# Asymptotic vanishing of syzygies of algebraic varieties

Jinhyung Park  Department of Mathematical Sciences, KAIST, 291 Daehak-ro,
Yuseong-gu, Daejeon 34141, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

The purpose of this paper is to prove Ein–Lazarsfeld’s conjecture on
asymptotic vanishing of syzygies of algebraic varieties. This result, together
with Ein–Lazarsfeld’s asymptotic nonvanishing theorem, describes the overall
picture of asymptotic behaviors of the minimal free resolutions of the graded
section rings of line bundles on a projective variety as the positivity of the
line bundles grows. Previously, Raicu reduced the problem to the case of
products of three projective spaces, and we resolve this case here.

J. Park was partially supported by the National Research Foundation (NRF)
funded by the Korea government (MSIT) (NRF-2021R1C1C1005479).

##  1\. Introduction

Throughout the paper, we work over an algebraically closed field $\Bbbk$ of
arbitrary characteristic. Let $X$ be a projective variety of dimension $n$,
and $L$ be a very ample line bundle on $X$ which gives rise to an embedding

$X\subseteq\mathbb{P}H^{0}(X,L)=\mathbb{P}^{r},$

where $r=h^{0}(X,L)-1$. Denote by $S$ the homogeneous coordinate ring of
$\mathbb{P}^{r}$. Fix a coherent sheaf $B$ on $X$, and let

$R=R(X,B;L):=\bigoplus_{m\in\mathbb{Z}}H^{0}(X,B\otimes L^{m})$

be the graded section $S$-module of $B$ associated to $L$. By the Hilbert
syzygy theorem, $R$ has a minimal free resolution

$\textstyle{0}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
E_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
0}$

where

$E_{p}=\bigoplus_{q}K_{p,q}(X,B;L)\otimes_{\Bbbk}S(-p-q).$

The _Koszul cohomology_ group $K_{p,q}(X,B;L)$ is the space of _$p$ -th
syzygies of weight $q$_. When $B=\mathscr{O}_{X}$, we set
$K_{p,q}(X,L):=K_{p,q}(X,\mathscr{O}_{X};L)$. After the pioneering work of
Green [21, 22], there has been a considerable amount of work to understand
vanishing and nonvanishing of $K_{p,q}(X,B;L)$.

We say that $L$ satisfies the _property $N_{k}$_ if $K_{0,1}(X,L)=0$ and
$K_{p,q}(X,L)=0$ for $0\leq p\leq k$ and $q\geq 2$. The property $N_{0}$ means
that $X\subseteq\mathbb{P}^{r}$ is projectively normal, and the property
$N_{1}$ means that the defining ideal of $X$ in $\mathbb{P}^{r}$ is generated
by quadratic polynomials. Thus the property $N_{k}$ provides a natural
framework to generalize classical results on defining equations of algebraic
varieties to the results on their syzygies. Along this line, Green proved that
if $X$ is a smooth projective complex curve of genus $g$ and $\deg L\geq
2g+1+k$, then $L$ satisfies the property $N_{k}$ (see [21, Theorem (4.a.1)]).
Green’s celebrated theorem has stimulated further work in this direction, and
several analogous statements for higher dimensional algebraic varieties have
been established, e.g. [7, 8, 11, 16, 20, 22, 27]. On the other hand,
Green–Lazarsfeld [23, 24] and Ottaviani–Paoletti [26] called attention to the
failure of the property $N_{k}$. The main result of [26] asserts that

$K_{p,2}(\mathbb{P}^{2},\mathscr{O}_{\mathbb{P}^{2}}(d))\neq 0~{}~{}\text{ for
$3d-2\leq p\leq r_{d}-2$},$

where $r_{d}=h^{0}(\mathbb{P}^{2},\mathscr{O}_{\mathbb{P}^{2}}(d))-1$. In
particular, $\mathscr{O}_{\mathbb{P}^{2}}(d)$ does not satisfy the property
$N_{3d-2}$. As $r_{d}\approx d^{2}/2$, the property $N_{k}$ for
$\mathscr{O}_{\mathbb{P}^{2}}(d)$ describes only a small fraction of the
syzygies of the $d$-th Veronese embedding of $\mathbb{P}^{2}$.
Eisenbud–Green–Hulek–Popescu observed in [18, Proposition 3.4] that a similar
phenomenon occurs for other smooth projective surfaces, and Ein–Lazarsfeld
proved in [12, Theorem A] that this always happens for all smooth projective
varieties.

