Let $X$ be a smooth complex surface of general type and $m>0$ an integer and $\eta\in Pic^0(X)$. If $X'$ is a smooth surface birational to $X$, then it is easy to show that $h^0(mK_X+\eta)=h^0(mK_{X'}+\eta)$ (I'm using the fact that $Pic^0(X)$ and $Pic^0(X')$ are canonically isomorphic). If one takes $X'$ to be the minimal model of $X$, then for $m>1$ by Kawamata-Viehweg vanishing $h^0(mK_X+\eta)=\chi(mK_X)$ is independent of $\eta\in Pic^0(X)$.

This argument does not work if $\dim X>2$ because minimal models in general are not smooth. My question is whether it is nevertheless true that $h^0(mK_X+\eta)$ is independent of $\eta$ for $m>1$.

I'm interested mainly in the case when the Albanese map of $X$ is generically finite.

ADDED: I would like to add some motivation. If $X$ is a variety with generically finite Albanese map and such that $\chi(K_X)>0$ the {\em paracanonical system} of $X$ (i.e. the family of effective divisors algebraically equivalent to $K_X$) contains a {\em main component}, that dominates a component of $Pic(X)$. This main component is a classical object of study.

As explained above, in the case of surfaces the analogous construction for $mK_X$, $m>1$ is ``boring'', since it gives a $\mathbb P^k$-bundle over acomponent of $Pic(X)$. So I am wondering whether for $\dim X>2$ things are similar, or, conversely, there is some interesting geometry also for $m>1$.


1 Answer 1


Actually, I think the above argument always works because the minimal model has mild singularities (or one can reason directly on $X$; see ref below).

Let $X$ be a smooth variety of general type and $X'=Proj R(K_X)$ its canonical model (which exists by BCHM). Since $X'$ has rational singularities we have a natural identification $Pic ^0(X)\cong Pic ^0(X')$. For any $m\geq 1$ it is easy to see that $H^0(mK_X+\eta)\cong H^0(mK_{X'}+\eta)$ (after replacing $X$ by a birational model, we may assume that $f:X\to X'$ is a morphism and $K_{X}=f^*K_{X'}+E$ where $E\geq 0$ is an effective exceptional $\mathbb Q$-divisor....). By Kawamata-Viehweg vanishing (2.17 of arXiv:9601026 Singularities of Pairs by J. Koll\'ar) we have $h^i(mK_{X'}+\eta)=0$ for any $i>0$, $m\geq 2$ and $\eta \in Pic ^0(X')$ so that $h^0(mK_{X'}+\eta )=\chi (mK_{X'}+\eta)=\chi (mK_{X'})$. See 2.12 of Hacon-Pardini J. reine angew. Math. 546 (2002) 177-199 for a related statement in any Kodaira dimension where the proof is done on the smooth variety $X$.


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