[论文解读] A Remark on Stability Conditions on Smooth Projective Varieties

Chunyi Li|arXiv (Cornell University)|Jan 30, 2026

Algebraic Geometry and Number Theory被引用 0

一句话总结

The paper proves that Db(X) admits Bridgeland stability conditions for any smooth projective X over C and outlines a four-step induction from elliptic curves to general varieties.

ABSTRACT

Let $X$ be a smooth projective variety over $\mathbb C$. In this paper, we prove that $\mathrm{D}^b(X)$, the bounded derived category of coherent sheaves on $X$, always admits stability conditions in the sense of Bridgeland.

研究动机与目标

• Prove existence of stability conditions on Db(X) for all smooth projective X
• Develop a transfer mechanism of stability from elliptic curves to projective spaces and beyond
• Introduce and utilize Bayer and Restriction properties to control inductions
• Leverage double Schubert polynomials and Soergel filtrations to manage structural filtrations

提出的方法

• Review Bridgeland stability definitions and deformation results
• Employ f♯/f♭ pushforward/pullback under finite morphisms to transport stability
• Construct and analyze sigma^{a,b} on E^n and their invariance properties
• Induce stability on P^n via quotient maps and embeddings
• Utilize double Schubert polynomials and Gr_w filtrations to structure filtrations
• Apply Stepwise reductions from E^n to (P^1)^n to P^n and then to X

实验结果

研究问题

• RQ1 Does Db(X) admit a Bridgeland stability condition for every smooth projective X over C?
• RQ2 How can stability conditions on products of elliptic curves be transferred to projective spaces and then to general smooth projective varieties?
• RQ3 What is the role of Bayer and Restriction properties in preserving stability under morphisms and embeddings?
• RQ4 Can known stability conditions on E^n be extended to X via group actions and filtrations?
• RQ5 How do Schubert polynomials and Soergel filtrations facilitate the stability construction?

主要发现

• There exists a stability condition on Db(X) for any smooth projective variety X over C.
• Stability conditions on products of elliptic curves E^n can be constructed and are invariant under certain group actions.
• Pushforward/pullback along finite morphisms can transfer stability conditions while preserving the support property.
• A Bayer property and a Restriction-N property framework help control stability under inductions and embeddings.
• A four-step strategy from E^n to P^n and then to X yields stability conditions on smooth projective varieties.