For the standard t-structure on $D(\mathcal{A})$, the heart is $D^{\leq 0} \cap D^{\geq 0} = \{A^\bullet : H^n(A) = 0 \text{ for } n \neq 0\} \cong \mathcal{A}$. The identification sends a complex to its $H^0$.

The heart of the perverse t-structure on $D^b_c(X)$ is the abelian category $\mathrm{Perv}(X)$ of perverse sheaves. Perverse sheaves are neither sheaves nor perverse; they are complexes of sheaves satisfying support and cosupport conditions with respect to a stratification.

For a torsion pair $(\mathcal{T}, \mathcal{F})$ in $\mathcal{A}$, the tilted heart in $D^b(\mathcal{A})$ is the subcategory of complexes $A^\bullet$ with $H^0(A) \in \mathcal{F}$, $H^{-1}(A) \in \mathcal{T}$, and $H^n(A) = 0$ for $n \neq 0, -1$. This new abelian category is called the tilt of $\mathcal{A}$ at $(\mathcal{T}, \mathcal{F})$.

For a weight structure on $\mathcal{T}$, the heart consists of objects that are both "non-negative" and "non-positive" in the weight sense. Unlike t-structure hearts, weight structure hearts are not necessarily abelian but are additive and have the property that every object of $\mathcal{T}$ has a "weight filtration."

The BGG category $\mathcal{O}$ for a semisimple Lie algebra can be realized as the heart of a certain t-structure on $D^b(\mathcal{O})$ (the standard one). Non-standard t-structures give "exotic" versions of category $\mathcal{O}$.

On $D^b(\mathrm{Coh}(\mathbb{P}^1))$, the standard heart is $\mathrm{Coh}(\mathbb{P}^1)$. Tilting at the torsion pair (torsion sheaves, vector bundles) gives a new heart equivalent to $\mathrm{mod}\text{-}k[\bullet \rightrightarrows \bullet]$ (representations of the Kronecker quiver).

A short exact sequence $0 \to A \to B \to C \to 0$ in the heart $\mathcal{A}$ corresponds to a distinguished triangle $A \to B \to C \to A[1]$ in $\mathcal{T}$ with all three vertices in $\mathcal{A}$. The connecting map $C \to A[1]$ lies in $\mathrm{Ext}^1_{\mathcal{A}}(C, A)$.

The $n$-th cohomology with respect to a t-structure is $H^n = H^0 \circ [-n] : \mathcal{T} \to \mathcal{A}$. This is a cohomological functor: it sends distinguished triangles to long exact sequences in $\mathcal{A}$.

$D^b(\mathrm{Coh}(\mathbb{P}^2))$ admits infinitely many t-structures with different hearts. The space of Bridgeland stability conditions parametrizes these t-structures (and their central charges).

The heart $\mathcal{A}$ determines the t-structure: $\mathcal{T}^{\leq 0} = \{X : H^n(X) = 0 \text{ for } n > 0\}$ and $\mathcal{T}^{\geq 0} = \{X : H^n(X) = 0 \text{ for } n < 0\}$. This follows from the axioms.

For a bounded t-structure with heart $\mathcal{A}$, the realization functor $\mathrm{real} : D^b(\mathcal{A}) \to \mathcal{T}$ extends the inclusion $\mathcal{A} \hookrightarrow \mathcal{T}$. It is always exact but not always an equivalence.

On a smooth variety $X$, the category of regular holonomic D-modules is the heart of a natural t-structure on $D^b_{\mathrm{rh}}(D_X)$. The Riemann-Hilbert correspondence identifies this heart with perverse sheaves.