A birational description of the minimal exponent

Technical Summary: A Birational Description of the Minimal Exponent

Problem Statement
The paper addresses the problem of providing a birational description of the minimal exponent, $e_\alpha(Z)$, of a hypersurface $Z$ in a smooth complex algebraic variety $X$. While the log canonical threshold, $\text{lct}(X, Z)$, admits a well-known description via the numerical data of a log resolution (specifically, the minimum of $(k_i+1)/a_i$ over exceptional divisors), the minimal exponent—a refinement defined by Saito using the Bernstein-Sato polynomial—lacks such a direct numerical characterization. Previous attempts to bound $e_\alpha(Z)$ using resolution data (e.g., $e_\alpha(Z) \ge \min (k_i+1)/a_i$) fail to yield equality because the right-hand side depends on the specific resolution chosen. The authors aim to characterize $e_\alpha(Z)$ using the cohomological properties of a log resolution, specifically through higher direct images of twisted sheaves of log differentials, analogous to the characterization of the log canonical threshold via multiplier ideals.

Methodology
The authors employ a framework combining the theory of mixed Hodge modules, $D$-modules, and the $V$-filtration. The core methodology involves:

1. Nearby Cycles and $V$-filtration: The minimal exponent is characterized via the $V$-filtration on the $D$-module $B_f$ associated with a local equation $f$ of $Z$. Specifically, $e_\alpha(Z) > \gamma$ (where $\gamma = p + \alpha$) is linked to the vanishing of specific graded pieces of the de Rham complex of the nearby cycles $\psi_{f, \alpha}(\mathcal{O}_X)$.
2. Log Resolution and Push-forward: Using a log resolution $\pi: Y \to X$ with exceptional divisor $E$ and strict transform $D = \pi^*(Z)$, the authors utilize Saito's strictness theorem to relate the nearby cycles on $X$ to those on $Y$: $\psi_{f, \alpha}(\mathcal{O}_X) \simeq \pi_+ \psi_{g, \alpha}(\mathcal{O}_Y)$, where $g = f \circ \pi$.
3. Filtered Resolutions: A central technical contribution is the construction of an explicit filtered resolution of the nearby cycles $\psi_{g, \alpha}(\mathcal{O}_Y)$ when $g$ defines a simple normal crossing (SNC) divisor. The authors construct a complex $C_G$ involving sheaves of log differentials $\Omega^\bullet_Y(\log E)$ twisted by divisors related to the $V$-filtration steps ($D_\alpha = \lceil \alpha D \rceil$).
4. Cohomological Characterization: By analyzing the graded pieces of the de Rham complex of these resolutions, the authors translate the condition $e_\alpha(Z) > p + \alpha$ into vanishing conditions and isomorphism properties of higher direct images $R^q\pi_*$ of twisted log differential forms.

Key Contributions and Results

• Theorem 1.1 (Integer Case): For a reduced hypersurface $Z$ and an integer $p > 0$, the condition $e_\alpha(Z) > p$ is equivalent to:

  1. The vanishing of higher direct images: $R^q\pi_*\Omega^i_Y(\log E) = 0$ for all $q \ge 1$ and $i \le p$.
  2. A codimension condition on the singular locus: $\text{codim}_Z(Z_{\text{sing}}) \ge 2p$ (or $\min\{3, 2p\}$).
     This result rephrases the condition of having $(p-1)$-rational singularities in terms of the log resolution of the pair $(X, Z)$.

• Theorem 1.2 (Non-Integer Case): For $\gamma = p + \alpha$ with $p \in \mathbb{Z}_{\ge 0}$ and $\alpha \in (0, 1) \cap \mathbb{Q}$, assuming $e_\alpha(Z) \ge p$:

  1. If $e_\alpha(Z) \ge p$, the morphism $R^q\pi_*\Omega^{n-p}_Y(\log E)(-E - \lfloor \alpha D \rfloor) \to R^q\pi_*\Omega^{n-p}_Y(\log E)(-E)$ is an isomorphism for $q \neq p$ and injective for $q=p$.
  2. $e_\alpha(Z) > p + \alpha$ if and only if this morphism is an isomorphism for $q=p$.

• Corollary 1.3 (Dual Formulation): Using Grothendieck duality, the authors provide a dual characterization: if $e_\alpha(Z) > p$, then $e_\alpha(Z) > p + \alpha$ if and only if $R^q\pi_*\Omega^p_Y(\log E)(\lfloor \alpha D \rfloor) = 0$ for all $q \ge 1$. This connects the minimal exponent to the triviality of certain cohomology groups of twisted log forms.

• Theorem 6.1 (Constancy in Families): The authors apply their characterization to prove that the minimal exponent is constant in a proper family of hypersurfaces that admit a simultaneous log resolution. This answers a question posed by Radu Laza.

Significance and Claims
The paper claims to provide a "more explicit description" of the minimal exponent that relies solely on the higher direct images of twisted sheaves of log differentials on a log resolution. The authors note that while this characterization is theoretically significant and allows for the proof of constancy in families, it is "not well-suited for computations" even in simple cases like ordinary singularities.

The work bridges the gap between the abstract definition of the minimal exponent via $V$-filtrations and concrete birational geometry. It extends the known characterization of the log canonical threshold (via multiplier ideals) to the broader context of the minimal exponent and higher Du Bois/rational singularities. The paper explicitly states that the main technical result is the construction of the filtered resolution for nearby cycles in the SNC case (Theorem 4.1), which serves as the engine for the main theorems. The authors view the characterization as having "rather theoretical interest," particularly in establishing properties like the constancy of the invariant in families, rather than as a practical computational tool.