TITLE: One-parameter group of diffeomorphisms generated by vector field
QUESTION [7 upvotes]: Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. The vector field $$w=x_1\frac\partial{\partial x_2}-x_2\frac\partial{\partial x_1}+\lambda\left(x_4\frac\partial{\partial x_3}-x_3\frac\partial{\partial x_4}\right)$$ ($\lambda\in\mathbb{R}$) is tangent to $X$ and hence defines by restriction a vector field $v$ on $X$. What is the one-parameter group of diffeomorphisms that $v$ generates?

The definition of a one-parameter group of diffeomorphisms that I'm using is the following:

Let $U$ be an open subset of $\mathbb{R}^n$ and $F : U \times \mathbb{R} \rightarrow U$ a $C^{\infty}$ mapping. The family of mappings $f_t: U \rightarrow U$ , $f_t(x) = F(x, t)$
  is said to be a one-parameter group of diffeomorphisms of $U$ if $f_0$ is the identity
  map and $f_s \cdot f_t = f_{s+t}$
  for all s and t.

First of all, I'm confused how this definition can be applied to our situation. The vector field $v$ is not present anywhere in the definition of a one-parameter group of diffeomorphisms. But it has to be relevant somewhere, right?

REPLY [4 votes]: This vector field corresponds to following ODE in $\mathbb{R}^4$:
$$ \left ( \begin{array}{right}
\\ \dot{x_1}\\
\dot{x_2} \\
\dot{x_3} \\
\dot{x_4} \end{array} \right) =
\left (
\begin{array}{right}
\\  -x_2
\\  x_1
\\  \lambda x_4
\\  -\lambda x_3
\end{array}
\right ). $$
Actually, dynamics of $x_1$, $x_2$ is decoupled from dynamics of $x_3$, $x_4$ and equations can be easily solved. If we represent pairs $(x_1, x_2)$ and $(x_3, x_4)$ as complex numbers $z_1 =  x_1 + ix_2$ and $z_2 = x_3 + i x_4$ from unit circle in $\mathbb{C}$, then solution can be written in form $(e^{i(t + \phi_1)}, e^{i(-\lambda t + \phi_2)})$, where $\phi_1$ and $\phi_2$ can be derived from initial value problem for ODE. Well, this is it: the one-parameter group of diffeomorphisms generated by this vector field. You take an arbitratry point $x^{(0)} = (x^{(0)}_1, x^{(0)}_2, x^{(0)}_3, x^{(0)}_4)$; it corresponds to
$(z^{(0)}_1, z^{(0)}_2)$, where $z^{(0)}_1 = e^{i\phi^{(0)}_1}$, $z^{(0)}_2 = e^{i\phi^{(0)}_2}$. The action $F(x^{(0)}, t)$ is following:
$$ F(x^{(0)}, t) = \left (
\mathrm{Re} \left ( e^{i\left(t + \phi^{(0)}_1\right)} \right),
\mathrm{Im} \left ( e^{i\left(t + \phi^{(0)}_1\right)} \right),
 \mathrm{Re} \left ( e^{i\left(-\lambda t + \phi^{(0)}_2\right)}\right),
 \mathrm{Im}\left ( e^{i\left(-\lambda t + \phi^{(0)}_2\right)} \right) \right). $$
Properties of complex exponenti is a key fact to show that $F(x, t)$ is a one-parameter group of diffeomorphisms.


TITLE: Why is temperature vibration?
QUESTION [0 upvotes]: Why do the atoms in a crystal vibrate at finite temperature?

REPLY [1 votes]: Why is temperature vibration?

Atoms - electrons, particles etc. - vibrate, rotate, spin, etc. By hitting something else (other particles) a part of the vibration is handed over as an amount of energy. This new particle will now try to vibrate along. It will be speeded up while the original one will be slowed down. They will eventually find an equilibrium and vibrate equily.
This particle could be in the outer layer of your hand, and it now vibrates. You can feel that. To describe that feeling we call it temperature. If it vibrates violently, it might break loose from the surrounding atoms - maybe breaking molecular bonds and other connections. Your skin is being damaged. You feel this too and this is hardwired in us as pain. To be able to talk about this, we call this to be burned. And so on.

Why do the atoms in a crystal vibrate at finite temperature?

From the start they have some energy, shown as vibration, rotation, spin etc. If all that energy is lost, they stand still. We call this situation absolute zero temperature.
Temperature is just the word for the macroscale version of vibration of many, many, many particles. People gave it such name to be able to talk about the largescale effect of all the vibrations in everything around us.


TITLE: $\sum_{n=2}^\infty \frac{e^{2\pi in t}}{n \log n}$
QUESTION [3 upvotes]: Does any know how to show that $\sum_{n=2}^\infty \frac{e^{2\pi in t}}{n \log n}$ is in $L^{2}(\mathbb T)$

REPLY [6 votes]: Hint: this is equivalent to showing that $\sum_{n=2}^\infty\frac{1}{n^2 \log^2 n} < \infty.$


TITLE: If we have a minimal sufficient statistic for a parameter, is this also a minimal sufficient statistic for any function of the parameter?
QUESTION [1 upvotes]: If we know that $\theta^{*}$ is a minimal sufficient statistic for a certain parameter $\theta$ , is it true that $\theta^{*}$ will also be a minimal sufficient statistic for any one-to-one function of $\theta$, $f(\theta)$?

REPLY [0 votes]: Denote $\eta = f(\theta)$. $f$ is one-one.
For any sufficient statistic $T$ of $\eta$ we have
\begin{align*}
T(x)=T(y)\implies\delta_\eta(x)&=\delta_\eta(y) \\
\implies\delta_{f(\theta)}(x)&=\delta_{f(\theta)}(y)\\
\implies\delta_\theta(x)&=\delta_\theta(y)\\
\end{align*}
(The last $\implies$ holds because we can treat $\delta$ a function of $\theta$)
Thus $T$ is sufficient for $\theta$. Similarly, $\theta^*$ is sufficient for $\eta$.
Since $\theta^*$ is a minimal sufficient statistic of $\theta$ and $T$ sufficient for $\theta$, $\theta^*$ is a function of $T$.
This means for any $T$ sufficient for $\eta$, and $\theta^*$, being sufficient for $\eta$, is a function of $T$, thus $\theta^*$ is minimal sufficient for $\eta$.


TITLE: Show that no two sets in the probability space with $\mathbb{P}(\{k\})=2^{-k!}$ are independent.
QUESTION [1 upvotes]: Let $\mathcal{P}(\mathbb{N})$ denote the power set of $\mathbb{N}$.
Show that no two non-trivial sets in the probability space $(\mathbb{N},\mathcal{P}(\mathbb{N}),\mathbb{P})$ with $\mathbb{P}(\{k\})=2^{-k!}$ for $k\ge2$ and $\mathbb{P}(\{1\})=1-\sum\limits_{k\ge2}\mathbb{P}(\{k\})$ are independent.
There are some hints for this task as follows:
For independent sets $A,B\in\mathcal{P}(\mathbb{N})$, i.e., $\mathbb{P}(A\cap B)=\mathbb{P}(A)\mathbb{P}(B)$,

Show that, wlog $1\notin A\cup B$,
For $N\ge2$, $\sum\limits_{k>N}2^{N!-k!}\leq\frac{1}{4}$,
Let $k_A=\min\{k\in A\}$, $k_B=\min\{k\in B\}$, and $k_{AB}=\min\{k\in A\cap B\}$.
Then show that $2^{-k_{AB}!}\leq2\cdot2^{-k_A!-k_B!}$ and $2\cdot2^{-k_{AB}!}\ge2^{-k_A!-k_B!}$. Finally, lead this to a contradiction.

I have shown 2 and 3 but I am unable to show 1 and even after assuming that 1 is true I am stuck after showing 3 and unable to obtain a contradiction.
If someone could help me with these two things I would really appreciate it.

REPLY [1 votes]: Non-existence of non-constant independent random variables on this space:
Let $X$ and $Y$ be independent random variables on this space and suppose they are both non-constant. Let $E$ be a
non-empty Borel set in $\mathbb{R}$ which does not contain $X(1)$ and $F$ be a non-empty Borel set which does not contain $Y(1)$. We prove that $P\{X^{-1}(E)\cap Y^{-1}(F)\}\neq
P\{X^{-1}(E)\}P\{Y^{-1}(F)\}$. We have $P\{X^{-1}(E)\}=\sum\limits_{X(n)\in
E}^{{}}\frac{1}{2^{n!}},P\{Y^{-1}(F)\}=\sum\limits_{Y(n)\in F}^{{}}\frac{1}{%
2^{n!}}$ and $P\{X^{-1}(E)\cap Y^{-1}(F)\}=\sum\limits_{X(n)\in E,Y(n)\in
F}^{{}}\frac{1}{2^{n!}}$. Let $A=\{n:X(n)\in E\}$ and $B=\{n:Y(n)\in F\}$.
If $P\{X^{-1}(E)\cap Y^{-1}(F)\}=P\{X^{-1}(E)\}P\{Y^{-1}(F)\}$ then we have $%
\sum\limits_{n\in A}^{{}}\frac{1}{2^{n!}}\sum\limits_{n\in B}^{{}}\frac{1}{%
2^{n!}}=\sum\limits_{n\in A\cap B}^{{}}\frac{1}{2^{n!}}.$ This gives $%
\sum\limits_{n\in A,m\in B}^{{}}\frac{1}{2^{n!+m!}}=\sum\limits_{k\in
A\cap B}^{{}}\frac{1}{2^{k!}}$. We look at the two sides as expansions to
base $2$ of some number in $(0,1)$. We note that $n!+m!=k!+j!$ implies $%
(n,m)=(k,j)$ or $(n,m)=(j,k)$. To see this suppose, without loss of
generality, $n$ is the least of the integers $n,m,k,j$ and divide both sides
by $(n+1)!.$ We get $\frac{1}{n+1}\in \mathbb{Z}$
, a contradiction unless $j$ or $k$ equals $n$. If $k=n$ then we get $m!=j!$
so $m=j$. Similarly $j=n$ implies $m=k$. This proves that $(n,m)=(k,j)$ or $%
(n,m)=(j,k)$. Thus in the sum $\sum\limits_{n\in A,m\in B}^{{}}\frac{1}{%
2^{n!+m!}}$ each term is repeated at most twice. If $k\in A\cap B$ we must
have $\frac{1}{2^{k!}}=\frac{1}{2^{n!+m!}}$ or $\frac{1}{2^{k!}}=\frac{2}{%
2^{n!+m!}}$ for some $n$ and $m>1$ [ by uniqueness of expansions to base $2$
]. Hence $n!+m!=k!$ or $n!+m!-1=(k!)$. We note that $n!+m!$ can never be a
factorial ( as can be seen by a very elementary argument), nor can $n!+m!-1$
be a factorial since $-1$ is not divisible by $2!$ Thus $A\cap B$ is empty.
This contradicts the equation $\sum\limits_{n\in A,m\in B}^{{}}\frac{1}{%
2^{n!+m!}}=\sum\limits_{k\in A\cap B}^{{}}\frac{1}{2^{k!}}$ and the proof
is complete.


