[Paper Review] Canonical singular hermitian metrics on relative logcanonical bundles

Hajime Tsuji|arXiv (Cornell University)|Aug 9, 2010

Advanced Algebra and Geometry23 references10 citations

TL;DR

This paper introduces supercanonical analytic Zariski decompositions (AZDs) — canonical singular hermitian metrics with minimal singularities and semipositive curvature — on relative log canonical bundles of KLT pairs with pseudoeffective log canonical divisors. By constructing these metrics via inductive extension of sections and multiplier ideal sheaves, the authors establish the invariance of logarithmic plurigenera under projective deformations and prove semipositivity of direct images of pluri log canonical systems, extending results from the canonical case to the log setting.

ABSTRACT

This supersedes 0704.0566. We prove the invariance of logarithmic plurigenera for a projective family of KLT pairs and the adjoint line bundle of KLT line bundles. The proof uses the canonical singular hermitian metrics on relative logcanonical bundles.

Motivation & Objective

• To define a canonical singular hermitian metric on the relative log canonical bundle of a KLT pair with pseudoeffective log canonical divisor.
• To construct a supercanonical AZD (analytic Zariski decomposition) with semipositive curvature and minimal singularities on the relative log canonical bundle.
• To prove the invariance of logarithmic plurigenera under projective deformations of KLT pairs.
• To extend the theory of canonical AZDs from the canonical case to the log canonical case, including adjoint line bundles.
• To establish semipositivity and local freeness of direct images of pluri log canonical systems via the constructed metrics.

Proposed method

• Constructs a singular hermitian metric $\hat{h}_{\text{can}}$ on $K_X$ for smooth projective varieties with pseudoeffective canonical bundle, using lower semicontinuous envelopes and extension across singular fibers.
• Applies the theory of multiplier ideal sheaves $\mathcal{I}(h^m)$ to ensure that $H^0(X, \mathcal{O}_X(mK_X) \otimes \mathcal{I}(\hat{h}_{\text{can}}^m)) \simeq H^0(X, \mathcal{O}_X(mK_X))$ for all $m \geq 0$, satisfying the AZD condition.
• Uses induction on $m$ and $\ell$ to extend sections from fibers to the total space, leveraging global generation and the asymptotic adjunction formula.
• Constructs metrics $h_{m,\ell}$ on $mM + H + \ell K_X + \Delta_1 + \cdots + \Delta_\ell$ with algebraic singularities and semipositive curvature via $L^2$-estimates and extension theorems.
• Applies Hölder’s inequality and limits over $\epsilon \to 0$ to show that $\hat{h}_{\text{can},D}(A,h_A)|X_s = O(\hat{h}_{\text{can},D,s}((A,h_A)|X_s))$, ensuring compatibility under degeneration.
• Extends the construction to KLT pairs by decomposing $kD = \Delta_1 + \cdots + \Delta_{k-1}$ with reduced simple normal crossing divisors and using adjunction theory for multiplier ideals.

Experimental results

Research questions

• RQ1 Can a canonical singular hermitian metric with minimal singularities and semipositive curvature be constructed on the relative log canonical bundle of a KLT pair with pseudoeffective log canonical divisor?
• RQ2 Does the variation of this supercanonical AZD under projective deformations imply the invariance of logarithmic plurigenera?
• RQ3 How can the theory of canonical AZDs be generalized from the canonical case to the log canonical case, including adjoint line bundles?
• RQ4 What conditions ensure that the direct image of a pluri log canonical system is locally free and carries a Griffiths semipositive metric?
• RQ5 Can the construction be extended to noncompact complex manifolds such as bounded domains in $\mathbb{C}^n$?

Key findings

• The supercanonical AZD $\hat{h}_{\text{can}}$ on $K_X$ for a smooth projective variety with pseudoeffective canonical bundle is uniquely determined, has semipositive curvature, and satisfies $H^0(X, \mathcal{O}_X(mK_X) \otimes \mathcal{I}(\hat{h}_{\text{can}}^m)) \simeq H^0(X, \mathcal{O}_X(mK_X))$ for all $m \geq 0$.
• The variation of $\hat{h}_{\text{can}}$ under projective deformations of smooth varieties with pseudoeffective canonical bundles gives a new proof of the invariance of plurigenera.
• For KLT pairs $(X_s, D_s)$ with pseudoeffective log canonical divisor, the supercanonical AZD $\hat{h}_{\text{can},D}$ on $K_{X/S} + A + D$ induces a canonical metric on each fiber with minimal singularities and semipositive curvature.
• The invariance of logarithmic plurigenera is established via the construction of supercanonical AZDs on relative log canonical bundles, generalizing the canonical case.
• The direct image sheaf $\mu_* \mathcal{O}_X(m(K_{X/S} + A + D))$ is locally free and carries a Griffiths semipositive metric, with curvature current semipositive in the sense of Demailly.
• The construction extends to noncompact complex manifolds such as bounded domains in $\mathbb{C}^n$, suggesting applications in the study of canonical metrics on noncompact Kähler manifolds.