Suppose $X$ is a normal projective variety over $\mathbb C$. In the case $X$ is smooth according to Hodge theory $h^1(X,O(X))$ is the dimension of the space of holomorphic $1$-forms on $X$ and this number is equal as well to the half of the first Betti number $b_1(X)/2$ .

I would like to know what happen in the case when $X$ is singular and normal.

1) Is there some relation (equality or inequality) between $h^1(X,O(X))$ and $b_1(X)$? For example does $b_1(X)=0$ imply $h^1(X,O(X))=0$?

2) Suppose that $h^1(X,O(X))=n$ is it true that there is a canonical $n$-dimensional space of $1$-forms on $X$, holomorphic outside of its singularities? (if yes, can something be said about their behaviour at singularities?)

Is there some pedagogical reference treating these questions?


(Although I have pretty much "retired" from Mathoverflow, I will answer this, since the answer is nice but probably not all that well known.)

Theorem. If $X$ is complex normal projective variety, then it is still true that $b_1(X)=2h^1(\mathcal{O}_X)$.

Proof. Let $\pi:\tilde X\to X$ be a desingularization. Since $X$ is normal, the fibres of $\pi$ are connected. Therefore $\pi_*\mathbb{Z}$ (with analytic topology) is connected. It follows that $H^1(X,\mathbb{Z})\to H^1(\tilde X, \mathbb{Z})$ is injective since it can be identified with the edge map for Leray. Therefore the, a priori mixed, Hodge structure on $H^1(X)$ is pure of type $\lbrace (1,0),(0,1)\rbrace$. Consequently $$b_1(X)=2\dim [H^1(X)]^{(0,1)}=2 \dim im[H^1(X,\mathcal{O}_X)\to H^1(\tilde X, \mathcal{O}_{\tilde X}] = 2h^1(\mathcal{O}_X)$$

Regarding your question 2, you can construct an Albanese map $X\to Alb(X)$ to the torus associated to the dual Hodge structure on $H_1(X)$. The space of $1$-forms on $Alb(X)$ will pullback to a space of the required dimension on $X$.

To address your comments: a general reference for mixed Hodge structures is the book by Peters and Steenbrink (although this may be bit a heavy). And yes, the argument does work in the Moishezon case, and a bit more generally. I guess that I may as well admit the above argument was extracted from paper in Duke from 1990; this contains some more details and elaborations as well.

  • $\begingroup$ Dear Donu many thanks for your answer (great that you have not retired from here completely :) )! In fact in the situation I am dealing with $X$ is Moishezon (and not necessarily projective). Do I understand correctly that your argument still works in such a case? $\endgroup$ – aglearner Apr 22 '13 at 17:56
  • $\begingroup$ Also, I would like to ask you if you can propose some reference where I could read about mixed Hodge structure (to get a more complete understanding of your answer). $\endgroup$ – aglearner Apr 22 '13 at 18:00
  • $\begingroup$ Dear @Donu, I am sad to learn that you have (almost) retired from this site. Why is that and would you consider changing your mind ? $\endgroup$ – Georges Elencwajg Apr 28 '13 at 22:03
  • $\begingroup$ Georges: this is a somewhat late response. I'd rather not elaborate here, but lack of time is certainly a big factor. $\endgroup$ – Donu Arapura May 2 '13 at 12:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.