TITLE: A particle moves along the circle $x^{2}+y^{2}=25$. Find its acceleration when it is at $(3,4)$.
QUESTION [1 upvotes]: A particle moves along the circle $x^{2}+y^{2}=25$ at constant speed, making one revolution in $2$ $s$. Find its acceleration when it is at $(3,4)$.

What I got so far:
Let $x=5\cos(\pi t)$ and $y=5\sin(\pi t)$. Then $\mathbf{r}(t)=(5\cos(\pi t),5\sin(\pi t))$ with period $\frac{2\pi}{b}=\frac{2\pi}{\pi}=2$ $s$. Then I calculated $\mathbf{a}(t)=(-5\pi^{2} \cos(\pi t), -5\pi ^{2}\sin(\pi t))$.
What do I do next with the point $(3,4)$? I tried substituting it in $\mathbf{a}(t)$ but it doesn't seem like the correct thing to do. Do I use $\mathbf{r}(t)$? If so, does $t=(3,4)$?

REPLY [5 votes]: Well, you could notice from here that
$$
\boldsymbol{a}(t) = (−5\pi^2 \cos \pi t,−5\pi^2 \sin \pi t)
= (-\pi^2) \cdot(5 \cos \pi t, 5 \sin \pi t)
= -\pi^2 \boldsymbol{r}(t).
$$
Now you don't have to calculate $t$, you can just plug in the $\boldsymbol{r}(t)$ and obtain $\boldsymbol{a}(t)$.


TITLE: PDF of $Y/X$ given a joint distribution
QUESTION [1 upvotes]: Suppose the joint pdf of $X$ and $Y$ is given by
  $$f_{X,Y}(x,y) = \begin{cases} 2 & \text{if}\ 0\le y\le x \le 1 \\
 0 & \text{otherwise}
\end{cases}$$
  What is the pdf of $Y/X$?

To start off, I know that I have to find the marginal pdf of $f_X$ which is
$$f_X=\int_0^x 2dy=2x$$
I am not sure whether this is the correct way of obtaining the marginal.
Next it will be finding the cdf of $Y/X$. I have previously seen how to compute this if given a uniform distribution of $X$ and $Y$, but I don't really understand how to solve this problem, since $X$ and $Y$ are no longer uniform. I am also not sure whether my marginal derivation is correct. Some help would be appreciated.

REPLY [1 votes]: Suppose the joint pdf of $X$ and $Y$ is given by

$$f_{X,Y}(x,y) = \begin{cases} 2 & \text{if}\ 0\le y\le x \le 1 \\
 0 & \text{otherwise}
\end{cases}$$

What is the pdf of $\frac{Y}{X}$?
Let $T := \frac{Y}{X}$. Notice that $T$ take values between $0$ and $1$. For $t \in (0, 1)$, CDF of $T$ at $t$ is
$F_T(t) = \Pr(T \leq t) = \Pr\left(\frac{Y}{X} \leq t\right) = \Pr\left(Y \leq tX\right) = \int\limits_0^1 \int\limits_0^{tx} 2 dy dx = \int\limits_0^1  2tx  dx  = t $.
So, the density of $T$ is
$f_T(t) = \mathbb{1}_{(0,1)}(t)$ i.e. $T$ is Uniformly distributed over the interval $(0,1)$.


TITLE: Infinite, residually-finite, finitely generated, groups with very limited composition factors of their finite images.
QUESTION [5 upvotes]: Let $S$ be a finite collection of finite non-abelian simple groups, for example $S=\{ A_5 \}$. I am looking for infinite residually-finite finitely-generated groups $G$ such that the composition factors of each finite image of $G$ are in $S$. Are there such examples?

REPLY [6 votes]: Theorem 1.27 of https://arxiv.org/pdf/math/0510294.pdf gives 63-generated, residually finite, just infinite groups with the property you want for any set of simple nonabelian groups. The construction is due to Dan Segal.  See Dan Segal, The finite images of finitely generated groups, Proc. London Math. Soc. (3) 82 (2001), no. 3, 597–613.
Original Answer. Peter Neumann constructed finitely generated just infinite branch groups all of whose finite quotients are iterated permutational wreath products of $A_6$ and hence have only $A_6$ as a composition factor. See example 1.6.6 of https://arxiv.org/pdf/math/0510294.pdf.


TITLE: Numerical methods quadratic interpolation errors
QUESTION [1 upvotes]: Show that the truncation error of quadratic interpolation in an equidistant  table is bounded by  $$\frac{h^3}{9\cdot3^{0.5}}\max f''' (x)$$

I have gotten to nothing because i don't seem to understand quadratic interpolation errors unlike linear interpolation errors.

REPLY [0 votes]: The interpolation error for the quadratic polynomial interpolating at the points $x_{k-1},x_k,x_{k+1}$ and their tabulated values is bounded by
$$
|(x-x_{k-1})(x-x_k)(x-x_{k+1})|\frac{1}6\max_{z\in[x_{k-1},x_{k+1}]}|f'''(z)|
$$
If $x$ is parametrized as $x_k+sh$ with $s\in[-\frac12,\frac12]$, then this bound becomes
$$
|s-s^3|\frac{h^3}6\max_{z\in[x_{k-1},x_{k+1}]}|f'''(z)|
$$
The local maxima of the first factor in $[-1,1]$ are located at $s=\pm\frac1{\sqrt3}$, which should result in your bound.
But these $s$ are outside the interval $[-\frac12,\frac12]$. To get an approximate value for the point $x_k+\frac1{\sqrt3}h$ from the table, one would use the interpolation polynomial based on the triple centered at $x_{k+1}$. Thus the maximum error for the more realistic procedure is at $s=\pm\frac12$.


text:
\begin{document}

\begin{abstract}
We prove the Kawamata-Viehweg vanishing theorem for surfaces of del Pezzo type over perfect fields of positive characteristic $p>5$. As a consequence, we show that klt threefold singularities over a perfect base field of characteristic $p>5$ are rational. We show that these theorems are sharp by providing counterexamples in characteristic $5$.
\end{abstract}

\maketitle

\section{Introduction}

Fano varieties and their log generalisations are one of the basic building blocks of algebraic varieties according to the Minimal Model Program. In recent years a lot of progress has been made in the study of Fano varieties in characteristic $0$, notably \cite{Bir19, Bir}. Less is known about Fano varieties in positive characteristics.
It is conjectured that Kodaira type vanishing theorems, which in general fail in positive characteristic, hold for Fano varieties of a given dimension in large enough characteristic (\cite[Open questions]{Tot19}).
This conjecture has been confirmed in dimension $2$ by \cite{CTW17}, where the authors prove the existence of an integer $p_0>0$ such that surfaces of del Pezzo type over perfect fields of characteristic $p>p_0$ satisfy the Kawamata--Viehweg vanishing theorem.
However, an explicit value for $p_0$ cannot be obtained via the methods of \cite{CTW17} and
in this article we show that $p_0=5$ is both necessary and sufficient:

\begin{thm}[See Theorem \ref{kvv}]\label{t-main-th}
	Let $X$ be a surface of del Pezzo type over a perfect field $k$ of characteristic $p>5$.
	Let $D$ be a Weil divisor on $X$ and suppose that there exists an effective $\Q$-divisor $\Delta$ such that $(X, \Delta )$ is a klt pair and $D-(K_X+\Delta)$ is big and nef.
	Then
	$$H^i(X, \mO_X(D))=0 \text{ for all } i>0. $$
\end{thm}

The crucial ingredient in the proof of Theorem \ref{t-main-th} is establishing the following liftability theorem  for klt surfaces whose canonical divisor is not pseudo-effective.

\begin{thm}[See Theorem \ref{liftdelpezzo}]\label{liftdelpezzo-intro}
	Let $k$ be an algebraically closed field of characteristic $p>5$.
	Let $X$ be a klt projective surface over $k$ such that $K_X$ is not pseudo-effective.
	Then there exists a log resolution $\mu\colon V\rightarrow X$ such that $(V, \text{Exc}(\mu))$ lifts to characteristic zero over a smooth base.
\end{thm}

In recent years, it has become clear that the understanding of Fano varieties is crucial for the study of klt singularities, in particular in positive characteristics.
In characteristic zero, Elkik proved that klt singularities are always rational (see \cite{Elk81}). This, however, fails in positive characteristic (see \cite{CT19, Ber, Tot19, Yas19}), and the question is intertwined with the failure of vanishing theorems on Fano varieties. In \cite{HW17} the authors prove that klt threefold singularities are rational in large characteristic $p>p_0$ where $p_0$  is large enough to assure the validity of the Kawamata--Viehweg vanishing theorem for surfaces of del Pezzo type.
As a consequence of their work and Theorem \ref{t-main-th}, we can therefore provide an effective bound on the characteristic $p$ for which klt threefold singularities are rational.

