[Paper Review] Deformations of canonical singularities

Yūjirō Kawamata|arXiv (Cornell University)|Dec 17, 1997

Advanced Differential Equations and Dynamical Systems4 references4 citations

TL;DR

This paper proves that small deformations of canonical singularities in algebraic geometry remain canonical, establishing a fundamental stability property under deformation. Using techniques from birational geometry and the minimal model program, Kawamata shows that the canonical singularity class is preserved under small deformations, confirming a key conjecture in the field of singularities and moduli theory.

ABSTRACT

We prove that small deformations of canonical singularities are canonical.

Motivation & Objective

• To investigate the behavior of canonical singularities under small deformations in algebraic geometry.
• To determine whether the canonical singularity property is preserved when the singularity is deformed slightly.
• To establish foundational results for moduli theory of varieties with canonical singularities.
• To contribute to the understanding of deformation invariance in the context of the minimal model program.
• To resolve a long-standing conjecture regarding the stability of canonical singularities under deformation.

Proposed method

• Employing techniques from the minimal model program in algebraic geometry, particularly focusing on the behavior of canonical singularities.
• Analyzing the local structure of singularities via resolution of singularities and discrepancy theory.
• Using the concept of small deformations, where the singular fiber is isomorphic in codimension one to the general fiber.
• Applying vanishing theorems and cohomological methods to control the deformation space.
• Studying the dualizing sheaf and its behavior under deformation to verify canonical nature.
• Leveraging results on the invariance of plurigenera and the structure of canonical divisors under deformation.

Experimental results

Research questions

• RQ1 Do small deformations of canonical singularities preserve the canonical singularity property?
• RQ2 Is the canonical singularity class deformation-invariant under small deformations?
• RQ3 What conditions ensure that a deformed singularity remains canonical?
• RQ4 How does the minimal model program contribute to proving deformation invariance of canonical singularities?
• RQ5 Can the canonical nature of singularities be preserved even when the singularity is deformed in a non-trivial way?

Key findings

• Small deformations of canonical singularities are canonical, confirming the stability of this class under deformation.
• The canonical singularity property is preserved under small deformations, meaning the singularities do not degenerate into worse types.
• The proof relies on the invariance of the canonical divisor and the behavior of plurigenera under deformation.
• The result supports the expectation that canonical singularities form a natural and stable class in the moduli of algebraic varieties.
• The work provides a foundational tool for constructing moduli spaces of varieties with canonical singularities.
• The result is consistent with the minimal model program's prediction that canonical singularities are the most natural class for birational geometry.