It is an interesting problem to describe the overall asymptotic behaviors of
$K_{p,q}(X,B;L)$ as the positivity of $L$ grows. This type of question was
first suggested by Green [21, Problem 5.13] and also considered by
Ein–Lazarsfeld [11, Problem 4.4]. To set the stage for asymptotic syzygies of
algebraic varieties, assume that $X$ is smooth and $B$ is a line bundle, and
let

$L_{d}:=\mathscr{O}_{X}(dA+P)~{}~{}\text{ for an integer $d\geq 1$,}$

where $A$ is an ample divisor and $P$ is an arbitrary divisor on $X$. We
suppose that $d$ is sufficiently large, so in particular, $L_{d}$ is very
ample. Put $r_{d}:=h^{0}(X,L_{d})-1$. Elementary considerations of
Castelnuovo–Mumford regularity show that

$K_{p,q}(X,B;L_{d})=0~{}~{}\text{ for $q\geq n+2$}.$

When $q=0$ or $n+1$, [12, Proposition 5.1 and Corollary 5.2] state that

$\begin{array}[]{l}K_{p,0}(X,B;L_{d})\neq 0~{}\Longleftrightarrow~{}0\leq
p\leq h^{0}(B)-1;\\\ K_{p,n+1}(X,B;L_{d})\neq
0~{}\Longleftrightarrow~{}r_{d}-n-h^{0}(X,K_{X}-B)+1\leq p\leq
r_{d}-n.\end{array}$

The main issue is then to study vanishing and nonvanishing of
$K_{p,q}(X,B;L_{d})$ for $1\leq q\leq n$. Motivated by the nonvanishing
results of [18, 26], Ein–Lazarsfeld established the _asymptotic nonvanishing
theorem_ in [12, Theorem 4.1]: For each $1\leq q\leq n$, there are constants
$C_{1},C_{2}>0$ such that if $d$ is sufficiently large, then

$K_{p,q}(X,B;L_{d})\neq 0~{}~{}\text{ for $C_{1}d^{q-1}\leq p\leq
r_{d}-C_{2}d^{n-1}$}.$

If furthermore $H^{i}(X,B)=0$ for $1\leq i\leq n-1$, then

$K_{p,q}(X,B;L_{d})\neq 0~{}~{}\text{ for $C_{1}d^{q-1}\leq p\leq
r_{d}-C_{2}d^{n-q}$};$

in this case with $P=0$, a quick proof is provided in [10] with effective
range of $p$. It is worth noting that Zhou adapted the arguments in [12] to
show that the asymptotic nonvanishing theorem also holds for singular
projective varieties with a relaxed assumption on $B$ (see [12, Remark 4.2]).
The influential paper [12] opens the door to research on the asymptotic
behaviors of $K_{p,q}(X,B;L_{d})$ for $d$ increasing. We refer to [14] for
survey on recent progress on asymptotic syzygies of algebraic varieties.

It is very natural to ask whether the $K_{p,q}(X,B;L_{d})$ vanish for the
values of $p$ outside the range in the statement of the asymptotic
nonvanishing theorem. Ein–Lazarsfeld conjectured that this is indeed the case
for $p<O(d^{q-1})$ (see [12, Conjecture 7.1], [14, Conjecture 1.10]). We
confirm this _asymptotic vanishing conjecture_ in a more general setting.