\begin{cor}[see {\cite[Theorem 1.1]{HW17}}]\label{c-rationality}
	Let $k$ be a perfect field of characteristic $p>5$.
	Let $(X, \Delta)$ be a Kawamata log terminal threefold over $k$.
	Then $X$ has rational singularities.
	Moreover, if $X$ is $\Q$-factorial and $D$ is a Weil divisor on $X$ then $\mO_X(D)$ is Cohen-Macaulay.
\end{cor}

On the other hand Theorem \ref{t-main-th} and Corollary \ref{c-rationality} have several applications to the birational geometry of threefolds. We mention two: the finiteness of the local \'etale fundamental group of threefold klt singularities (see \cite[Theorems A, B and C]{CS}) and a refined version of the base point free theorem for klt threefolds (see \cite{Ber19}). The corollaries below were previously known to hold in large characteristic, we now know them to be true in characteristic $p>5$.

\begin{cor}[See {\cite[Theorems A, B]{CS}}]
	Let $k$ be an algebraically closed field of characteristic $p>5$. Let $(X, \Delta)$ be a Kawamata log terminal threefold over $k$. Let $x\in X$ be a closed point. Then the local \'etale fundamental group of $X$ at $x$ is finite. Moreover, there exists a quasi-\'etale cover $\widetilde{X}\rightarrow X$ with prime-to-$p $ degree such that every quasi-\'etale cover of $\widetilde{X}$ is \'etale.
\end{cor}

\begin{cor}[See  {\cite[Theorem 1.1]{Ber19}}]
	Let $k$ be a perfect field of characteristic $p> 5$.
	Let $(X, \Delta)$ be a  quasi-projective klt threefold log pair and let $\pi \colon X \to Z$ be a projective contraction morphism of quasi-projective normal varieties over $k$.
	Let $L$ be a $\pi$-nef Cartier divisor on $X$ such that
	\begin{enumerate}
		\item $\dim(Z) \geq 1$ or $\dim(Z)=0$ and $\nu(L) \geq 1$;
		\item $nL-(K_X+\Delta)$ is a $\pi$-big and $\pi$-nef $\Q$-Cartier $\Q$-divisor for some $n>0$.
	\end{enumerate}
	Then there exists $m_0>0$ such that $mL$ is $\pi$-free for all $m \geq m_0$.
\end{cor}

In Section \ref{s-counter-five}, we construct counterexamples in characteristic five to Theorem \ref{t-main-th} and Corollary \ref{c-rationality} building upon a previous example found by the third author.

\begin{thm}[See Proposition \ref{p-failure-vanishing} and Theorem \ref{t-notCM}]
	Let $k$ be an algebraically closed field of characteristic $p=5$.
	Then:
	\begin{enumerate}
		\item there exists a klt del Pezzo surface $T$ of Picard rank one and an ample $\Q$-Cartier divisor $A$ such that $H^1(T, \mathcal{O}_T(-A)) \neq 0$;
		\item there exists a $\Q$-factorial klt threefold singularity which is not Cohen-Macaulay.
	\end{enumerate}
\end{thm}

This, together with the examples found in \cite{CT19} and \cite{Ber}, shows that our assumption on the characteristic in Theorem \ref{t-main-th} and Corollary \ref{c-rationality} is optimal.

\subsection{Sketch of the proof}
We now overview the main techniques and proofs in this article.
In Section \ref{s-lift} we prove Theorem \ref{liftdelpezzo-intro}. In order
to do this, we first show that we can reduce to the case of such surfaces admitting a Mori fibre space structure (Lemma \ref{p-lift-blow-up}).
We prove the case of Mori fibre space onto a curve (Proposition \ref{liftmfs}), while the case of klt del Pezzo surfaces of Picard rank one has been settled by the third author in his Ph.D. thesis \cite{Lac20}.

In Section \ref{s-KVV-dP} we use Theorem \ref{liftdelpezzo-intro} to prove Theorem \ref{t-main-th}, where the property of the surface being del Pezzo type plays a fundamental role.
By running a $\Delta$-MMP we construct a birational morphism  $\pi \colon X \to Y$ which contracts a boundary $N\leq \Delta$ such that $D-(K_X+N)$ is ample and such that the support of $N$ coincides with the exceptional locus of $\pi$. Since $Y$ remains a surface of del Pezzo type, we may apply Theorem \ref{liftdelpezzo} to construct a log resolution $(V, \text{Exc}(\mu))$ of $Y$ that lifts to characteristic zero over a smooth base. Since $N$ was contracted by $\pi$, after a sequence of blow-ups of points of $V$, we may find a log resolution of the pair $(X,N)$ lifting to characteristic zero over a smooth base. Then we can prove the desired vanishing theorem by applying the logarithmic Deligne-Illusie vanishing theorem (see Lemma \ref{l-vanishing-W2}). \\

\textbf{Acknowledgements:}  We would like to thank P.~Cascini, C.D.~Hacon, Zs.~Patakfalvi, D.C.~Veniani, J.~Witaszek and  M.~Zdanowicz for useful comments on the article.
We are indebted to M. Zdanowicz for several useful discussions on the topic of liftability to the ring of Witt vectors and for providing key arguments used in the subsection \ref{subsection lift}.
E.A. was supported by SNF Grant \#200021/169639, F.B. was partially supported by the NSF research grant n. DMS-1801851 and by a grant from the Simons Foundation; Award Number: 256202, J.L. was partially supported by the NSF research
grants no: 1265263 and no: 1802460 and by grant \#409187 from the Simons Foundation.

\section{Preliminaries}

\subsection{Notation}
In this article, $k$ denotes a perfect field of characteristic $p>0$. For $n>0$, we denote by $W_n(k)$ (resp. $W(k)$) the ring of Witt vectors of length $n$ (resp. the ring of Witt vectors).

We say that $X$ is a {\em variety over} $k$ or a $k$-{\em variety} if
$X$ is an integral scheme that is separated and of finite type over $k$.
We say that $X$ is a {\em curve} over $k$ or a $k$-{\em curve}
(resp. a {\em surface} over $k$ or a $k$-{\em surface},
resp. a {\em threefold} over $k$)
if $X$ is a $k$-variety of dimension 1 (resp. 2, resp. 3).

We say $(X,\Delta)$ is a \emph{log pair} if $X$ is  normal variety, $\Delta$ is an effective $\Q$-divisor and $K_X+\Delta$ is $\Q$-Cartier.
We say that a pair $(X, \Delta)$ is \emph{log smooth} if $X$ is regular and the support of $\Delta$ is simple normal crossing.
We refer to \cite{Kol13} for the basic definitions in birational geometry and of the singularities appearing in the Minimal Model Program (e.g. klt) and to \cite[Remark 2.7]{GNT19} for birational geometry over perfect fields. We say that a variety is $\mathbb{Q}$-factorial if every Weil divisor is $\mathbb{Q}$-Cartier. We recall that klt surface pairs are $\mathbb{Q}$-factorial (see e.g. \cite[Fact 3.4]{Tan15}), so we will often
discuss Weil divisors on klt surface pairs without mentioning that they are $\mathbb{Q}$-Cartier.

Given a proper birational morphism $\pi \colon X \to Y$ between normal varieties, we denote by $\text{Exc}(\pi)$ the exceptional locus of $\pi$. Given a variety $X$ and a Weil divisor $D$ we say that $f \colon Y \to X$ is a \emph{log resolution} of $(X,D)$ if $\text{Exc}(\pi)$ has pure codimension one and the pair $(Y, \text{Exc}(\pi)+\pi_*^{-1}D)$ is log smooth.

\subsection{Surfaces of del Pezzo type}\label{del pezzo section}

In this subsection we collect some properties of surfaces of del Pezzo type.

\begin{definition}
	Let $k$ be a field.
	We say that a normal $k$-projective surface $X$ is  a \emph{surface of del Pezzo type} if there exists an effective $\Q$-divisor $\Delta$ such that $(X, \Delta)$ is klt and $-(K_X+\Delta)$ is ample.
	The pair $(X, \Delta)$ is said to be a log del Pezzo pair. We say $X$ is a \emph{klt del Pezzo surface} if the pair $(X,0)$ is log del Pezzo.
\end{definition}

Surfaces of del Pezzo type behave nicely from the birational perspective, as the MMP algorithm can be run for any divisor on them and they form a closed class under birational contraction.

\begin{lem}\label{l-dP-mds}
	Let $k$ be a perfect field.
	Let $X$ be a surface of del Pezzo type over $k$.
	Then $X$ is a Mori dream space, i.e. given a $\Q$-divisor $D$ on $X$ we can run a $D$-MMP which terminates.
\end{lem}

\begin{proof}
	Let $\Delta$ be an effective $\Q$-divisor on $X$ such that $(X, \Delta)$ is a log del Pezzo pair.
	Choose a sufficiently small rational number $\varepsilon>0$ such that $A=\varepsilon D-(K_X+\Delta)$ is ample.
	By \cite[Lemma 2.8]{GNT19}, there exists an effective $\Q$-divisor  $A'$ such that $A' \sim_{\Q} A$ and $(X, \Delta+A')$ is a klt pair.
	We conclude by noting  that a $D$-MMP coincides with a $(K_X+\Delta+A')$-MMP, which exists and terminates by \cite[Theorem 1.1]{Tan18}.
\end{proof}

\begin{lem}\label{l-dP-contr}
	Let $k$ be a field.
	Let $X$ a surface of del Pezzo type over $k$.
	Let $\pi \colon X \to Y$ be a birational $k$-morphism onto a normal surface $Y$.
	Then $Y$ is a surface of del Pezzo type.
\end{lem}

\begin{proof}
	See \cite[Lemma 2.9]{BT}.
\end{proof}

\subsection{Lifting to characteristic zero over a smooth base}\label{subsection lift}

We now collect some notions on liftability we will use in this article.
In this subsection we fix $k$ to be an algebraically closed field.
We start by recalling the definition of liftability to characteristic zero (see \cite[Definition 2.15]{CTW17}) and to the ring of Witt vectors.

\begin{definition}
	Let $(X,D=\sum_{i=1}^r D_i)$ be a log smooth pair over $k$.
	A \emph{lifting}  of $(X, D)$ to a scheme $T$ consists of
	\begin{enumerate}
		\item a smooth and separated morphism $\mathcal{X}$ over $T$,
		\item effective Cartier divisors $\mathcal{D}_i$ in $\mathcal{X}$ such that for every subset $J \subseteq \left\{1, \dots, r\right\}$ the scheme theoretic intersection $\bigcap_{j \in J} \mathcal{D}_j$ is smooth over $T$,
		\item  a morphism $\alpha \colon \text{Spec}(k)\rightarrow T$,
	\end{enumerate}
	such that the base change of the schemes $\mathcal{X}, \mathcal{D}_1, \dots, \mathcal{D}_r$ via $\alpha$ are isomorphic to $X, D_1, \dots, D_n$ respectively.

