Sheaves of t-structures and valuative criteria for stable complexes

By D. Abramovich, Alexander Polishchuk. 2003

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In this work more questions arise than answers given, for which of course we do not apologize. The core of this paper is concerned with the construction of a ``constant'' t-structure on the bounded derived category of coherent sheaves $D(X\times S)$, with $X$ and $S$ smooth varieties, given a nondegenerate t-structure on $D(X)$ with noetherian heart. While we believe this construction and the methods involved should be generally useful in studying derived categories, we present here but one application: we prove a valuative criterion for separation and properness for the collection P(1) of stable objects of phase 1 under a numerical, locally finite and noetherian Bridgeland-Douglas stability condition $(Z,\cP)$ on $D(X)$, where $X$ is a smooth projective variety. As an immediate result, the number of long-exact sequences in the first named author's output is no longer an embarrassment.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7afd87eb5e2b629870d2cff91b1fd7c43e0bc012

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Citations in the corpus

MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations (2013, background): “= E0 s → E 1 s → E 2 s → ··· → E m s = Es is the Harder-Narasimhan ﬁltration ofEs. The proof is based on the notion of constant family of t-structures due to Abramovich and Polishchuk, constructed in [AP06] (in case Sis smooth) and [Pol07] (in general). Throughout the remainder of this section, we w”
The Minimal Model Program for the Hilbert Scheme of Points on P^2 and Bridgeland Stability (2012, methodology): “s is proved by Bayer-Macr`ı in [BM]. Fix a triple of numbers deﬁning a Chern character (r,c,d) ❀(r,cL,dL2) on P2 (rand care integers, and d−c2/2 is an integer). Corollary 5.13 (Abramovich-Polishchuk, [AP]). For each point (s,t) in the upper-half plane, the moduli stack MP2(r,c,d) of semi-stable obje”
Moduli stacks and invariants of semistable objects on K3 surfaces (2007, background): “In [1], there is a partial result for Problem 3.”
Derived categories of coherent sheaves (2006, unknown): “ht on this. After that we need to construct good moduli spaces of stable objects, where good might mean for example that the moduli space is a projective scheme. We refer to Abramovich and Polishchuk [1] for some recent progress in this direction. The ﬁnal step in the proof of Theorem 3.1 was very m”
Lectures on Bridgeland Stability (2016, methodology): “Once a coarse moduli space exists, then separatedness and properness of the moduli space is a general result by Abramovich and Polishchuk [AP06], which we will review in Section 8.”
Stability conditions on $K3$ surfaces (2003, background): “The central charge Z defines a stability function on A and it is easy to check that the corresponding semistable objects are precisely the nonzero objects of the subcategories P(φ) for 0 φ 6 1.”
Gluing Stability Conditions (2009, unknown): “…means that D1 and D2 are triangulated subcategories in D such that Hom(E2, E1) = 0 for every E1 ∈ D1 and E2 ∈ D2, and for every object E ∈ D there exists an exact triangle (0.1) E2 → E → E1 → E2[1] This work of the second author was partially supported by the NSF grant DMS-0601034. with E1 ∈…”
Bridgeland-stable moduli spaces for $K$-trivial surfaces (2007, background): “· · · → K → E→ Q→ K[1] → E[1] → Q[1] → · · ·”
Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface (2014, background): “H(Y, IZ(H +KY )) ∼= Ext(OY , IZ(H +KY )) ∼= Ext Db(Y )(OY [1], IZ(H +KY )) ∼= Ext1−i A(ω,β)(IZ(H),OY [1]) ∨,”