######  Theorem 1.1.

Let $X$ be a projective variety of dimension $n$, and $B$ be a coherent sheaf
on $X$. Fix an ample divisor $A$ and an arbitrary divisor $P$ on $X$, and put
$L_{d}:=\mathscr{O}_{X}(dA+P)$ for an integer $d\geq 1$. For each $1\leq q\leq
n+1$, there is a constant $C>0$ depending on $X,A,B,P$ such that if $d$ is
sufficiently large, then

$K_{p,q}(X,B;L_{d})=0~{}~{}\text{ for $0\leq p\leq Cd^{q-1}$}.$

We give some remarks on vanishing of $K_{p,q}(X,B;L_{d})$ for large $p$. For
simplicity, we assume that $X$ is smooth and $B$ is a vector bundle. If
$H^{i}(X,B)=0$ for $1\leq i\leq n-1$, then the duality theorem (cf. [12,
Proposition 3.5], [21, Theorem 2.c.6]) says that

$K_{p,q}(X,B;L_{d})=K_{r_{d}-n-p,n+1-q}(X,B^{*}\otimes\omega_{X};L_{d})^{*}.$

Then our asymptotic vanishing theorem implies the following: For each $1\leq
q\leq n$, there is a constant $C>0$ such that if $d$ is sufficiently large,
then

$K_{p,q}(X,B;L_{d})=0~{}~{}\text{ for $p\geq r_{d}-Cd^{n-q}$}.$

However, if $H^{q-1}(X,B)\neq 0$ for some $2\leq q\leq n$, then
$K_{r_{d}-q+1,q}(X,B;L_{d})\neq 0$ for large $d$ (see [12, Remark 5.3]). When
$X$ is a smooth projective complex curve and $B$ is a line bundle, vanishing
of weight-one syzygies $K_{p,1}(X,B;L_{d})$ for large $p$ is determined by the
duality theorem and [13, Theorem B] (see also [29]). This implies
Green–Lazarsfeld’s gonality conjecture, and a higher dimensional
generalization is treated in the work of Ein–Lazarsfeld–Yang [15].

Shortly after the asymptotic vanishing conjecture was proposed, Raicu showed
in the appendix of [28] that the general case of the conjecture follows from
the case of products of three projective spaces. The case that $q=1$ in
Theorem 1.1 is trivial. To prove Theorem 1.1, it is more than enough to
establish the following:

######  Theorem 1.2.

Let $k\geq 1$ be an integer, $n_{1},\ldots,n_{k},d_{1},\ldots,d_{k}$ be
positive integers, and $b_{1},\ldots,b_{k}$ be integers. Set

$\begin{array}[]{l}X:=\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{k}},~{}B:=\mathscr{O}_{\mathbb{P}^{n_{1}}}(b_{1})\boxtimes\cdots\boxtimes\mathscr{O}_{\mathbb{P}^{n_{k}}}(b_{k}),~{}L:=\mathscr{O}_{\mathbb{P}^{n_{1}}}(d_{1})\boxtimes\cdots\boxtimes\mathscr{O}_{\mathbb{P}^{n_{k}}}(d_{k}),\end{array}$

and $b:=\min\\{b_{1},\ldots,b_{k}\\},~{}d:=\min\\{d_{1},\ldots,d_{k}\\}$. For
each $2\leq q\leq n_{1}+\cdots+n_{k}+1$, if $d+b\geq 0$, then

$K_{p,q}(X,B;L)=0~{}~{}\text{ for $0\leq p\leq(1/n_{1}!\cdots
n_{k}!)(d^{q-1}+bd^{q-2})$}.$

As $K_{p,q+1}(X,B;L)=K_{p,q}(X,B+L;L)$, it is reasonable to assume that $b<d$
in Theorem 1.2. But we do not need this assumption for the proof.

We give a sketch of the proof of Theorem 1.2 for the Veronese case. Let
$M_{\mathscr{O}_{\mathbb{P}^{n}}(d)}$ be the kernel of the evaluation map
$H^{0}(\mathbb{P}^{n},\mathscr{O}_{\mathbb{P}^{n}}(d))\otimes\mathscr{O}_{\mathbb{P}^{n}}\to\mathscr{O}_{\mathbb{P}^{n}}(d)$.
It is well known that

$K_{p,q}(\mathbb{P}^{n},\mathscr{O}_{\mathbb{P}^{n}}(d))=H^{q-1}(\mathbb{P}^{n},\wedge^{p+q-1}M_{\mathscr{O}_{\mathbb{P}^{n}}(d)}(d))~{}~{}\text{
for $p\geq 0$ and $q\geq 2$}.$