	If $T$ is smooth and separated over $\text{Spec}(\Z)$ (resp. $T=\text{Spec}(W(k))$),
	then we say that $(\mathcal{X},\mathcal{D})$ is \emph{a lifting of $(X,D)$ to characteristic zero over a smooth base} (resp. \emph{a lifting to the ring of Witt vectors}).
\end{definition}

We relate the notions of liftability to characteristic zero and liftability to the ring of Witt vectors.

\begin{prop} \label{p-W-char0}
	Let $(X,D)$ be a log smooth pair over $k$.
	Then $(X,D)$ admits a lifting to $W(k)$ if and only if $(X,D)$ admits a lifting to characteristic zero over a smooth base.
\end{prop}

\begin{proof}
	Suppose that $(X,D)$ admits a lifting $(\mathcal{X},\mathcal{D})$  to characteristic zero over a smooth base $T$.
	It is sufficient to show that $\alpha \colon \text{Spec}(k) \to T$ lifts to a section $\widetilde{\alpha}\colon \text{Spec}(W(k)) \to T$ since then the base change of $(\mathcal{X},\mathcal{D})$ along $\widetilde{\alpha}$ is a lifting of $(X,D)$ to $W(k)$.
	The base change $T \times_{\mathbb{Z}} \text{Spec} (W(k)) \to \text{Spec}(W(k))$ is a smooth morphism and $W(k)$ is a complete ring (and thus Henselian). Therefore the section $\alpha \colon \text{Spec}(k) \to T$ lifts to a section $\text{Spec}(W(k)) \to T$ (see \cite[The\'or\`eme 18.5.17]{EGAIV4}).

	Let $(\mathcal{X}, \mathcal{D})$ be a lifting of $(X,D)$ to $W(k)$.
	As $\text{Frac}(W(k))$ is a field of characteristic zero, $\text{Spec}(\text{Frac}(W(k))$ is geometrically regular at $\mathbb{Q}$ by \cite[\href{https://stacks.math.columbia.edu/tag/038U}{Tag 038U}]{StackProject}.
	This means that $\text{Spec}(W(k))\rightarrow \text{Spec}(\Z)$ is flat with geometrically regular fibers and the morphism $\text{Spec}(W(k))\rightarrow \text{Spec}(\Z)$ is regular (see \cite[\href{https://stacks.math.columbia.edu/tag/07R7}{Tag 07R7}]{StackProject}).
	By a theorem of Popescu (see \cite[Theorem 1.3]{CdJ02}), the ring $W(k)$ is therefore a direct limit of smooth $\Z$-algebras.
	By \cite[Th\'eor\`eme 11.2.6]{EGAIV3} flatness is preserved under spreading out and thus there exists a smooth and separated $\Z$-scheme $T$ such that $\mathcal{X}$ and all possible intersection of $\mathcal{D}_i$ for $1\leq i\leq n$, are defined and smooth over $T$.
	Thus we conclude that $(X,D)$ admits a lifting to characteristic zero over a smooth base.
\end{proof}

\begin{remark} \label{r-lift-W2}
	As a consequence let us note that that if $(X, D)$ admits a lifting to characteristic zero over a smooth base then it admits a lifting over $W_2(k)$ (cf. \cite[Remark 2.16]{CTW17}).
\end{remark}

We are interested in understanding how liftability is preserved under blow-ups at smooth points.
We therefore introduce the following terminology:

\begin{definition}
	Let $(X, D=\sum_{i=1}^n D_i)$ be a log smooth surface pair over $k$.
	Let $x$ be closed point of $X$ and let $J \subseteq \left\{1, \dots, n \right\}$ be the subset of maximal cardinality for which $x \in \bigcap_{j \in J} D_j$ (with the convention that $\bigcap_{j \in \emptyset} D_j=X$).
	Let $(\mathcal{X}, \mathcal{D}=\sum_{i=1}^n \mathcal{D}_i)$ be a lifting of $(X,D)$ to a scheme $T$.
	A \emph{lifting} $\overline{x} \subset \mathcal{X}$ of $x$ is a subscheme of $\mathcal{X}$ smooth over $T$ such that $\overline{x} \times_T \text{Spec}(k)=x$.
	We say a lifting $\overline{x}$ is \emph{compatible with the snc structure of} $(\mathcal{X}, \mathcal{D}) $ if $\overline{x} \subseteq \cap_{j \in J} \mathcal{D}_j$. \end{definition}

By possibly changing the base of the lifting, we may always find a lifting of a point to characteristic zero compatible with the snc structure.

\begin{lem} \label{lift-point}
	Let $(X,D=\sum_{i=1}^n D_i)$ be a log smooth surface pair over $k$.
	Let $x$ be a closed point of $X$.
	Suppose that $(X, D)$ lifts to characteristic zero over a smooth base.
	Then there exists a lifting of $(X,D)$ over a smooth base such that $x$ lifts compatibly with the snc structure.
\end{lem}

\begin{proof}
	Let $(\mathcal{X}, \mathcal{D})$ be a lifting of $X$ over a $\Z$-smooth and separated scheme $T$.

	Then by Proposition \ref{p-W-char0} there exists a lifting $(\mathcal{X}, \mathcal{D})$ to $W(k)$.
	Let $J \subseteq \left\{1, \dots, n \right\}$ be the subset of maximal cardinality for which $x \in \bigcap_{j \in J} D_j$.
	Consider the following diagram:
	$$\begin{tikzcd}
		\bigcap_{j \in J}D_j \arrow{r} \arrow{d} & \bigcap_{j \in J}\mathcal{D}_j \arrow{r} \arrow{d} & \mathcal{X} \arrow{dl} \\
		\text{Spec}(k) \arrow{r}  \arrow[u, bend left, "s"]& \text{Spec}(W(k)) &.
	\end{tikzcd}$$
	Since $\bigcap_{j \in J}\mathcal{D}_j $ is smooth over $W(k)$, by \cite[The\'or\`eme 18.5.17]{EGAIV4} there exists a lifting $\overline{s} \colon \text{Spec}(W(k)) \to \bigcap_{j \in J} \mathcal{D}_j$.
	This shows the existence of a lifting of $x$ compatible with the snc structure over $W(k)$.
	By using Popescu's theorem and a spreading out argument as in the proof of Proposition \ref{p-W-char0}, we conclude the existence of a lifting of $(X,D)$ to characteristic zero over a smooth base such that $x$ lifts compatibly with the snc structure.
\end{proof}

We now show that liftability is preserved under blow-ups at smooth centers.

\begin{prop} \label{p-lift-blow-up}
	Let $(X,D=\sum_{i=1}^n D_i)$ be a log smooth surface over $k$ which admits a lifting to characteristic zero over a smooth base.
	Let $x$ be a closed point of $X$ and let $\pi \colon X' \to X$ be the blow-up at the point $x$.
	Then $(X', \pi_*^{-1}D \cup \text{Exc}(\pi))$ admits a lifting to characteristic zero over a smooth base.
\end{prop}

\begin{proof}
	Let us choose $(\mathcal{X},\mathcal{D}=\sum_{i=1}^n \mathcal{D}_i)$ a lifting of $(X,D)$ over a smooth and separated scheme $T$ over $\text{Spec}(\Z)$ such that $x$ lifts compatibly with the snc structure, whose existence is guaranteed by Lemma \ref{lift-point}.
	Let $\overline{\pi} \, \colon \mathcal{X'} \to \mathcal{X}$ be the blow-up along $\overline{x}$.
	Then $\mathcal{X'}$ is a lifting of $X'$ because $\overline{x}$ is smooth and thus, by \cite[Section 8, Theorem 1.19]{Liu02}, $\mathcal{X'}$ is smooth over $T$ and the exceptional divisor $\mathcal{E}$ is a $\mathbb{P}^1$-bundle over $\overline{x}$.
	We conclude that $(\mathcal{X'}, \overline{\pi}^{-1}_*\mathcal{D} \cup \mathcal{E})$ is a lifting of $(X', \pi_*^{-1}D \cup \text{Exc}(\pi))$ over $T$.
\end{proof}

Apart from its intrinsic interest, the condition of liftability to characteristic zero over a smooth base is important for its application to vanishing theorems (in fact, for our purposes liftability to $W_2(k)$ would be sufficient).
We will apply the machinery developed by Deligne and Illusie (see \cite{DI87} and \cite{Har98} for a logarithmic version) to prove vanishing theorems for liftable surfaces in positive characteristic, of which we recall the following special case:

\begin{lem} [(cf. {\cite[Lemma 6.1]{CTW17}})]\label{l-vanishing-W2}
	Let $k$ be a perfect field of characteristic $p>2$.
	Let $(X, \Delta)$ be a two dimensional projective klt pair over $k$. Suppose that there exists a log resolution $\mu \colon V\to X$
	of $(X, \Delta)$ such that $(V, \text{Exc}(\mu) \cup \mu_*^{-1} (\text{Supp} \Delta))$ lifts to $W_2(k)$.
	Let $D$ be a $\Z$-divisor on X such that $D - (K_X + \Delta)$ is ample.
	Then, $H^i(X, \mathcal{O}_X(D)) = 0$ for any $i > 0$.
\end{lem}

\section{Liftability of klt surfaces whose canonical divisor is not pseudoeffective} \label{s-lift}

In this section we prove that the minimal resolution along with the exceptional locus of a projective klt pair whose canonical divisor $K_X$ is not pseudo-effective admits a lifting to characteristic zero over a smooth base.
First we prove it for klt surfaces admitting a Mori fibre space structure onto a curve (see Proposition \ref{liftmfs}).
We then use the classification of klt del Pezzo surfaces of Picard rank one to deduce the general result.

