On the moduli space of simple sheaves on singular K3 surfaces (2403.00735v2)

Abstract: Mukai proved that the moduli space of simple sheaves on a smooth projective K3 surface is symplectic, and in \cite{FM2} we gave two constructions allowing one to construct new locally closed Lagrangian/isotropic subspaces of the moduli from old ones. In this paper, we extend both Mukai's result and our construction to reduced projective K3 surfaces; for the former we need to restrict our attention to perfect sheaves. There are two key points where we cannot get a straightforward generalization. In each, we need to prove that a certain differential form on the moduli space of simple, perfect sheaves vanishes, and we introduce a smoothability condition to complete the proof.

• The paper establishes sufficient conditions for the smooth deformation of polarized singular K3 surfaces using Ext¹ surjectivity and H² vanishing.
• It extends Tziolas’s formal smoothability criteria to polarized surfaces with ADE singularities, integrating Burns and Wahl’s framework.
• Explicit quartic K3 surface examples underscore the geometric implications and validate the established deformation criteria.

Smoothability of Singular K3 Surfaces with Polarizations

Introduction

This paper predominantly addresses the smoothability of singular K3 surfaces adorned with polarizations, focusing on a subset with isolated locally complete intersection (lci) singularities. Singular K3 surfaces occupy a pivotal position in algebraic geometry, not only for their structural beauty but also for their connections to various mathematical and physical theories. The notion of smoothability, which pertains to the potential to deform a singular variety into a smooth one through algebraic or analytic perturbations, is crucial for understanding the moduli space of such surfaces. As an extension, introducing polarization, an ample line bundle, into the scenario significantly narrows down the considered deformations, hence offering a more intricate view of the subject.

Theoretical Framework and Key Results

The central contribution of this paper involves providing sufficient conditions for the existence of a deformation of a polarized singular K3 surface into a smooth polarized K3 surface. In essence, the paper extends Tziolas's condition for formal smoothability, inherently linked to Burns and Wahl's criteria for singularity smoothing, to polarized K3 surfaces with ADE singularities.

Notably, the paper establishes a vital criterion for the polarized smoothability of singular K3 surfaces: the surjectivity of the specific morphism from the Ext1 functor of the Atiyah sequence associated with the ample line bundle to the global sections of the tangent sheaf's first Ext group. This condition, coupled with the vanishing of the second cohomology of the tangent sheaf, affirmatively answers the paper's leading questions on smoothability.

Furthermore, the examination explores the geometric meanings and implications of various morphisms and groups in the scenario of deformations. For instance, it brings to light how certain Ext groups and cohomology groups act as obstruction and tangent spaces to deformations of different kinds, extending from the surface itself to its embedded pairs.

Another remarkable aspect of this work is its tangible application to K3 surfaces as quartics in projective space. By illustrating explicit examples, the paper demonstrates the practical vitality and limitations of the established criteria. The examples meticulously chosen not only underline the conditions' necessity and sufficiency for smoothability but also shed light on their intrinsic geometric significance.

Implications and Future Directions

The results presented in this paper have broad-reaching implications for the paper of K3 surfaces, algebraic geometry, and potentially string theory and mathematical physics where such surfaces frequently emerge. Beyond presenting concrete conditions for smoothability, the paper propels the discourse on the intricate relationship between singularities and their deformations in the context of algebraic surfaces.

Future research avenues could explore further refinements of smoothability criteria, perhaps by incorporating additional geometric or topological structures. Moreover, the exploration of analogous conditions for higher-dimensional varieties or for K3 surfaces with more complex singularities could provide exciting new insights. The interplay between smoothability and derived categories, automorphisms, and Kähler geometries remains another fertile ground for investigation, potentially unveiling deeper aspects of K3 surfaces and their moduli.

Conclusion

This paper significantly advances the understanding of polarized singular K3 surfaces by characterizing their smoothability through algebraic deformations. By elucidating concrete conditions for such deformations, the paper enhances the toolbox available for algebraic geometers exploring K3 surfaces and contributes to the broader comprehension of complex algebraic surfaces' structure and classification.