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Let $X$ be a compact Kahler space(Kahler variety), then it is known that the Albanese morphism $alb_X:X\to Alb(X)$ is a well-defined morphism when $X$ has at worst rational singularities.

It there some examples such that $X$ has worse than rational singularities and Albanese morphism is not well defined.

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    $\begingroup$ Proposition A.6 of kurims.kyoto-u.ac.jp/~motizuki/… on p. 72 says that any proper normal integral scheme $X$ over a perfect field $k$ with a $k$-rational point admits an Albanese morphism $X\to Alb(X)$. If you allow non-normal singularities, then you can probably find (very singular) curves which do not have an Albanese variety. $\endgroup$
    – Ariyan Javanpeykar
    Commented Dec 31, 2017 at 11:01
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    $\begingroup$ What @AriyanJavanpeykar writes is true. However, my guess is that the OP is not asking about the Albanese morphism, but rather the "birational Albanese morphism", i.e., for a specified smooth $\mathbb{C}$-point $x_0$, an initial object in the category of rational transformations $f:X\dashrightarrow A$ to Abelian varieties that are regular at $x_0$ and that map $x_0$ to the origin. In this case there are many birational Albanese morphisms that are not regular, e.g., when $X$ is a projective cone over a smooth plane cubic. $\endgroup$
    – Jason Starr
    Commented Dec 31, 2017 at 15:32
  • $\begingroup$ @AriyanJavanpeykar, thank you for the interesting comments. However you didn't mention your example $\endgroup$
    – 1984
    Commented Dec 31, 2017 at 22:28
  • $\begingroup$ @JasonStarr, Thank you, interesting comment. I am interested in Albanese morphism $\endgroup$
    – 1984
    Commented Dec 31, 2017 at 22:30
  • $\begingroup$ @JasonStarr , I think you was right, these two notion are the same in characteristic zero for normal projective varieties with only rational singularities , in fact, Serre introduced it for quas-Albanese morphism , but I don't have example to have good picture on both of them. See Lemma 8.1 of Y. Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. Reine Angew. Math. 363 (1985), 1–46. 2 $\endgroup$
    – 1984
    Commented Jan 1, 2018 at 8:24

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