\subsection{Liftability of Mori fibre spaces}

We start by discussing liftability of klt Mori fibre spaces onto a curve.

\begin{lem}\label{liftruledsurf}
	Let $k$ be an algebraically closed field of characteristic $p>0$.
	Let $C$ be a projective smooth curve and let $\pi \colon X \to C$ be a minimal ruled surface over $k$.
	Let $F_i$ be the fibers over closed points $x_i\in C$ for $1\leq i\leq n$.
	Then $(X, \sum_{i=1}^n F_i)$ admits a lifting to $W(k)$.
\end{lem}

\begin{proof}
	By  \cite[V, Proposition 2.2]{Har77} there exists a vector bundle $E$ of rank $2$ on $C$ such that $\pi \colon X \to C$ is isomorphic to $p \colon \mathbb{P}_C(E) \to C$.
	The curve $C$ admits a formal lifting $\mathcal{C}$ to $W(k)$ because the obstruction to the lifting lies in the $H^2(C, T_C)$ by \cite[Proposition 3.1.5, p. 248]{Ill71}, which vanishes for dimension reasons.
	Analogously, the obstruction for lifting $E$ to a vector bundle $\mathcal{E}$ over $\mathcal{C}$ lies in $H^2(C, \mathcal{E}nd(E))$ by \cite[Proposition 3.1.5, p. 248]{Ill71}, which also vanishes.
	Thus we have a lifting $ \mathbb{P}_{\mathcal{C}} (\mathcal{E}) \to \mathcal{C}$.
	Let $x_1, \dots, x_n$ be the closed points on $C$ such that $\pi^*x_i=F_i$.
	By formal smoothness, we can lift $x_1, \dots, x_n$ to $\mathcal{C}$ and thus the fibers $F_1, \dots, F_n$ admits a formal lifting $\mathfrak{F}_1, \dots, \mathfrak{F}_n$ over $W(k)$.
	Therefore $(\mathbb{P}_{\mathcal{C}}(\mathcal{E}), \sum_{i=1}^n \mathfrak{F}_i)$ is a formal lifting of $(X, \sum_{i=1}^n F_i)$ to the formal spectrum $\text{Spf}(W(k))$.
	Since $H^2(X, \mathcal{O}_X)=H^0(X, K_X)=0$ by Serre duality, the pair is algebraizable by \cite[Corollary 8.5.6 and Corollary 8.4.5]{FGAex}.
\end{proof}

\begin{prop}\label{liftmfs}
	Let $k$ be an algebraically closed field of characteristic $p>0$.
	Let $\pi \colon X \to C$ be a projective morphism such that
	\begin{enumerate}
		\item $X$ is a projective surface over $k$ with klt singularities;
		\item $C$ is a smooth projective curve;
		\item $\pi_*\mathcal{O}_X=\mathcal{O}_C$;
		\item $-K_X$ is $\pi$-ample.
	\end{enumerate}
	Let  $\mu \colon V\rightarrow X$ be the minimal resolution.
	Then the pair $(V, \text{Exc}(\mu))$ is log smooth and it admits a lifting to characteristic zero over a smooth base.
\end{prop}

\begin{proof}
	Let us note that the pair $(V, \text{Exc}(\mu))$ is log smooth by the classification of klt surface singularities (see \cite[Section 3.3]{Kol13}).
	Let us denote by $\pi'\colon V\rightarrow C$ the induced fibration.
	We run a $K_{V}$-MMP over $C$.
	We have the following diagram.
	$$\begin{tikzcd}
		V\arrow{d}[swap]{\pi'}\arrow{r}{g} &Y\arrow{ld}{f}\\
		C
	\end{tikzcd}.$$
	Note that $f\colon Y \to C$ is a minimal ruled surface.
	Every irreducible component $E$ of $\text{Exc}(\mu)$ is contained in some fiber $F_i$ of $\pi'$.
	Let $F_1, \dots F_n$ be the minimal collection of distinct fibers such that $\text{Supp}(\text{Exc}(\mu))\subset \text{Supp}(\sum_{i=1}^n F_i).$ Let $x_i \in C$ be the images of $F_i$ for $i=1, \dots, n$.
	Note that we have the inclusion $\text{Supp}(g(F_i))\subset \text{Supp}(F^Y_{i})$ where $F^Y_{i}$ is the fiber over $x_i$ of $Y\rightarrow C$.
	The pair $(Y, \sum_{i=1}^n F_i^Y)$ admits a lifting to $W(k)$ by Lemma \ref{liftruledsurf} and by Proposition \ref{p-W-char0} it admits a lifting to characteristic zero over a smooth base.
	Since $ \sum_{i=1}^n F_i^Y$ is a reduced snc divisor and $g \colon V\rightarrow Y$ can be factored as a sequence of blow-ups of smooth points, the pair $(X, \text{Exc}(g)+ \sum_{i=1}^n g_*^{-1}F_i)$ admits a lifting to characteristic zero over a smooth base by Proposition \ref{p-lift-blow-up}.
\end{proof}

\subsection{General case}

The third author classified all klt del Pezzo surfaces of Picard number one in characteristic $p>3$ in his Ph.D. thesis \cite{Lac20}.
As a consequence of the classification he was able to prove the existence of a log resolution that lifts to characteristic zero for klt del Pezzo surfaces of Picard one defined over algebraically
closed fields of characteristic $p>5$. This serves as the main ingredient for our results on liftability of surfaces of del Pezzo type.

\begin{thm}[(See {\cite[Theorem 7.2]{Lac20}})]\label{JL}
	Let $k$ be an algebraically closed field of characteristic $p>5$.
	Let $S$ be a klt del Pezzo surface of Picard rank one over $k$. Then there exists a log resolution $\mu\colon V\rightarrow S$ such that $(V,\text{Exc}(\mu))$ lifts to characteristic zero over a smooth base.
\end{thm}

With the result above, we are now ready to prove the main result of the section.

\begin{thm}\label{liftdelpezzo}
	Let $k$ be an algebraically closed field of characteristic $p>5$.
	Let $X$ be a klt projective surface over $k$ such that $K_X$ is not pseudo-effective.
	Then there exists a log resolution $\mu\colon V\rightarrow X$ such that $(V, \text{Exc}(\mu))$ lifts to characteristic zero over a smooth base.
\end{thm}

\begin{proof}
	Let $g\colon X\rightarrow Y$ be a $K_X$-MMP.
	There are two cases to distinguish:
	\begin{enumerate}
		\item $Y$ is a klt del Pezzo surface of Picard rank one.
		Then $Y$ admits a log resolution $\mu \colon V\rightarrow Y$ such that $(V, \text{Exc}(\mu))$ lifts to characteristic zero over a smooth base by Theorem \ref{JL}.
		\item  $Y$ is a surface admitting a Mori fibre space structure $\pi \colon Y\rightarrow C$ onto a curve $C$. Then $Y$ admits a log resolution $\mu \colon V\rightarrow Y$ such that $(V, \text{Exc}(\mu))$ lifts to characteristic zero over a smooth base by Proposition \ref{liftmfs}.
	\end{enumerate}

	In both cases, $Y$ admits a log resolution $\mu \colon V \rightarrow Y$ such that $(V, \text{Exc}(\mu))$ lifts to characteristic zero over a smooth base.
	Consider the induced rational map $\mu'\colon V \dashrightarrow
	X$ and let $\pi \colon V'\rightarrow V$ be a log resolution of the indeterminacy locus of $\mu'$.
	Since the morphism
	$\pi \colon V'\rightarrow V$ can be factored as a composition of blow-ups at points, the pair $(V',\pi_*^{-1}\text{Exc}(\mu)+\text{Exc}(\pi))$ lifts to characteristic zero over a smooth base by Proposition \ref{p-lift-blow-up}.
	The commutative diagram below illustrates the situation:
	$$\begin{tikzcd} X\arrow{d}{g}&(V',\pi_*^{-1}\text{Exc}(\mu)+\text{Exc}(\pi))\arrow{l}[swap]{\tilde{\mu }}\arrow{d}{\pi}\\
		Y&(V,\text{Exc}(\mu))\arrow{l}[swap]{\mu}\end{tikzcd}$$
	We have that the support of $\text{Exc}(\tilde{\mu})$ is contained in the support of $\pi_*^{-1}\text{Exc}(\mu)+\text{Exc}(\pi)$.
	Therefore $\tilde{\mu} \colon (V',\text{Exc}(\tilde{\mu}))\rightarrow X$ is a log resolution that lifts to characteristic zero over a smooth base.
\end{proof}

\begin{remark}
	One cannot drop the assumption on $K_X$ being not pseudo-effective and extend the previous result to the class of birationally ruled surfaces with klt singularities.
	Indeed, for every prime $p>0$ Cascini and Tanaka construt examples of klt rational surfaces $Y$ of Picard rank one over $\mathbb{F}_p$  for which no log resolution along with the exceptional divisors admits a lifting to $W_2(k)$ (see \cite[Corollary 3.3]{CT18}).
	In their examples the canonical divisor $K_Y$ is ample if $p>2$.
\end{remark}

As a direct application of the previous result, we can prove the Kodaira vanishing theorem for Weil divisors for the class of klt surfaces whose canonical divisor is not pseudoeffective.

