Reference request: Slope Stability, Bridgeland Stability

Question

Just wanted to ask for references about these topics:

• Smooth projective curves
• Coherent & Quasicoherent sheaves and their connection to bundles
• Degree of a line bundle on a curve
• Derived category of coherent sheaves
• Harder–Narasimhan stability (filtration)
• Bounded t-structures
• Bridgeland stability

I have a background in homological algebra, sheaf theory & cohomology, derived categories and basic algebraic geometry (very little about schemes).

I've studied from Gelfand & Manin's book and Kashiwara & Schapira's book/notes on sheaves. Right now I'm trying to get through Ravi Vakil FOAG notes and Hartshorne's book but they're very dense, even if I've already worked out some of the exercises about sheaves.

I wonder if there is some reference that encompasses these topics in a reasonable way. The aim is to have a good understanding of what Bridgeland stability is, and why do we need to work in the derived category of coherent sheaves.

Answer 1

• The first three topics are covered well in most introductory texts on algebraic geometry, for instance Hartshorne will do for those.
• For derived categories and their relationship with algebraic geometry, I recommend Huybrechts' Fourier Mukai Transforms in Algebraic Geometry. The first chapters, perhaps up through the Bondal-Orlov reconstruction theorem, will give you a good taste of the relevance of $\mathbf{D}^{\mathrm{b}}(X)$ in algebraic geometry.
• For HN filtrations you can see the first chapter of Huybrechts-Lehn's The Geometry of Moduli Spaces of Sheaves.
• For bounded t-structures, I think you can read this in Gelfand-Manin.
• For Bridgeland stability, there is a good survey article of Huybrechts here. You should also read Bridgeland's original papers on the subject: 1 and 2.