I have already known how to get the heart of a bounded t-structure on $D^b(P^n)$ by Macri`s paper,
<https://arxiv.org/abs/math/0411613>.


However I cannot purpose analogously on $D^b(P^1 \times P^2)$.


How to get the heart of a bounded t-structure on $D^b(P^1 \times P^2)$?



(Bridgeland stability conditions)How to get the heart of a bounded t-structure on $D^b(P^1 \times P^2)$?
I assume you are looking for an analogous result of [your previous question](https://mathoverflow.net/questions/382331/bridgeland-stability-conditions-how-can-i-get-the-heart-of-a-bounded-t-structu).


In this case I think you could use Corollary 2.7 of Orlov's paper ["PROJECTIVE BUNDLES, MONOIDAL TRANSFORMATIONS, AND DERIVED CATEGORIES OF COHERENT SHEAVES"](https://iopscience.iop.org/article/10.1070/IM1993v041n01ABEH002182)



> 
> If there exists a complete exceptional set in the derived category D^{b}(M) then the derived category D^{b}(E) also posseses a complete exceptional set
> 
> 
> 


Here M is a smooth projective variety, E is a rank r vector bundle over M and $p:\mathbb{P}(E)\to M$ is the associated projective bundle.


If the complete exceptional collection over $D^{b}(M)$ is $\{E\_{0},\dots,E\_{n}\}$ then the complete exceptional collection over $E$ is given by $\{p^{\ast}E\_0\otimes \mathcal{O}\_{E}(-r+1),\dots p^{\ast}E\_n\otimes \mathcal{O}\_{E}(-r+1),\dots, p^{\ast}E\_{0},\dots, p^{\ast}E\_{n}\}$.


In our particular case $M=\mathbb{P}^{2}$ and $E$ produces the trivial projective bundle $\mathbb{P}^{1}\times \mathbb{P}^{2}$ over $\mathbb{P}^{2}$, for example we can consider E to be the rank 2 vector bundle $\mathcal{O}\_{\mathbb{P}^{2}}\oplus \mathcal{O}\_{\mathbb{P}^{2}}$ over $\mathbb{P}^{2}$ (so $\mathbb{P}(E)=\mathbb{P}^{1}\times\mathbb{P}^{2}$ ), the complete exceptional collection over $\mathbb{P}^{2}$ we can take to be $\{ \mathcal{O}\_{\mathbb{P}^{2}},\mathcal{O}\_{\mathbb{P}^{2}}(1),\mathcal{O}\_{\mathbb{P}^{2}}(2) \}$ by Beilinson's result and so by Orlov's result you have a complete exceptional collection on $D^{b}(\mathbb{P}^{1}\times \mathbb{P}^{2})$ given by $\{ p^{\ast}\mathcal{O}\_{\mathbb{P}^{2}} \otimes \mathcal{O}\_{\mathbb{P}^{1}\times \mathbb{P}^{2}}(-1),p^{\ast}\mathcal{O}\_{\mathbb{P}^{2}}(1) \otimes\mathcal{O}\_{\mathbb{P}^{1}\times \mathbb{P}^{2}}(-1),p^{\ast}\mathcal{O}\_{\mathbb{P}^{2}}(2)\otimes \mathcal{O}\_{\mathbb{P}^{1}\times \mathbb{P}^{2} }(-1), p^{\ast}\mathcal{O}\_{\mathbb{P}^{2}} ,p^{\ast}\mathcal{O}\_{\mathbb{P}^{2}}(1) ,p^{\ast}\mathcal{O}\_{\mathbb{P}^{2}}(2) \}$.


Now it would be good if we could use Macri's result about Ext-exceptional collections. We can simply shift this in an adequate way.


Denote the exceptional collection above as $\{E\_{0},\dots,E\_{5}\}$, then the collection $\{ E\_{0}[5],E\_{1}[4],\dots,E\_{5}\}$ is Ext-exceptional because $Hom^{-k}(E\_{i}[-i+5],E\_{j}[-j+5])=Hom(E\_{i}[-i+5],E\_{j}[-j+5-k])=0$ since $-i+5=-j+5-k$ if and only if $i>j$ and $\{E\_{0},\dots, E\_{5}\}$ is exceptional.


So using Macri's result we have that $\langle E\_0[5],E\_{1}[4], \dots, E\_{5} \rangle$ is the heart of a bounded t-structure on $D^{b}(\mathbb{P}^{1}\times \mathbb{P}^{2})$