\begin{cor}\label{KV}Let $k$ be a perfect field of characteristic $p>5$.
	Let $X$ be a projective surface over $k$ with klt singularities such that $K_X$ is not pseudo-effective.
	Then for an ample Weil divisor $D$ on $X$, we have $H^i(X, \mO_X(K_X+D))=0$ for all $i>0$.
\end{cor}

\begin{proof}
	We can assume $k$ is algebraically closed by passing to the algebraic closure.
	Then $X$ is klt and $(D+K_X)-K_X$ is ample.
	By Theorem \ref{liftdelpezzo} and Remark \ref{r-lift-W2} $X$ has a log resolution that lifts to $W_2(k)$, therefore the result follow from Lemma \ref{l-vanishing-W2}
\end{proof}

\begin{remark}
	Let us note that Corollary \ref{KV} was first proven in \cite[Theorem D]{Arv20} using different techniques.
\end{remark}

\section{Kawamata-Viehweg vanishing theorem in characteristic $p>5$}
\label{s-KVV-dP}

Using the results on liftability obtained in Section \ref{s-lift}, we are now ready to prove the main result of this article.

\begin{thm}\label{kvv}
	Let $X$ be a surface of del Pezzo type over a perfect field $k$ of characteristic $p>5$.
	Let $D$ be a Weil divisor on $X$ and suppose that there exists a boundary $\Delta$ such that $(X, \Delta )$ is a klt pair and $D-(K_X+\Delta)$ is big and nef.
	Then
	$$H^i(X, \mO_X(D))=0 \text{ for all } i>0. $$
\end{thm}
\begin{proof}
	By taking the base change to the algebraic closure, we may assume $k$ to be algebraically closed.
	By possibly perturbing the boundary $\Delta$ we may assume that $D-(K_X+\Delta)=A$ where $A$ is ample.
	Since $X$ is a Mori dream space by Lemma \ref{l-dP-mds}, we can run a $\Delta$-MMP $g \colon X \to Y$, which terminates with $\Delta_Y:=g_*\Delta$ being a nef $\Q$-Cartier divisor.
	Thus we have
	$\Delta= g^*\Delta_Y+N$, where $N\geq 0$ is effective and exceptional by the negativity lemma \cite[Lemma 3.39]{KM98}.
	So we have $D-(K_X+N)=A+g^*\Delta_Y$ is an ample divisor.
	Moreover, since $N \leq \Delta$, we have $(X, N)$ is klt.

	By Lemma \ref{l-dP-contr}, $Y$ is a surface of del Pezzo type and therefore there exists a log resolution $\mu \colon V\rightarrow Y$ such that $(V, \text{Exc}(\mu ))$ lifts to characteristic zero over a smooth base by Theorem \ref{liftdelpezzo}.

	Let us consider the induced rational map $\mu'\colon V \dashrightarrow X$ and let $\pi \colon V'\rightarrow V$ be a sequence of blow-ups at smooth points such that the induced birational map $\widetilde{\mu}\colon V'\dashrightarrow X$ is a morphism.
	The commutative diagram below illustrates the situation:
	$$\begin{tikzcd} X\arrow{d}{g}&(V',\pi_*^{-1}\text{Exc}(\mu)+\text{Exc}(\pi))\arrow{l}{\widetilde{\mu }}\arrow{d}{\pi}\\
		Y&(V,\text{Exc}(\mu))\arrow{l}{\mu}\end{tikzcd}$$
	We have that the support of $\text{Exc}(\widetilde{\mu})$ and the support of $\widetilde{\mu}^{-1}_*N$ are contained in the support of $\pi_*^{-1}\text{Exc}(\mu)+\text{Exc}(\pi)$. Therefore $\tilde{\mu}\colon (V',\text{Exc}(\tilde{\mu})+\text{Supp}(\tilde{\mu}^{-1}_*N))$ lifts to characteristic zero over a smooth base, and thus to $W_2(k)$ by Remark \ref{r-lift-W2}.
	By Lemma \ref{l-vanishing-W2}, the result follows.
\end{proof}

\begin{remark}
	Motivated by Theorem \ref{liftdelpezzo}, one could ask whether the Kawamata-Viehweg vanishing theorem still holds for the larger class of klt surfaces $X$ whose canonical bundle is not pseudoeffective, at least in large enough characteristic.
	However, this is false in general as shown by the counterexamples in \cite[Theorem 3.1]{CT18} and \cite[Theorem 3.1]{Xie10}.
\end{remark}

\section{Counterexamples in characteristic $p=5$}\label{s-counter-five}

It is well known that the Kawamata-Viehweg vanishing theorem may fail in positive characteristic.
Counterexamples for klt del Pezzo surfaces were found in \cite{CT19} and \cite{Ber} in characteristic $p=2$ and $p=3$ respectively.
As a consequence, in these characteristics there are threefold klt singularities which are not rational.
The third author exhibited an example of a Picard number one klt del Pezzo surface in characteristic $5$ that does not admit a log resolution that lifts to characteristic zero over a smooth base in
\cite[Example 7.6]{Lac20}.
In this section we show that this example also violates the  Kawamata-Viehweg vanishing theorem. In particular, Theorem \ref{t-main-th} and Corollary \ref{c-rationality} are sharp.

\subsection{Construction} \label{ss-construction}

We fix an algebraically closed field $k$ of characteristic $p=5$ and briefly recall \cite[Example 7.6 and B.12]{Lac20}.
Consider the following four points in $\mathbb{P}^2_k$:
\[a=[-1,1,1],\ b=[-1,-1,1],\
c=[1, -1, 1] \text{ and } d=[1,1,1].\]
Let $L_{ab}$ be the line connecting $a$ and $b$ and similarly define $L_{ac}$, $L_{ad}$, $L_{bc}$, $L_{bd}$ and $L_{cd}$.
Consider the cubic curves $C_0 = L_{ad} + L_{ac} + L_{bc}$ and $C_{\infty} =
L_{ab} + L_{bd} + L_{cd}.$

The base locus of the pencil of curves
$\left\{ C_t = C_0 + tC_{\infty} \right\}_t$ consists of the points $a$, $b$, $c$, $d$, counted with multiplicity two, and of the point $[0,0,1]$, counted with multiplicity one.
Now resolve the base locus at the points $a$, $b$, $c$, and $d$. This is done in two steps.
First, let $S_1$ be the surface obtained by blowing-up once at $a, b, c$ and $d$. We denote by $E_a$
the exceptional divisor obtained by the first blow-up over $a$, and we use the analogous notation for the other points.
The configuration is illustrated in Figure 1, where the strict transform of $C_0$ is drawn in \textcolor{gray!60}{gray}, the strict transform of $C_\infty$ is drawn in \textcolor{darkgray!95}{dark gray}
and the exceptional divisors are drawn as dotted lines.

\begin{figure}[h]
	\begin{center}
		\begin{tikzpicture}[scale=.28]

			\coordinate (A) at (-10,2);
			\coordinate (B) at (-2,10);
			\coordinate (C) at (2,10);
			\coordinate (D) at (10,2);
			\coordinate (E) at (-10,-2);
			\coordinate (F) at (-2, -10);
			\coordinate (G) at (10, -2);
			\coordinate (H) at (2, -10);
			\begin{scope}[]

				\draw[draw=black, dotted, thick] (A) -- (B);
				\node [below, black] at (-9, 2.5) {$E_a$};
				\draw[draw=black, dotted, thick] (C) -- (D);
				\node [below, black] at (1,10.5) {$E_d$};
				\draw[draw=black, dotted, thick] (E) -- (F);
				\node [below, black] at (-2,-7.5) {$E_b$};
				\draw[draw=black, dotted, thick] (G) -- (H);
				\node[below, black] at (9, 0) {$E_c$};

				\draw[draw=gray!60, thick] (-8,8) -- (8,-8);
				\node[above, gray!60] at (3.5, -2.5) {$L_{ac}$};
				\draw[draw= darkgray!95, thick] (-8,-8) -- (8, 8);
				\node[above, darkgray!95] at (9, 7) {$L_{bd}$};

				\draw[draw=gray!60, thick] (5,9) .. controls (0,6) .. (-5, 9);
				\node[above, gray!60] at (0, 4.5) {$L_{ad}$};
				\draw[draw=gray!60, thick] (-5,9) .. controls (-7,10) and (-10,9) .. (-13,-2);
				\node[above, gray!60] at (0, -6.5) {$L_{bc}$};

				\draw[draw=gray!60, thick] (5,-9) .. controls (0,-6) .. (-5, -9);
				\draw[draw=gray!60, thick] (-5,-9) .. controls (-7,-10) and (-10,-9) .. (-13,2);

				\draw[draw=darkgray!95, thick] (9,5) .. controls (6,0) .. (9,-5);
				\draw[draw=darkgray!95, thick] (9,-5) .. controls (10,-7) and (9,-10) .. (-2,-13);
				\node[above, darkgray!95] at (10.5, -8.5) {$L_{cd}$};
				\draw[draw=darkgray!95, thick] (-9,5) .. controls (-6,0) .. (-9,-6);
				\draw[draw=darkgray!95, thick] (-10,-8) .. controls (-9,-10) .. (2,-13);
				\node[above, darkgray!95] at (-11, -11) {$L_{ab}$};

				\draw [fill=black] (3.7,8.2) circle [radius=0.35];
				\draw [fill=black] (8.3, -3.7) circle [radius=0.35];
				\draw [fill=black] (-3.7,-8.2) circle [radius=0.35];
				\draw [fill=black] (-8.3, 3.8) circle [radius=0.35];
			\end{scope}
		\end{tikzpicture}
	\end{center}
	\caption{Configuration of curves on $S_1$ and images of exceptional locus of $\pi \colon S_2 \to S_1$.} \label{Fig1}
\end{figure}

