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Are there any good tricks to construct a heart of a t-structure? (I'm thinking on the derived category of coherent sheaves of some variety)

I'll start with the only one I know. If $(T,F)$ is a torsion pair on an abelian category $A$ then you can form the tilt inside $D(A)$.

By definition, the tilt is complexes in $D(A)$ such that the minus one cohomology lies in $T$ and the zeroth cohomology lies in $F$ and all other cohomology vanishes. Unfortunately this method is only good for hearts concentrated in two degrees.

An example of such a torsion pair is if you take $Ab$ the category of abelian groups, $T$ torsion groups and $F$ free groups. In any abelian group you can find the largest torsion subgroup and the quotient will be torsion free. This can be generalised to arbitrary integral domains and from that to arbitrary integral schemes, although the geometric meaning of the tilt (if there is one) escapes me.

Another example is Bridgeland's category of perverse coherent sheaves, which can be defined as a tilt. I would be very interested in generalisations of Bridgeland's category. There is also a notion of perverse coherent sheaf by Bezrukavnikov, but from what I understand it is only useful with Artin stacks, as the jump in the dimension of the coarse space given by stabiliser groups is what allows interesting perversities. Comments on these latter perverse sheaves would also be very much appreciated, as I don't really understand the construction.

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up vote 11 down vote accepted

This will be a short overview on techniques I am familiar with. For simplicity, I will talk about bounded t-structures, which are determined by their heart $\mathcal{A} = D^{\le 0} \cap D^{\ge 0}$; and on the bounded derived category of coherent sheaves $D^b(X)$ on a variety/stack.

  1. Tilting is in principle extremely powerful: $\mathcal{A}_1$ is obtained by tilting from $\mathcal{A}_2$ whenever objects of $\mathcal{A}_1$ have only two cohomologies with respect to $\mathcal{A}_2$, e.g. $\mathcal{A}_1 \subset \langle \mathcal{A}_2, \mathcal{A}_2[1]\rangle$. (See e.g. Lemma 1.1.2 in http://front.math.ucdavis.edu/0606.5013)

  2. Tilting can be iterated. As an example, using 1. it is a not too difficult exercise to see that Bezrukavnikov's t-structures of perverse coherent sheaves can be constructed by iterated tilting. When $D^b(X)$ is equivalent to a derived category of quiver representations, iterated tilting can often be descibed by iterated quiver mutations.

  3. It is not very difficult to construct torsion pairs. For example, when $\mathcal{A}$ is Noetherian, then any subcategory $\mathcal{T} \subset \mathcal{A}$ that is closed under extensions and quotients is the torsion part of a torsion pair $(\mathcal{T}, \mathcal{F} = \mathcal{T}^{\perp})$. Alternatively, any notion of stability condition on a heart $\mathcal{A}$, say, induced via a slope function: $\mathcal{T}_{> \mu_0}$ is the extension-closed subcategory generated by stable objects $E$ with $\mu(E) > \mu_0$. (See http://front.math.ucdavis.edu/0307.5164, section 6.)

  4. Given a semi-orthogonal decomposition of a triangulated category, one can construct a t-structure on the full category by gluing t-structures on the components - this is all in the original BBD.

  5. It is much easier to construct (unbounded) t-structures in the unbounded derived category $D_{qc}(X)$ of quasi-coherent sheaves. (Any subcategory closed under [1], extensions and small coproducts is the $D^{\le 0}$-part of a t-structure.) Sometimes one can use this to construct bounded t-structures on $D^b(X)$ by showing that they restrict - see e.g. section 2 of http://front.math.ucdavis.edu/0606.5013. But in general t-structures on $D_{qc}(X)$ do not descend, and even if they do, it might be hard to prove.

  6. As an example of the latter techniques, when $G$ acts freely on $X$, then $G$-invariant t-structures on $X$ are in 1:1-correspondence with t-structures on the quotient $X/G$ satisfying an additional assumption: tensoring with $f_* \mathcal{O}_X$ is right-exact.

  7. Any derived equivalence $D^b(X) \cong D^b(\mathcal{A})$ induces a t-structure on $D^b(X)$ by pull-back - I guess you already knew that!

I realize that I talked mostly about tilting - I do think it's a very powerful method.

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7. yes, I already knew that ;) but the rest is incredibly informative: beautiful! –  Jacob Bell Aug 2 '12 at 22:57
    
about iterated tilting: can you confirm if I understand? The first heart is given by those complexes with $H^0 \in T$, $H^{-1} \in F$. If I then have another torsion pair $(T',F')$ on the tilt, the second heart is given by those complexes with $H^0_{\mathcal{H}} \in T'$, $H^{-1}_{\mathcal{H}} \in F'$. Where the clumsy $H^\cdot_{\mathcal{H}}$ stands for the "non-standard" cohomology with respect to the first heart. –  Jacob Bell Aug 2 '12 at 23:00
    
Correct. In other words: to tilt, you only need to assume that $\mathcal{A}\subset D$ is the heart of a bounded t-structure, you don't need $D \cong D^b(\mathcal{A})$. –  Arend Bayer Aug 2 '12 at 23:09
    
yes, that's the right way to put it. thanks again, I'll wait a little bit just in case someone else wants to chip in before accepting your answer. –  Jacob Bell Aug 2 '12 at 23:14
    
There is a problem with iterating tilting --- after the very first tilting the heart may be (and frequently is) non-Noetherian! So, the trick as in 3. does not work. –  Sasha Aug 3 '12 at 7:58
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