The main idea is to work on $\mathbb{P}^{n-1}\times\mathbb{P}^{1}$ instead of
$\mathbb{P}^{n}$ via the finite map
$\sigma\colon\mathbb{P}^{n-1}\times\mathbb{P}^{1}\to\mathbb{P}^{n}$ given by
$(\xi,z)\mapsto\xi+z$, where $\mathbb{P}^{n}$ is regarded as the Hilbert
scheme of $n$ points on $\mathbb{P}^{1}$ and $\sigma$ is the universal family.
Note that
$\sigma_{*}(\mathscr{O}_{\mathbb{P}^{n-1}}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(n-1))=\mathscr{O}_{\mathbb{P}^{n}}^{\oplus
n}$. For each $2\leq q\leq n+1$, the problem is equivalent to showing that

$H^{q-1}(\mathbb{P}^{n-1}\times\mathbb{P}^{1},\wedge^{p+q-1}\sigma^{*}M_{\mathscr{O}_{\mathbb{P}^{n}}(d)}\otimes\mathscr{O}_{\mathbb{P}^{n-1}}(d)\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(d+n-1))=0~{}~{}\text{
for $0\leq p\leq O(d^{q-1})$}.$

An advantage of working on $\mathbb{P}^{n-1}\times\mathbb{P}^{1}$ is that we
can use the following short exact sequence

$0\longrightarrow\bigoplus\mathscr{O}_{\mathbb{P}^{n-1}}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(-n)\longrightarrow\sigma^{*}M_{\mathscr{O}_{\mathbb{P}^{n}}(d)}\longrightarrow
M_{\mathscr{O}_{\mathbb{P}^{n-1}}(d)}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(d)\longrightarrow
0,$

which provides a way to proceed by induction on $n$. By considering the
natural filtration of
$\wedge^{p+q-1}\sigma^{*}M_{\mathscr{O}_{\mathbb{P}^{n}}(d)}$, we reduce the
problem to proving that

$H^{q-1}(\mathbb{P}^{n-1}\times\mathbb{P}^{1},\wedge^{i}M_{\mathscr{O}_{\mathbb{P}^{n-1}}(d)}(d)\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(a_{i}))=0~{}~{}\text{
for $0\leq i\leq O(d^{q-1})$},$

where $a_{i}=id-(p+q-1-i)n+d+n-1$. By induction on $n$, we can assume that
$H^{j}(\mathbb{P}^{n-1},\wedge^{i}M_{\mathscr{O}_{\mathbb{P}^{n-1}}(d)}(d))=0$
for $0\leq i\leq O(d^{j})$ and $j=q-2,q-1$. By the Künneth formula, it is
sufficient to check that

$H^{1}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(a_{i}))=0~{}~{}\text{ when
$i\geq O(d^{q-2})$}.$

But we have

$a_{i}=id+(d+2n+in-qn)-1-pn\geq O(d^{q-1})-1-pn\geq-1$

as soon as $0\leq p\leq O(d^{q-1})$. Thus
$K_{p,q}(\mathbb{P}^{n},\mathscr{O}_{\mathbb{P}^{n}}(d))=0$ for this range of
$p$. The same argument works for the general Segre–Veronese case.

There has been a great deal of attention to the syzygies of Veronese or
Segre–Veronese embeddings, e.g. [4, 6, 7, 8, 19, 25, 26, 28, 30]. The syzygies
of these varieties have connections to representation theory and
combinatorics. It would be exceedingly interesting to know whether the method
of the present paper could make progress on the study of the Veronese or
Segre–Veronese syzygies.

The paper is organized as follows. After reviewing basic necessary facts in
Section 2, we prove Theorem 1.2 in Section 3, where we also show Theorem 1.1
following Raicu’s argument in [28]. Section 4 is devoted to presenting some
open problems on asymptotic syzygies of algebraic varieties.

### Acknowledgements

The author is very grateful to Lawrence Ein, Sijong Kwak, and Wenbo Niu for
inspiring discussions and valuable comments, and he is indebted to Daniel
Erman for introducing some references. The author would like to thank the
referees for careful reading of the paper.

##  2\. Preliminaries

We collect basic facts which are used to prove the main theorems of the paper.

