ with respect to the dualizing sheaf $\omega_Y$:
  $$\text{level}_{Rf_*\mathcal{O}_{\tilde{Y}}(\lfloor c \cdot G \rfloor)}(\mathcal{O}_Y) = \text{level}_{\mathcal{J}(\omega_Y, \mathcal{I}^c)}(\omega_Y)$$

C. Proposition 1.2 (Main Application)

For a pair $(Y, \mathcal{I}^c)$ where $Y$ is normal, quasi-affine, and an LCI:

1. The value $\text{level}_{\mathcal{J}(\omega_Y, \mathcal{I}^c)}(\omega_Y)$ is finite.
2. The pair has rational singularities if and only if this level equals 1.
3. If the level is $>1$, it quantifies the "distance" from rationality.

D. Computability

The authors note that for affine LCI varieties, this invariant is computable using existing computer algebra systems (e.g., Macaulay2 packages like ThickSubcategories). This provides a concrete numerical tool for studying singularities of pairs.

E. Kollár–Kovács Pairs (Appendix A)

The authors extend their results to Kollár–Kovács rational pairs (reduced pairs $(Y, D)$ with coefficients 1). They establish a splitting criterion and a derived characterization (Theorem A.2) analogous to the Schwede–Takagi case, noting that these two notions of rationality are distinct.

4. Significance and Impact

• New Invariant: The paper introduces a categorical invariant (level) that measures the failure of rationality for pairs. Unlike previous invariants that only detect if a singularity is rational, this provides a quantitative measure of non-rationality.
• Bridging Geometry and Category Theory: It successfully adapts homotopical generation techniques (previously used for schemes) to the more complex setting of pairs, overcoming the lack of adjunction units.
• Computational Potential: By restricting to affine LCI varieties, the authors make these abstract derived concepts computable, opening the door for experimental algebraic geometry in the study of pair singularities.
• Theoretical Extension: It generalizes the Lank–Venkatesh characterization of rational singularities to the Schwede–Takagi framework, unifying the derived categorical approach to singularities across different contexts (varieties vs. pairs).

In summary, the paper provides a rigorous derived categorical framework to not only detect but also measure the rationality of Schwede–Takagi pairs, offering a new numerical lens through which to view the geometry of singularities in the Minimal Model Program.