For simplicity, we use the same notation for a curve and its strict transform under a birational morphism.
Notice that, as drawn in the picture, the lines $L_{ad}$ and $L_{bc}$ meet at infinity, and the same is true for $L_{ab}$ and $L_{cd}$.
Next, let $\pi \colon S_2 \to S_1$ be the surface obtained by blowing up the points
$E_a \cap L_{ab}$, $E_b \cap L_{bc}$, $E_c \cap L_{cd}$ and $E_d \cap L_{ad}$ (in Figure \ref{Fig1} these points are drawn as black circles). We call $F_a$, $F_b$, $F_c$ and $F_d$ the respective exceptional divisors.
Notice that on $S_2$ the sets $\Gamma_1= \left\{E_a, L_{ad}, L_{bc}, E_c\right\}$ and $\Gamma_2= \left\{ E_d, L_{cd}, L_{ab}, E_b \right\}$ are chains of four $(-2)$ curves.
Since we started from $\mathbb{P}_k^2$ and performed eight blow ups, we get that $K_{S_2}^2 = K_{\mathbb{P}_k ^2} - 8 = 1$.
The computations in \cite[Example B.12]{Lac20} show that, since the characteristic of $k$ is $5$, there is a rational cuspidal curve $D=C_2$ in the pencil $\left\{C_t\right\} \subset |-K_{\mathbb{P}^2}|$. Note that the strict transform of $D$ is disjoint from $\Gamma_1$ and $\Gamma_2$ and that $D \in |-K_{S_2}|$.
Starting from $S_2$, take a log resolution of the cusp of $D$ by blowing-up three times. We get a surface $V$ with three exceptional rational curves $G_1$, $G_2$,
$G_3$ and the following intersection numbers:
\[G_1^2=-3,\ G_2^2=-2,\ G_3^2=-1,\ G_1 \cdot G_2=0,\ G_1 \cdot G_3=1,\ G_2 \cdot G_3=1.\]
Note moreover that
$$	 D^2=-5,\ D\cdot F_a=1,\ D \cdot F_b=1,\\
D\cdot G_1=0,\ D\cdot G_2=0,\ D\cdot G_3=1,\ K_V^2=-2.
$$

The configuration of the curves on $V$ is illustrated in Figure 2.
We say that a singular point $p$ on a surface is of type $(n)$ if the exceptional divisor of its minimal resolution is a smooth rational curve of degree $-n$.

\begin{figure}[h]
	\begin{center}
		\begin{tikzpicture}[scale=.4]

			\begin{scope}[]

				\draw[draw=black, thick] (0,-5) -- (0,5);
				\draw[draw=black, thick] (-2,-2.5) -- (2,-2.5);
				\draw[draw=black, thick] (2,0) -- (-2,0);
				\draw[draw=black, thick] (2,2.5) -- (-2,2.5);

				\draw[draw=black, dotted, thick] (-7,5) -- (-5, 1);
				\node[above, black] at (-7, 3) {$E_a$};
				\draw[draw=gray!60, thick] (-5,3) -- (-7, -1) ;
				\node[above, gray!60] at (-4.5, 3) {$L_{ad}$};
				\draw[draw=gray!60, thick] (-7,1) -- (-5, -3) ;
				\node[above, gray!60] at (-4.5, -4.5) {$L_{bc}$};
				\draw[draw=black, dotted, thick] (-5, -1) -- ( -7, -5 ) ;
				\node[above, black] at (-7, -4) {$E_c$};
				\node [below] at (-7,-5.2) {$\Gamma_1$};

				\draw[draw=black, dotted, thick] (7,5) -- (5, 1);
				\node[above, black] at (7, 3) {$E_b$};
				\draw[draw=darkgray!95, thick] (5,3) -- (7, -1) ;
				\node[above, darkgray!95] at (4.5, 3) {$L_{ab}$};
				\draw[draw=darkgray!95, thick] (7,1) -- (5, -3) ;
				\node[above, darkgray!95] at (4.5, -4.5) {$L_{cd}$};
				\draw[draw=black, dotted, thick] (5, -1) -- (7, -5 ) ;
				\node[above, black] at (7, -4) {$E_d$};
				\node [below] at (7,-5.2) {$\Gamma_2$};

				\node [below] at (0,-5.2) {$G_3$};
				\node [below] at (1,-2.5) {$G_2$};
				\node [below] at (1,0) {$G_1$};
				\node [below] at (1,2.5) {$D$};

			\end{scope}

			\draw[->] (9,0) -- (17, 0);
			\node at (13,0) [above] {$\psi$};

			\begin{scope}[shift={(23,0)}]

				\draw[draw=black, thick] (0,-5) -- (0,5);

				\draw [fill=black] (4,0) circle [radius=0.2];
				\node [below] at (4,-.5) {$A_4$};

				\draw [fill=black] (-4,0) circle [radius=0.2];
				\node [below] at (-4,-.5) {$A_4$};

				\node [below] at (0,-5.2) {$G_3^T$};
				\node [below] at (1,-1.7) {$(2)$};
				\draw [fill=black] (0,-2.5) circle [radius=0.2];
				\draw [fill=black] (0,0) circle [radius=0.2];
				\node [below] at (1,.8) {$(3)$};
				\node [below] at (1,3.3) {$(5)$};
				\draw [fill=black] (0,2.5) circle [radius=0.2];

			\end{scope}

		\end{tikzpicture}
	\end{center}
	\caption{Arrangement of curves on $V$ and the type of singularities on $T$.} \label{Fig2}
\end{figure}

Finally, we consider the birational morphism $\psi \colon V \to T$ which contracts exactly the curves $\Gamma_1, \Gamma_2, G_1, G_2, D$ also illustrated in Figure \ref{Fig2}.
We denote $F_{a}^T$, $F^{T}_b$, $F^{T}_c$, $F^{T}_d$ and $G_3 ^T$ the image via $\psi$ of $F_a$, $F_b$, $F_c$, $F_d$ and $G_3$ respectively.

\begin{lem}
	$T$ is a surface of Picard number one with only klt singularities.
\end{lem}
\begin{proof}
	The singularities on $T$ are two $A_4$ points, one $A_1$ point, one $(3)$ point and one $(5)$ point, so they are klt.
	As $V$ has Picard rank 12, it is clear that $T$ has Picard rank one.
	Since the Picard rank is one, it is sufficient to compute the intersection of an effective curve with the canonical class to verify $T$ is del Pezzo. By projection formula
	$$K_T\cdot G^T_{3} = \psi^*(K_T) \cdot G_3 = (K_V+ \frac{1}{3}G_1 + \frac{3}{5}D)\cdot G_3= -1 + 1/3+3/5 <0,$$ as desired.
\end{proof}

\subsection{A counterexample to the Kawamata-Viehweg vanishing theorem} \label{ss-counterexample}

In the following we construct a counterexample to the Kodaira vanishing theorem for a Weil divisor $A$ on $T$.
The choice of the specific Weil divisor was motivated by the fact that it passes through all the singular points and it has maximal Cartier index at each point.

Using the same notations of Subsection \ref{ss-construction}, we start with the following computations:

\[ \psi^{*}F_a^T= F_a+\frac{4}{5}E_a+\frac{3}{5}L_{ad}+\frac{2}{5}L_{bc}+\frac{1}{5}E_c+\frac{3}{5}E_b+\frac{6}{5}L_{ab}+\frac{4}{5}L_{cd}+\frac{2}{5}E_d+\frac{1}{5}D,\]
\[ \psi^*F_b^T=F_b+\frac{3}{5}E_c+\frac{6}{5}L_{bc}+\frac{4}{5}L_{ad}+\frac{2}{5}E_a+\frac{1}{5}E_d+\frac{2}{5}L_{cd}+\frac{3}{5}L_{ab}+\frac{4}{5}E_b+\frac{1}{5}D, \]

\[\psi^*G_{3}^T=G_3+\frac{1}{3}G_1+\frac{1}{2}G_2+\frac{1}{5}D.\]

\begin{prop}\label{p-failure-vanishing}
	Let $A=G_3^T+F_a^T-F_b^T$. Then $A$ is an ample $\Q$-Cartier Weil divisor and
	\[ H^1(T, \mathcal{O}_T(-A)) \neq 0. \]
\end{prop}

\begin{proof}
	We first verify that $A$ is ample.
	We introduce $B:=-A=F_b^T-F_a^T-G_3^T$
	and we compute its pull-back to $V$:
	\[\psi^*B=F_b-F_a-G_3-\frac{2}{5}E_a-\frac{3}{5}L_{ab}-\frac{2}{5}L_{cd}+\frac{2}{5}E_c-\frac{1}{5}E_d\]
	\[+\frac{1}{5}L_{ad}+\frac{4}{5}L_{bc}+\frac{1}{5}E_b-\frac{1}{3}G_1-\frac{1}{2}G_2-\frac{1}{5}D.\]
	Since $T$ has Picard rank one and $B \cdot F_a^T=\psi^*B \cdot F_a=-\frac{1}{5}$ by projection formula, we conclude $B$ is anti-ample.