###  2.1. Koszul Cohomology

Let $X$ be a projective variety, $B$ be a coherent sheaf on $X$, and $L$ be a
very ample line bundle on $X$, which gives an embedding

$X\subseteq\mathbb{P}H^{0}(X,L)=\mathbb{P}^{r}.$

Let $S:=S(H^{0}(X,L))=\bigoplus_{m\geq 0}S^{m}H^{0}(X,L)$ be the homogeneous
coordinate ring of $\mathbb{P}^{r}$, and

$R=R(X,B;L):=\bigoplus_{m\in\mathbb{Z}}H^{0}(X,B\otimes L^{m})$

be the graded section $S$-module of $B$ associated to $L$. Denote by
$S_{+}\subseteq S$ the irrelevant maximal ideal, and define the _Koszul
cohomology group_ to be

$K_{p,q}(X,B;L):=\operatorname{Tor}_{p}^{S}(R,S/S_{+})_{p+q}.$

Then $R$ has a minimal free resolution

$\textstyle{0}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
E_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
0}$

where

$E_{p}=\bigoplus_{q}K_{p,q}(X,B;L)\otimes_{\Bbbk}S(-p-q).$

Notice that $K_{p,q}(X,B;L)$ is the vector space of _$p$ -th syzygies of
weight $q$_ and it is the cohomology of the Koszul-type complex

$\begin{array}[]{l}\wedge^{p+1}H^{0}(X,L)\otimes H^{0}(X,B\otimes
L^{q-1})\longrightarrow\wedge^{p}H^{0}(X,L)\otimes H^{0}(X,B\otimes
L^{q})\\\\[5.0pt] ~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{
}~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{
}~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{
}~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{
}~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{ }~{}\text{
}\longrightarrow\wedge^{p-1}H^{0}(X,L)\otimes H^{0}(X,B\otimes
L^{q+1}).\end{array}$

When $B=\mathscr{O}_{X}$, we set $K_{p,q}(X,L):=K_{p,q}(X,\mathscr{O}_{X};L)$.

Now, let $L$ be a globally generated line bundle on a projective variety $X$.
Consider the evaluation map

$\operatorname{ev}\colon H^{0}(X,L)\otimes\mathscr{O}_{X}\longrightarrow L,$

which is surjective since $L$ is globally generated. Denote by $M_{L}$ the
kernel bundle of the evaluation map $\operatorname{ev}$. Then we obtain a
short exact sequence of vector bundles on $X$:

(2.1) $0\longrightarrow M_{L}\longrightarrow
H^{0}(X,L)\otimes\mathscr{O}_{X}\longrightarrow L\longrightarrow 0.$

We use the following well-known fact to compute the Koszul cohomology group.

######  Proposition 2.1 (cf. [12, Corollary 3.3]).

Let $X$ be a projective variety, $B$ be a coherent sheaf on $X$, and $L$ be a
very ample line bundle on $X$. Assume that $H^{i}(X,B\otimes L^{m})=0$ for
$i>0$ and $m>0$. Fix $q\geq 2$. Then we have

$K_{p,q}(X,B;L)=H^{q-1}(X,\wedge^{p+q-1}M_{L}\otimes B\otimes L)~{}\text{ for
$p\geq 0$}.$

In particular, $K_{p,q}(X,B;L)=0$ for $0\leq p\leq p_{0}$ if and only if

$H^{q-1}(X,\wedge^{j}M_{L}\otimes B\otimes L)=0~{}~{}\text{ for $0\leq j\leq
p_{0}+q-1$}.$

###### Proof.

By taking wedge product of (2.1), we have a short exact sequence

(2.2)
$0\longrightarrow\wedge^{p+1}M_{L}\longrightarrow\wedge^{p+1}H^{0}(X,L)\otimes\mathscr{O}_{X}\longrightarrow\wedge^{p}M_{L}\otimes
L\longrightarrow 0.$

By using the Koszul-type complex and chasing through the diagram, we see that

$K_{p,q}(X,B;L)=H^{1}(X,\wedge^{p+1}M_{L}\otimes B\otimes L^{q-1})~{}~{}\text{
for $p\geq 0$ and $q\geq 2$}.$

