Let $X$ be a smooth projective $3$-fold over $\mathbb{C}$. Let $K_0(X)$ be its Grothendieck group, consider the Euler form defined as: $\chi(M,N): K_0(X)\times K_0(X)\rightarrow\mathbb{Z}$ by $(M,N)\mapsto\sum_{i=0}^3(-1)^idim_\mathbb{C}Ext_{O_X}^i(M,N)$
Then define the so called numerically Grothendieck group $K^{num}(X)$ as $K_0(X)/ker\chi(M,-)$. Then restrict $\chi$ to $K_0^{num}(X)$. Now I would like to compute the matrix form of $\chi$ on $K^{num}_0(X)$ with appropriate basis of $K_0^{num}(X)$ and study some properties of $\chi$ (say rank, signature etc). But I have the difficulties on choosing basis. If $X$ is a smooth projective surface, I can choose $[k_p],[O_{C_i}],[O_X]$ where $k_p$ is skyscraper sheaf on $X$, $C_i\in Pic(X)$. But I don't know the appropriate basis of $K_0^{num}(X)$ for three-fold.
In the work of A.Kuznetsov in his paper: Derived category of Fano three-fold: https://arxiv.org/pdf/0809.0225.pdf He showed that for a smooth projective variety $X$ with $dimX\leq 3$, one can choose the basis of chow groups: $A^p(X)$(where $p$ is the codimension of cycles on $X$), say denoted by $Z_p^1,Z_p^2,\ldots,Z_p^m$ such that $0\leq p\leq dim X$ to form basis of $K_0^{num}(X)$ as structure sheaves on those cycles: $[O_{Z_p}^i]$ with $1\leq i\leq m, 0\leq p\leq dim X$. Then he gave an example of Fano $3$-fold of picard rank $1$, the basis of $K_0^{num}(X)$ is given by $\{[O_X],[O_S],[O_H],[k_P]\}$, where $S\in Pic(X)$ is the generator. $H$ is a line on $X$.
My question is what should be the appropriate basis of general smooth projective $3$-fold ? For example, if $Pic(X)$ has two generators $S_1,S_2$, is the basis of $K_0^{num}(X)$ formed by $\{[O_X],[O_{S_1}],[O_{S_2}],[O_{H_1}],[O_{H_2}],[O_{H_{1,2}}],[k(P)]\}$, where $H_1=S_1^2, H_2=S_2^2, H_{1,2}=S_1\cdot S_2$?
My second question is what about if $dim(X)=n>3$, is there any appropriate basis to choose?
Of course, If $X$ admits the full exceptional collection in derived category of $X$, I can just choose them, but in general, we dont have such good collections.
