[Paper Review] Properness criteria for families of coherent analytic sheaves

Matei Toma|arXiv (Cornell University)|Oct 4, 2017

Algebraic Geometry and Number Theory21 references3 citations

TL;DR

This paper extends Langton's valuative criterion for properness of moduli spaces of coherent sheaves to complex analytic geometry, providing new criteria for compactness of moduli spaces of semistable sheaves on compact complex manifolds. It introduces a strong degree function and uses Artin approximation and combinatorial techniques to handle irrational polarizations and non-projective bases, proving that moduli spaces are compact under suitable boundedness and stability conditions.

ABSTRACT

We extend Langton's valuative criterion for families of coherent algebraic sheaves to a complex analytic set-up. As a consequence we derive a set of sufficient conditions for the compactness of a moduli space of semistable sheaves over a compact complex manifold. This applies also to some cases appearing in complex projective geometry not covered by previous results.

Motivation & Objective

• To generalize Langton’s valuative criterion for properness from algebraic to complex analytic settings.
• To establish sufficient conditions for the compactness of moduli spaces of semistable coherent analytic sheaves on compact complex manifolds.
• To address cases with irrational polarizations and non-projective base manifolds where previous results fail.
• To provide a criterion for separatedness of moduli spaces in the analytic category.
• To extend stability theory beyond projective geometry using Gauduchon metrics and degree functions.

Proposed method

• Introduces a strong degree function on the Grothendieck group of coherent sheaves, satisfying continuity, positivity, and finiteness conditions.
• Uses Artin approximation to bypass non-properness issues in the relative Douady space.
• Applies a combinatorial argument to compensate for the lack of rationality in polarization classes.
• Employs a devissage argument on the special fiber via iterated cokernel reduction to achieve semistable replacement.
• Relies on boundedness of isomorphism classes of coherent sheaves, as defined in Tomassini (2016).
• Adapts Langton’s original proof strategy while modifying key steps to fit the analytic and non-projective context.

Experimental results

Research questions

• RQ1 Can Langton’s valuative criterion for properness be extended to families of coherent analytic sheaves over compact complex manifolds?
• RQ2 How can one ensure compactness of moduli spaces of semistable sheaves when the polarization is irrational or the base is non-projective?
• RQ3 What conditions on degree functions and boundedness ensure properness in the analytic setting?
• RQ4 How does the notion of semistability behave under flat deformations in the absence of projectivity?
• RQ5 What is the role of Artin approximation and combinatorial techniques in replacing non-semistable fibers?

Key findings

• A new valuative criterion for properness is established for families of coherent analytic sheaves over compact complex manifolds, extending Langton’s result to the analytic category.
• The moduli space of semistable sheaves over a compact complex manifold is compact if the sheaves are bounded and semistable with respect to a strong degree function.
• For one-dimensional bases, any flat family with general semistable fibers admits a replacement with only semistable fibers, even for irrational polarizations.
• The criterion for separatedness is established via a devissage argument on the special fiber, showing that the graded Jordan-Hölder factors are preserved under fiberwise modification.
• The proof relies on Artin approximation to avoid issues with the relative Douady space and uses a combinatorial method to handle non-rational polarizations.
• The results apply to cases in complex projective geometry not covered by earlier theorems, particularly when the polarization is irrational.