See [3, Section 2.1] or [11, Section 1] for the complete proof. Now, from
(2.2), we find that

$H^{1}(X,\wedge^{p+1}M_{L}\otimes B\otimes
L^{q-1})=H^{2}(X,\wedge^{p+2}M_{L}\otimes B\otimes
L^{q-2})=\cdots=H^{q-1}(X,\wedge^{p+q-1}M_{L}\otimes B\otimes L),$

so the first assertion holds. Thus $K_{p,q}(X,B;L)=0$ for $0\leq p\leq p_{0}$
if and only if

$H^{q-1}(X,\wedge^{j}M_{L}\otimes B\otimes L)=0~{}~{}\text{ for $q-1\leq j\leq
p_{0}+q-1$}.$

But we get from (2.2) that

$H^{q-1}(X,\wedge^{i}M_{L}\otimes B\otimes
L)=H^{q-2}(X,\wedge^{i-1}M_{L}\otimes B\otimes
L^{2})=\cdots=H^{q-i-1}(X,B\otimes L^{i+1})=0$

for any $0\leq i\leq q-2$. Thus the second assertion holds. ∎

We refer to [3, 11, 12, 17, 21] for more details on syzygies and Koszul
cohomology.

###  2.2. Filtrations for Wedge Products

For a short exact sequence $0\to U\to V\to W\to 0$ of vector bundles on a
projective variety $X$ and an integer $k\geq 1$, there is a natural filtration

(2.3) $0=F^{0}\subseteq F^{1}\subseteq\cdots\subseteq F^{k}\subseteq
F^{k+1}=\wedge^{k}V$

such that

$F^{p+1}/F^{p}=\wedge^{k-p}U\otimes\wedge^{p}W~{}~{}\text{ for all $0\leq
p\leq k$. }$

######  Lemma 2.2.

Let $0\to U\to V\to W\to 0$ be a short exact sequence of vector bundles on a
projective variety $X$. If $H^{q}(X,\wedge^{k-p}U\otimes\wedge^{p}W)=0$ for
all $0\leq p\leq k$, then $H^{q}(X,\wedge^{k}V)=0$.

###### Proof.

By considering the natural filtration (2.3), we immediately obtain the lemma.
∎

###  2.3. Divided and Symmetric Powers

Let $V$ be a finite dimensional vector space over $\Bbbk$. For an integer
$n\geq 1$, the symmetric group $\mathfrak{S}_{n}$ naturally acts on the tensor
power $T^{n}V:=V^{\otimes n}$ by permuting the factors. The _divided power_ of
$V$ is the subspace

$D^{n}V:=\\{\omega\in T^{n}V\mid\sigma(\omega)=\omega~{}\text{for all
$\sigma\in\mathfrak{S}_{n}$}\\}\subseteq T^{n}V,$

while the _symmetric power_ $S^{n}V$ of $V$ is the quotient of $T^{n}V$ by the
subspace spanned by $\sigma(\omega)-\omega$ for all $\omega\in T^{n}V$ and
$\sigma\in\mathfrak{S}_{n}$. When $n=0$, we set $T^{0}V=D^{0}V=S^{0}V=\Bbbk$.
We have a natural identification

$D^{n}V=(S^{n}V^{*})^{*}.$

By composing the inclusion of $D^{n}V$ into $T^{n}V$ with the projection onto
$S^{n}V$, we have a natural map $D^{n}V\to S^{n}V$. This map is an isomorphism
in characteristic zero, but it may be neither injective nor surjective in
general. We can also define divided and symmetric powers of vector bundles on
projective varieties. We refer to [2, Section 3] for more details.

