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Serre's vanishing theorem (SV) states that, on a projective variety $X$ with a choice of ample line bundle $\mathcal{O}_X(1)$, for any coherent sheaf $F$, we have $$H^i(X,F(m))=0,\quad m>>0$$ for every $i>0$ (i.e. there's an $m$ big enough such that the vanishing occurs for any bigger multiple $m'>m$ of the polarization and for every index $i$), where $F(m)$ denotes the twist $F\otimes\mathcal{O}_X(m)$.

If $D_*:=\{ D_m\}_{m\in\mathbb{N}}$ is a sequence of divisors on the projective variety $X$, we say that $D_*$ satisfies SV on $X$ if, for every coherent sheaf $F$ on $X$, $$H^i(X,F(D_m))=0,\quad m>>0$$ for every $i>0$, where as usual $F(D_m)=F\otimes\mathcal{O}_X(D_m)$.

I'm mostly interested in the case in which:

($\star$) $X$ is a normal surface over $\mathbb{C}$.

Q1. Is there some "general" and "nontrivial" criterion for a sequence $D_*$ to satisfy SV? E.g. $D_*:=$ a subsequence of $\{m\cdot D_0\}$ for a fixed ample $D_0$ clearly satisfies it, but it's a trivial consequence of the usual SV.

Also, is there some such property that is valid for every $X$? (For example, could it be something like: $D_*$ eventually forms a subsequence of a sequence of ample divisors which is "increasingly positive" in some specified sense?)

Narrowing down the question (and assuming ($\star$), if it is of any use):

Q2. Assume that $D_*$ is a sequence of (not necessarily effective) ample divisors such that $D_{m+1}-D_m$ is effective. Does $D_*$ satisfy SV?

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    $\begingroup$ I think Q2 is easily seen to be false, eg. $X$ the blow up of $P^2$ at a point $p$ with exc. div $E$ and $D_m=(m+1)H-E$ where $H$ is the pullback of a general hyperplane. Then we have a short exact sequence $0\to O_X(D_m+2E)\to O_X(D_m+3E)\to O_{P^1}(-2)\to 0$ and since $H^1(O(-2))\ne 0$, then either $H^1(O_X(D_m+3E))\ne 0$ or $H^2(O_X(D_m+2E))\ne 0$. I guess that one could hope that if $D_m$ are ample and $\lim D_m^{\dim Z}\cdot Z\to \infty$ for any subvariety $Z$, then $D_*$ satisfies SV (?). $\endgroup$
    – Hacon
    Jul 7, 2016 at 23:32
  • $\begingroup$ @Hacon: thank you for the counterexample. I suspected that Q2 might not be true. What if we require that $D_{m+1}-D_m$ has strictly positive self intersection? It still does not imply (say by Nakai-Moishezon) that it is ample right? $\endgroup$
    – Qfwfq
    Jul 7, 2016 at 23:39
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    $\begingroup$ Not sure I follow. In my example $(D_{m+1}-D_m)^2=H^2>0$. If you require $(D_m^{\dim Z}\cdot Z / H^m\cdot Z)>\epsilon m$ for any subvariety, then using Anhern-Siu type arguments you should be able to show that $D_m=mH+P$ where $H$ is an ample Q-divisor and P is nef and then deduce that $D_*$ is SV by a combination of SV and Fujita vanishing....not sure if this is useful (?). $\endgroup$
    – Hacon
    Jul 7, 2016 at 23:56

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Claim: $D_*$ is SV iff for any ample divisor $H$ there exists $m(H)$ such that if $m\geq m(H)$, then $D_m-H$ is nef.

Suppose $D_*$ is SV. Fix $A$ very ample. Since $D_*$ is SV, then $H^i(D_m-H-jA)=0$ for all $i>0$ and $0\leq j\leq \dim X +1$. By Castelnuovo-Mumford regularity $D_m-H$ is generated by global sections and in particular nef.

Suppose that for any ample divisor $H$ there exists $m(H)$ such that if $m\geq m(H)$, then $D_m-H$ is nef. Let $F$ be a coherent sheaf on $X$ and $A$ an ample line bundle on $X$, then by Fujita vanishing there exists an integer $t_0$ such that $H^i(F(tA+P))=0$ for any $t\geq t_0$ and $P$ a nef line bundle. But then, for any $m\geq m(t_0A)$, we have $D_m-t_0A$ is nef and hence $H^i(F(D_m))=0$.

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