Sorry for my poor English.


Let $X$ be a reducible projective variety. 


**My question is:**


1. How can I compute the dualizing sheaf of $X$ and express it in an explicit way?
2. Is there a method to get dualizing sheaf of whole reducible variety $X$ from the information of dualizing sheaves of its irreducible components?


Currently I'm not concern the general case, but I want a few accessible concrete examples such as:


1. reducible hypersurfaces,
2. union of toric varieties glued at isomorphic orbits(Alexeev calls it stable toric variety).


The reason why I concern is to understand the limit in moduli spaces of stable pairs. 



Dualizing sheaf of reducible variety?
With regards part 2. 


Let's assume that you have two components $X\_1$ and $X\_2$ (or even unions of components) such that $X\_1 \cup X\_2 = X$=. Let $I\_1$ and $I\_2$ denote the ideal sheaves of $X\_1$ and $X\_2$ in $X$.


Set $Z$ to be the *scheme* $X\_1 \cap X\_2$, in other words, the ideal sheaf of $Z$ is $I\_1 + I\_2$. 


It is easy to see you have a short exact sequence
$$0 \to I\_1 \cap I\_2 \to I\_1 \oplus I\_2 \to (I\_1 + I\_2) \to 0$$
where the third map sends $(a,b)$ to $a-b$.


The nine-lemma should imply that you have a short exact sequence


$$0 \to O\_X \to O\_{X\_1} \oplus O\_{X\_2} \to O\_Z \to 0$$


If you Hom this sequence into the dualizing complex of $X$, you get a triangle
$$\omega\_Z^. \to \omega\_{X\_1}^. \oplus \omega\_{X\_2}^. \to \omega\_{X}^. \to \omega\_Z^.[1]$$


You can then take cohomology and, depending on how things intersect (and what you understand about the intersection), possibly answer your question. 


If $X\_1$ and $X\_2$ are hypersurfaces with no common components (which should imply everything in sight is Cohen-Macualay) then these dualizing complexes are all just sheaves (with various shifts), and you just get a short exact sequence
$$0 \to \omega\_{X\_1} \oplus \omega\_{X\_2} \to \omega\_{X} \to \omega\_{Z} \to 0$$


Technically speaking, I should also probably push all these sheaves forward onto $X$ via inclusion maps.


