[Paper Review] A Propos de l'Existence de Fibrés Stables sur les Surfaces

Andrè Hirschowitz, Yves Laszlo|ArXiv.org|Oct 29, 1993

Point processes and geometric inequalities3 citations

TL;DR

This paper establishes the existence of stable vector bundles of rank $ r \geq 2 $ on any smooth, projective, connected surface over an algebraically closed field of characteristic zero, given arbitrary first Chern class and sufficiently large second Chern class. Using deformation-theoretic techniques and moduli space analysis, the authors prove that such stable bundles exist under these conditions, resolving a foundational question in algebraic geometry of surface vector bundles.

ABSTRACT

We prove the existence (in characteristic 0) on every polarized (smooth, projective and connected) surface of stable bundles of rank $r\geq 2$, arbitrary first Chern class and large enough $c_2$.

Motivation & Objective

• To investigate the existence of stable vector bundles on algebraic surfaces, a central problem in moduli theory of vector bundles.
• To determine conditions under which stable bundles of rank $ r \geq 2 $ exist on smooth, projective surfaces.
• To extend known results on stable bundles from curves to surfaces, particularly in the context of polarized surfaces.
• To establish that for any given first Chern class and sufficiently large second Chern class, stable bundles exist over any such surface.
• To provide a foundational result in the moduli theory of vector bundles on surfaces by proving existence under natural geometric constraints.

Proposed method

• Employing deformation theory to analyze the local structure of the moduli space of semistable bundles on a surface.
• Using the theory of slope stability and Harder-Narasimhan filtrations to construct stable bundles from semistable ones.
• Analyzing the dimension of the space of extensions of stable bundles to control the existence of stable bundles with prescribed Chern classes.
• Applying cohomological criteria to ensure the existence of non-trivial extensions that yield stable bundles.
• Using the fact that for large enough $ c_2 $, the moduli space of semistable bundles is non-empty and has components of expected dimension.
• Working over algebraically closed fields of characteristic zero to ensure the validity of standard cohomological tools and deformation techniques.

Experimental results

Research questions

• RQ1 Under what conditions do stable vector bundles of rank $ r \geq 2 $ exist on a smooth, projective surface?
• RQ2 Can stable bundles be constructed with arbitrary first Chern class and sufficiently large second Chern class on any polarized surface?
• RQ3 Is the moduli space of semistable bundles on a surface non-empty when $ c_2 $ is large enough to allow stable deformations?
• RQ4 Does the existence of stable bundles on surfaces follow from general principles of deformation theory and cohomological vanishing in characteristic zero?
• RQ5 Can one guarantee the existence of stable bundles without restricting the surface or the Chern classes, provided $ c_2 $ is large?

Key findings

• Stable vector bundles of rank $ r \geq 2 $ exist on every smooth, projective, connected surface over a field of characteristic zero.
• For any given first Chern class $ c_1 $, and for sufficiently large $ c_2 $, stable bundles exist regardless of the specific surface or polarization.
• The moduli space of semistable bundles is non-empty and has components of expected dimension when $ c_2 $ is large.
• The construction relies on deformation-theoretic techniques and cohomological criteria to ensure the existence of stable extensions.
• The result holds in characteristic zero, where standard tools of algebraic geometry (e.g., Serre duality, Riemann-Roch) apply without obstruction.
• The paper establishes a foundational existence result that enables further study of moduli spaces of stable bundles on surfaces.