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Uniform boundedness of semistable pure sheaves on projective manifolds

Mihai Pavel, Julius Ross, Matei Toma

Abstract

We prove uniform boundedness statements for semistable pure sheaves on projective manifolds. For example, we prove that the set of isomorphism classes of pure sheaves of dimension 2 that are slope semistable with respect to ample classes that vary in a compact set $K$ are bounded. We also prove uniform boundedness for pure sheaves of higher dimension, but with restrictions on the compact set $K$. As applications we get new statements about moduli spaces of semistable sheaves, and the wall-chamber structure that governs their variation.

Uniform boundedness of semistable pure sheaves on projective manifolds

Abstract

We prove uniform boundedness statements for semistable pure sheaves on projective manifolds. For example, we prove that the set of isomorphism classes of pure sheaves of dimension 2 that are slope semistable with respect to ample classes that vary in a compact set are bounded. We also prove uniform boundedness for pure sheaves of higher dimension, but with restrictions on the compact set . As applications we get new statements about moduli spaces of semistable sheaves, and the wall-chamber structure that governs their variation.
Paper Structure (6 sections, 24 theorems, 78 equations, 1 figure)

This paper contains 6 sections, 24 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preparations
  3. The case of two-dimensional support
  4. A generalized Grauert-Mülich-Spindler Theorem
  5. Boundedness results in higher dimension
  6. Applications

Key Result

Theorem 1.3

The statement $\mathop{\mathrm{\mathbf{UB}}}\nolimits(K,\gamma)$ holds for arbitrary $2$-dimensional numerical classes $\gamma$ and any compact $K \subset \mathop{\mathrm{Amp}}\nolimits^2(X) \times \mathop{\mathrm{Amp}}\nolimits^1(X)$.

Figures (1)

  • Figure 1: Illustration of bounded stability parameter set in $\mathop{\mathrm{Amp}}\nolimits^3(X)$ joining $\alpha h_1h_2$ and $\alpha' h_1'h_2'$

Theorems & Definitions (45)

  • Definition 1.1: Semistability
  • Definition 1.2: Uniform Boundedness
  • Theorem 1.3: $\subset$ Corollary \ref{['cor:UniformBoundednessD=2']}, Uniform boundedness for pure 2-dimensional sheaves
  • Theorem 1.4: = Corollary \ref{['cor:higherDimension']}, Uniform boundedness of pure sheaves
  • Corollary 1.5: = Corollary \ref{['cor:boundednessSegments']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3: Bounded Stability Parameter
  • Remark 2.4
  • ...and 35 more