[Paper Review] Vanishing and semipositivity theorems for semi-log canonical pairs

Osamu Fujino|arXiv (Cornell University)|Sep 2, 2015

Algebraic Geometry and Number Theory25 references3 citations

TL;DR

This paper establishes effective vanishing and semipositivity theorems for semi-log canonical pairs without relying on variation of mixed Hodge structure. It proves an effective vanishing theorem for direct images of log pluricanonical bundles and applies it to derive a semipositivity theorem for relative log pluricanonical bundles over curves, which implies the projectivity of moduli spaces of stable varieties.

ABSTRACT

We prove an effective vanishing theorem for direct images of log pluricanonical bundles of projective semi-log canonical pairs. As an application, we obtain a semipositivity theorem for direct images of relative log pluricanonical bundles of projective semi-log canonical pairs over curves, which implies the projectivity of the moduli spaces of stable varieties. It is worth mentioning that we do not use the theory of variation of (mixed) Hodge structure.

Motivation & Objective

• To establish an effective vanishing theorem for direct images of log pluricanonical bundles on projective semi-log canonical pairs.
• To prove a semipositivity theorem for direct images of relative log pluricanonical bundles over curves, without using variation of mixed Hodge structure.
• To provide a new, Hodge-theory-free proof of the projectivity of moduli spaces of stable varieties.
• To offer a simplified and accessible proof of the basic semipositivity theorem in Fujino (2012), independent of advanced Hodge-theoretic machinery.
• To complement Fujino (2012) by making its results more accessible through alternative, direct methods.

Proposed method

• Proves an effective vanishing theorem (Theorem 1.1) for $ f_*\mathcal{O}_X(D) \otimes \mathcal{O}_Y(lL) $ with explicit cohomological vanishing bounds based on Castelnuovo–Mumford regularity.
• Applies the Kollár–Ohsawa type vanishing theorem for semi-log canonical pairs (Theorem 3.1) as the core technical ingredient.
• Uses the theory of mixed Hodge structure on cohomology with compact support to establish vanishing for simple normal crossing pairs.
• Constructs iterated fiber products $ X^{(s)} $ and applies birational modifications via blow-ups to resolve singularities while preserving log canonical properties.
• Employs a generically isomorphic inclusion $ g_*\omega_Z(\Delta_Z) \subset \omega_{X^{(s)}}(D^{(s)}) $ to relate canonical sheaves across resolutions.
• Applies global generation via $ \mathcal{M} = \omega_C \otimes \mathcal{L}^{\otimes 2} $ to deduce nefness of $ f_*\omega_{X/C}(D) $ without Hodge theory.

Experimental results

Research questions

• RQ1 Can effective vanishing theorems for semi-log canonical pairs be established without relying on variation of mixed Hodge structure?
• RQ2 Is the semipositivity of direct images of relative log pluricanonical bundles over curves provable via vanishing theorems alone?
• RQ3 Can the projectivity of moduli spaces of stable varieties be established independently of Hodge-theoretic methods?
• RQ4 What is the minimal set of assumptions under which the basic semipositivity theorem holds for simple normal crossing pairs?
• RQ5 Can the proof of Fujino (2012, Theorem 1.9) be made self-contained and Hodge-theory-free?

Key findings

• An effective vanishing theorem (Theorem 1.1) is established: $ H^i(Y, f_*\mathcal{O}_X(D) \otimes \mathcal{O}_Y(lL)) = 0 $ for $ i > 0 $ and $ l \geq (k-1)(n+1-t) - t + 1 $, with global generation for $ l \geq (k-1)(n+1-t) - t + 1 + n $.
• The semipositivity theorem (Theorem 1.2) is proven: $ f_*\mathcal{O}_X(k(K_{X/Y} + \Delta)) $ is nef for flat morphisms $ f: X \to Y $ over a smooth curve $ Y $, under semi-log canonical and generation conditions.
• The basic semipositivity theorem (Theorem 1.4) is reproven without Hodge theory: $ f_*\omega_{X/C}(D) $ is nef for a simple normal crossing pair $ (X,D) $ with $ f: X \to C $ flat and every stratum dominant over $ C $.
• The projectivity of complete subspaces of the coarse moduli space of stable varieties (Theorem 1.3) is established via Kollár’s criterion and the semipositivity result.
• The proof of the basic semipositivity theorem is simplified and made independent of graded polarizable admissible variation of mixed Hodge structure.
• The construction of a resolution $ g: Z \to X^{(s)} $ with $ \Delta_Z $ a simple normal crossing divisor ensures compatibility of canonical sheaves and enables global generation arguments.