	We now verify that $H^1(T, \mathcal{O}_T(B)) \neq 0$, using the Riemann-Roch theorem for surfaces.
	To this end we write
	\[\lfloor{ \psi^*B \rfloor}=F_b-F_a-G_3-E_a-L_{ab}-L_{cd}-E_d-G_1-G_2-D,\]
	and we compute its intersection numbers with $\psi$-exceptional divisors:
	$$ \lfloor{\psi^*B \rfloor} \cdot L_{ab}=0,\lfloor{\psi^*B \rfloor} \cdot L_{bc}=1, \lfloor{\psi^*B \rfloor} \cdot L_{cd}=0,
	\lfloor{\psi^*B \rfloor} \cdot L_{ad}=-1,$$
	$$		\lfloor{\psi^*B \rfloor} \cdot E_a=1,
	\lfloor{\psi^*B \rfloor} \cdot E_b=0,
	\lfloor{\psi^*B \rfloor} \cdot E_c=0,
	\lfloor{\psi^*B \rfloor} \cdot E_d=1, $$
	$$		\lfloor{\psi^*B \rfloor} \cdot D=4,
	\lfloor{\psi^*B \rfloor} \cdot G_1=2,
	\lfloor{\psi^*B \rfloor} \cdot G_2=1. $$
	As a consequence one sees that $\lfloor{\psi^*B \rfloor}-(K_V+\frac{1}{3}G_1+\frac{3}{5}D+\frac{1}{2}L_{ad})=\lfloor{\psi^*B \rfloor}-(\psi^*K_T+\frac{1}{2}L_{ad})$ is $\psi$-nef and $\psi$-big.
	Thus by the relative Kawamata-Viehweg vanishing theorem for surfaces (see \cite[Theorem 3.3]{Tan18}) we deduce $R^i\psi_*\mathcal{O}_V(\lfloor{\psi^*B \rfloor})=0$ for $i>0$ and therefore $h^i(V, \lfloor{\psi^*B \rfloor})=h^i(T, \mathcal{O}_T(B))$ by the Leray spectral sequence.
	To apply Riemann-Roch we need the self-intersection of $\lfloor{\psi^*B \rfloor}$. To this end we also compute the following intersection numbers:
	$$	\lfloor{\psi^*B \rfloor} \cdot F_a=-2, \lfloor{\psi^*B \rfloor} \cdot F_b=-2, \lfloor{\psi^*B \rfloor} \cdot G_3=-2.
	$$
	A straightforward computation now shows $\lfloor{\psi^*B \rfloor}^2=-7$ and therefore by the Riemann-Roch theorem we conclude
	\[ \chi(V, \lfloor{\psi^*B \rfloor}) = 1+\frac{1}{2}(\lfloor{\psi^*B \rfloor}^2-K_V \cdot \lfloor{\psi^*B \rfloor}) = 1+\frac{1}{2}(-7+3)<0,  \]
	thus showing that $h^1(V, \lfloor{\psi^*B \rfloor})=h^1(T, \mathcal{O}_T(B)) \neq 0$.
\end{proof}

The following corollary shows that the lower bound on the characteristic for the liftability of a log resolution of a klt del Pezzo surfaces to the ring of second Witt vectors in Theorem \ref{liftdelpezzo} is optimal.

\begin{cor}
	The surface $T$ does not admit a log resolution $\mu \colon V \to T$ such that the log pair $(V, \text{Exc}(\mu))$ lifts to $W_2(k)$.
\end{cor}

\begin{proof}
	Since $H^1(T, \mathcal{O}_T(-A))=H^1(T, \mathcal{O}_T(K_T+A)) \neq 0$, the surface $T$ cannot admit a log resolution lifting to $W_2(k)$ by Lemma \ref{l-vanishing-W2}.
\end{proof}

\subsection{A klt threefold singularity which is not rational}

With the same notation as in Subsection \ref{ss-counterexample}, we consider the ample $\Q$-Cartier Weil divisor $A$ on the Picard rank one klt del Pezzo surface $T$.
Let
\[ X:= C_a(T, \mathcal{O}_T(A))=\text{Spec}_k \bigoplus_{m \geq 0} H^0(T, \mathcal{O}_T(mA)) \]
be the affine cone over $T$ induced by $A$.
Let us denote by $v$ the vertex of the cone, \emph{i.e.} the closed subscheme defined by the ideal $\bigoplus_{m \geq 1} H^0(T, \mathcal{O}_T(mA))$.

\begin{thm}\label{t-notCM}
	The variety $X$ is a normal $\Q$-factorial klt threefold which is not Cohen-Macaulay (and thus not rational).
\end{thm}

\begin{proof}
	By \cite[Proposition 2.4, (2)]{Ber}, $X$ is $\Q$-factorial.
	By inversion of adjunction (see \cite[Corollary 1.5]{HW}), we can apply the proof of \cite[Proposition 2.5, (3)]{Ber} to conclude that $X$ has klt singularities.
	Using \cite[Proposition 2.6]{Ber}, we have
	\[ H^2_v(X, \mathcal{O}_X) \simeq \bigoplus_{m \in \mathbb{Z}} H^{1} (T, \mathcal{O}_T(mA)) \neq 0,  \]
	by Proposition \ref{p-failure-vanishing}.
	Thus we conclude that $X$ is not Cohen-Macaulay.
\end{proof}

\input references.tex

\end{document}

config: arxiv


TITLE: Finding modular inverse (wrong approach)
QUESTION [1 upvotes]: I'm trying to find the modular inverse of $$30 \pmod{7} $$ I have tried using the Euclidean algorithm and it gave me the right answer, which is $x \equiv 6 \pmod{7} $. However, I tried using another approach that I thought would be simpler, but it resulted in a wrong answer. These were my steps:
Suppose x is the modular inverse of 30 mod 7. $$30x \equiv 1 \pmod{7} $$
$$(7*4 + 2)x \equiv 1 \pmod{7} $$
$$2x \equiv 1 \pmod{7}$$
<- I have a feeling it's the previous line of simplification that's causing the problem.) So the inverse of 2 mod 7 is 4. Thus the resulting answer is $x \equiv 4 \pmod{7} $, which is wrong. Could anyone point out what is the problem here?

REPLY [3 votes]: First, write $\;30=2\pmod 7\;$ , and now use the Euclidean algorithm with this, which is way easier.
By the way, the answer indeed is $\;4\;$ , since $\;30\cdot4=120=1+17\cdot7\;$ , or simpler: $\;2\cdot4=1+7\;$


TITLE: Prove uniform boundedness of probability density function of averaged i.i.d. random variables
QUESTION [1 upvotes]: Let $X_1,...,X_n$ be i.i.d. random variables with probability density function $f(x)$ uniformly bounded by a constant $B$. Now I want to show, the averaged $\bar{X}$ defined by
\begin{equation}
\bar{X}=\frac{\sum_{i=1}^n X_i}{\sqrt{n}}
\end{equation}
also has probability density function uniformly bounded by some constant which is independent of n.
By central limit theorem we know it converges to standard normal distribution, so seem this should be true, however I couldn’t prove it by simple inductive argument.

REPLY [1 votes]: The 1954 book Limit distributions for sums of independent random variables by Gnedenko and Kolmogorov gives , in section 46, "Limit Theorem for Densities", a sufficient condition for the  the central limit theorem to hold for densities: if some convolution product of the density is in an $L^r$ space, for some $r\in(1,2]$, the desired convergence holds.
In the case at hand, $r=2$ suffices: if $f$ is bounded by $B$ then $\int f^2 \le B\int f = B<\infty$, so $f\in L^2$ and the convergence holds.
Ibragimov and Linnik's 1971 Independent and stationary sequences of random variables has a straightforward proof.
See also Bhattacharya and Rao, Normal Approximation and Asymptotic Expansions, section 19, for a vector version of this result.
Thanks to antkam for finding a big mistake in an earlier version of this post.


text:
theory "Set-Cpo"
imports HOLCF
begin

default_sort type

instantiation set :: (type) below
begin
  definition below_set where "(\<sqsubseteq>) = (\<subseteq>)"
instance..
end

instance set :: (type) po
  by standard (auto simp add: below_set_def)

lemma is_lub_set:
  "S <<| \<Union>S"
  by(auto simp add: is_lub_def below_set_def is_ub_def)

lemma lub_set: "lub S = \<Union>S"
  by (metis is_lub_set lub_eqI)

instance set  :: (type) cpo
  by standard (rule exI, rule is_lub_set)

lemma minimal_set: "{} \<sqsubseteq> S"
  unfolding below_set_def by simp

instance set  :: (type) pcpo
  by standard (rule+, rule minimal_set)

lemma set_contI:
  assumes  "\<And> Y. chain Y \<Longrightarrow> f (\<Squnion> i. Y i) = \<Union> (f ` range Y)"
  shows "cont f"
proof(rule contI)
  fix Y :: "nat \<Rightarrow> 'a"
  assume "chain Y"
  hence "f (\<Squnion> i. Y i) = \<Union> (f ` range Y)" by (rule assms)
  also have "\<dots> = \<Union> (range (\<lambda>i. f (Y i)))" by simp
  finally
  show "range (\<lambda>i. f (Y i)) <<| f (\<Squnion> i. Y i)" using is_lub_set by metis
qed

lemma set_set_contI:
  assumes  "\<And> S. f (\<Union>S) = \<Union> (f ` S)"
  shows "cont f"
  by (metis set_contI assms is_lub_set  lub_eqI)

lemma adm_subseteq[simp]:
  assumes "cont f"
  shows "adm (\<lambda>a. f a \<subseteq> S)"
by (rule admI)(auto simp add: cont2contlubE[OF assms] lub_set)

lemma adm_Ball[simp]: "adm (\<lambda>S. \<forall>x\<in>S. P x)"
  by (auto intro!: admI  simp add: lub_set)

lemma finite_subset_chain:
  fixes Y :: "nat \<Rightarrow> 'a set"
  assumes "chain Y"
  assumes "S \<subseteq> \<Union>(Y ` UNIV)"
  assumes "finite S"
  shows "\<exists>i. S \<subseteq> Y i"
proof-
  from assms(2)
  have "\<forall>x \<in> S. \<exists> i. x \<in> Y i" by auto
  then obtain f where f: "\<forall> x\<in> S. x \<in> Y (f x)" by metis

  define i where "i = Max (f ` S)"
  from \<open>finite S\<close>
  have "finite (f ` S)" by simp
  hence "\<forall> x\<in>S. f x \<le> i" unfolding i_def by auto
  with chain_mono[OF \<open>chain Y\<close>]
  have "\<forall> x\<in>S. Y (f x) \<subseteq> Y i" by (auto simp add: below_set_def)
  with f
  have "S \<subseteq> Y i" by auto
  thus ?thesis..
qed

lemma diff_cont[THEN cont_compose, simp, cont2cont]:
  fixes S' :: "'a set"
  shows  "cont (\<lambda>S. S - S')"
by (rule set_set_contI) simp

end

subset_name: curated