Let $X$ be an elliptic curve and $D(X)$ the bounded derived category of $Coh(X)$, coherent sheaves on $X$. If $(D^{\leq 0}, D^{>0})$ is a bounded $t$-structure, then can we already say that the heart $D^{\leq 0}\cap D^{>0}[1]$ is equivalent to $Coh(X)$ (by a shift) ?
$\begingroup$
$\endgroup$
3
-
$\begingroup$ I'm not an expert in this area, but if you're interested in questions about what one can say about general bounded $t$-structures on familiar categories, one example of work I happen to be aware of is Antieau, Gepner, and Heller's K-theoretic obstructions to bounded $t$-structures. From the title, they might not address your actual question, but perhaps some consdierations they make might be relevant. $\endgroup$– Tim CampionCommented Feb 10, 2021 at 15:37
-
$\begingroup$ There's also a recognition principle due to Lurie -- see Prop 1.3.3.7 of HA which might only be directly applicable in the affine case. This was used to great effect for example by Gheorghe, Wang, and Xu in motivic homotopy theory, with applications to recent breakthroughs in computing homotopy groups of spheres. $\endgroup$– Tim CampionCommented Feb 10, 2021 at 15:41
-
$\begingroup$ Unless I'm being stupid, perverse sheaves give an example of a non-standard $t$-structure with the same heart. So I guess what I'm driving at is that there is a kind of principle that $t$-structures are "sometimes" determined by their hearts (contrary to your hope that lots of $t$-structures have the same heart) but that this principle is not at all universally true, so I find your question interesting. $\endgroup$– Tim CampionCommented Feb 10, 2021 at 15:49
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
4
One can construct t-structures on the bounded derived category of coherent sheaves on a smooth projective curve (or higher-dimensional variety) by tilting, see Bayer's notes, Prop. 3.6.1, and the corresponding hearts are not shifts of each other, because the tilt shifts some objects but not all of them.
Another way to obtain t-structures is to apply autoequivalences of the derived category to the standard heart, see Exercise 11 in Bayer's notes for the elliptic curve case.
Generally t-structures on bounded derived categories of coherent sheaves are studied in the context of the Bridegland stability conditions, and stability conditions for elliptic curves have been classified in Section 9 of Bridgeland's original paper.
-
1$\begingroup$ Oh, in the original question is was asked whether the equivalence between hearts is given by a shift. However, it is certainly interesting to know whether all these hearts are equivalent "abstractly". Do you know this? $\endgroup$ Commented Feb 12, 2021 at 12:48
-
$\begingroup$ I think it's a much more subtle and not so well studied question whether various hearts are equivalent as abelian categories. This is definitely false for $\mathbf{P}^1$ (standard heart has infinitely many simples, but the Kronecker quiver has two), but I don't know about elliptic curve and the tilted hearts. $\endgroup$ Commented Feb 12, 2021 at 15:13
-
$\begingroup$ This paper is exactly how i came up with the question. More precisely when i tried to understand the proof of the fact (in the Proof of Theorem 9.1.) that indecomposable implies semistable he uses things (serre duality, $Ext^1_{Coh(X)}(B,A) = Hom_{D(X)}^1(B,A))$ that i only know when the heart is already $Coh(X)$ (by a shift). $\endgroup$ Commented Feb 12, 2021 at 17:22
-
1$\begingroup$ @user12344321: Serre duality is a statement about objects in derived categories, it's not for a specific heart, see e.g. arxiv.org/pdf/1509.09115.pdf, 1.6. $\endgroup$ Commented Feb 12, 2021 at 20:10