A tilted heart is the abelian category obtained by changing a bounded t-structure using a
torsion pair in its heart. For a torsion pair 
in 
, one has 
and every object has a short exact
sequence with a torsion subobject in 
and quotient in 
. The tilt is the extension closure ![<F[1],T>](/images/equations/TiltedHeart/Inline6.svg)
. Its objects have cohomology
in degrees 
,
respectively in 
and 
.
For coherent sheaves on a polarized threefold, the first slope tilt puts all strictly positive Harder-Narasimhan slopes in 
and all nonpositive slopes in 
. A second tilt uses the tilt slope on this new heart. The
strict-positive/nonpositive boundary convention matters for the resulting double-tilt
heart used in large-volume stability.
