Abstract

In Bayer and Macrì (J Am Math Soc 27(3):707–752, 2014), the first author and Macrì constructed a family of nef divisors on any moduli space of Bridgeland-stable objects on a smooth projective variety X. In this article, we extend this construction to the setting of any separated scheme Y of finite type over a field, where we consider moduli spaces of Bridgeland-stable objects on Y with compact support. We also show that the nef divisor is compatible with the polarising ample line bundle coming from the GIT construction of the moduli space in the special case when Y admits a tilting bundle and the stability condition arises from a theta -stability condition for the endomorphism algebra. Our main tool generalises the work of Abramovich–Polishchuk (J Reine Angew Math 590:89–130, 2006) and Polishchuk (Mosc Math J 7(1):109–134, 2007): given a t-structure on the derived category D_c(Y) on Y of objects with compact support and a base scheme S, we construct a constant family of t-structures on a category of objects on Y times S with compact support relative to S.

Highlights

  • This group has finite rank under very mild assumptions

  • Given separated schemes S and Y of finite type, we define what it means for an object of D(Y × S) to have left-compact support, see Definition 2.1.3

  • Given a t-structure on the category Dc(Y ) of objects with compact support, we construct a constant family of t-structures in the derived category of objects on Y × S with left-compact support (Theorem 4.3.1) and show that it satisfies the open heart property (Proposition 4.4.3)

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Summary

Families of t-structures for compact support

The proof of Theorem 1.2.1 relies on extending the work of Abramovich–Polishchuk [6] and Polishchuk [52] to the setting of objects with compact support. Given separated schemes S and Y of finite type, we define what it means for an object of D(Y × S) to have left-compact support, see Definition 2.1.3. This rather ad-hoc definition is more restrictive than requiring an object to have proper support over. Given a t-structure on the category Dc(Y ) of objects with compact support, we construct a constant family of t-structures in the derived category of objects on Y × S with left-compact support (Theorem 4.3.1) and show that it satisfies the open heart property (Proposition 4.4.3). The positivity statements from Theorem 1.2.1 follow as in [13], where a key step invokes the open heart property for the newly constructed t-structure for objects on Y × C with left-compact support

Comparison with θ -stability
Additional background and outlook
Running assumptions and notation
Derived category with left-compact support
Compact and left-compact support
Localisation
Objects with proper support over the base
Integral functors
On t-structures
Sheaves of t-structures over the base
On resolution of the diagonal
A family of t-structures
On graded S-modules in an abelian category
The Noetherian property
Sheaf of t-structures over an arbitrary base
Extending t-structures to the quasi-coherent setting
Sheaf of t-structures over an affine base
Construction over an arbitrary base
The open heart property
Numerical Bridgeland stability conditions for compact support
Numerical Grothendieck groups
Stability conditions for compact support
The linearisation map
Positivity
A geometric condition to ensure proper support
On schemes admitting a tilting bundle
Stability conditions for quiver algebras
On schemes with a tilting bundle
Comparison of flat families
Comparison of line bundles
Full Text
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