Constant families of t-structures on derived categories of coherent sheaves

By Alexander Polishchuk. 2006

123 citations recorded by the source corpus.

We generalize the construction given in math.AG/0309435 of a "constant" t-structure on the bounded derived category of coherent sheaves $D(X\times S)$ starting with a t-structure on $D(X)$. Namely, we remove smoothness and quasiprojectivity assumptions on $X$ and $S$ and work with t-structures that are not necessarily Noetherian but are close to Noetherian in the appropriate sense. The main new tool is the construction of induced t-structures that uses unbounded derived categories of quasicoherent sheaves and relies on the results of \cite{AJS}. As an application of the "constant" t-structures techniques we prove that every bounded nondegenerate t-structure on $D(X)$ with Noetherian heart is invariant under the action of a connected group of autoequivalences of $D(X)$. Also, we show that if $X$ is smooth then the only local t-structures on $D(X)$, i.e., those for which there exist compatible t-structures on $D(U)$ for all open $U\subset X$, are the perverse t-structures considered in math.AG/0005152.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5416a038d9720460252d927e3f3d63e5a26e89fb

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Citations in the corpus

Polynomial Bridgeland stability conditions and the large volume limit (2007, background): “This is shown by the following lemma, which is a slight reformulation of a lemma in [32]:”
Inducing stability conditions (2007, methodology): “We also thank Yukinobu Toda for useful conversations and for pointing out the reference [ 44 ].”
The space of stability conditions on the local projective plane (2009, unknown): “F = Coh0(X)\P(( 1; 0]) (see e.g. [ Pol07 , Lemma 1.1.2]).”
On the structure of instability in moduli theory (2014, background): “upoid of Gm-equivariant objects on X×A1. Unwinding the descent data, this is equivalent to a graded OX[t]- module object F∈Dqc(X)♥, where the t-structure on Dqc(X) is induced from that on Db(X) as in [P2], which satisﬁes additional conditions: 1) that F has bounded cohomology sheaves which are coher”
Gluing Stability Conditions (2009, background): “We start by stating a particular case of this construction (see section 3.1 of [15] for a related construction).”
Projectivity and birational geometry of Bridgeland moduli spaces (2012, unknown): “The main advantage of our construction is that we can show its positivity property directly, without using GIT; instead, the proof is based on a categorical construction by Abramovich and Polishchuk [1,72].”