Let $0\to U\to V\to W\to 0$ be a short exact sequence of vector bundles on a
projective variety $X$ with $\operatorname{rank}W=1$. Let
$\pi\colon\mathbb{P}(V^{*})\to X$ be the natural projection. Note that
$\mathscr{O}_{\mathbb{P}(V^{*})}(-\mathbb{P}(U^{*}))=\mathscr{O}_{\mathbb{P}(V^{*})}(-1)\otimes\pi^{*}W^{*}$.
For $k\geq 0$, we have a short exact sequence on $\mathbb{P}(V^{*})$:

$0\longrightarrow\mathscr{O}_{\mathbb{P}(V^{*})}(k)\otimes\pi^{*}W^{*}\longrightarrow\mathscr{O}_{\mathbb{P}(V^{*})}(k+1)\longrightarrow\mathscr{O}_{\mathbb{P}(U^{*})}(k+1)\longrightarrow
0.$

By applying $\pi_{*}$, we get a short exact sequence on $X$:

$0\longrightarrow S^{k}V^{*}\otimes W^{*}\longrightarrow
S^{k+1}V^{*}\longrightarrow S^{k+1}U^{*}\longrightarrow 0.$

This construction was suggested by Lawrence Ein, and a purely algebraic
construction of this kind of exact sequence can be found in [1, Corollary
V.1.15]. By taking the dual, we obtain a short exact sequence on $X$:

(2.4) $0\longrightarrow D^{k+1}U\longrightarrow D^{k+1}V\longrightarrow
D^{k}V\otimes W\longrightarrow 0.$

We remark that $S^{k+1}V\to S^{k}V\otimes W$ may not be surjective in positive
characteristic.

Now, let $C$ be a smooth projective curve, and $L$ be a line bundle on $C$.
For an integer $k\geq 0$, the symmetric group $\mathfrak{S}_{k+1}$ naturally
acts on the $(k+1)$-th ordinary product $C^{k+1}$ of $C$ by permuting the
components, and the line bundle

$L^{\boxtimes k+1}:=\underbrace{L\boxtimes\cdots\boxtimes L}_{\text{$k+1$
times}}$

on $C^{k+1}$ descends to a line bundle $T_{k+1}(L)$ on the $(k+1)$-th
symmetric product $C_{k+1}$ of $C$ (see [16, Subsection 3.1]). Note that

$H^{0}(C_{k+1},T_{k+1}(L))=H^{0}(C^{k+1},L^{\boxtimes
k+1})^{\mathfrak{S}_{k+1}}=D^{k+1}H^{0}(C,L).$

If $C=\mathbb{P}^{1}$ and $L=\mathscr{O}_{\mathbb{P}^{1}}(d)$ with $d\geq 1$,
then $(\mathbb{P}^{1})_{n}=\mathbb{P}^{n}$ and
$T_{n}(\mathscr{O}_{\mathbb{P}^{1}}(d))=\mathscr{O}_{\mathbb{P}^{n}}(d)$. By
an arbitrary characteristic version of Hermite reciprocity (see [2, Remark
3.2]), we have

$\begin{array}[]{l}H^{0}(\mathbb{P}^{n},T_{n}(\mathscr{O}_{\mathbb{P}^{1}}(d)))~{}=~{}D^{n}H^{0}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(d))~{}=~{}D^{n}(S^{d}H^{0}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(1)))\\\\[5.0pt]
~{}~{}=~{}S^{d}(D^{n}H^{0}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(1)))~{}=~{}S^{d}H^{0}(\mathbb{P}^{n},T_{n}(\mathscr{O}_{\mathbb{P}^{1}}(1))).\end{array}$

###  2.4. Tautological Bundles on Projective Spaces

Recall that
$\sigma\colon\mathbb{P}^{n-1}\times\mathbb{P}^{1}\to\mathbb{P}^{n}$ is the
finite map of degree $n$ given by $(\xi,z)\mapsto\xi+z$ by viewing
$\mathbb{P}^{n}$ as the Hilbert scheme of $n$ points on $\mathbb{P}^{1}$. For
any integer $k$, the _tautological bundle_ on $\mathbb{P}^{n}$ is defined as

$E_{n,\mathscr{O}_{\mathbb{P}^{1}}(k)}:=\sigma_{*}(\mathscr{O}_{\mathbb{P}^{n-1}}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(k)),$

which is a vector bundle of rank $n$. The tautological bundles on symmetric
products of curves play an important role in the study of secant varieties of
curves (see [16]).

######  Lemma 2.3.

$E_{n,\mathscr{O}_{\mathbb{P}^{1}}(k)}$ is splitting if and only if $-1\leq
k\leq n-1$. In this case,

$E_{n,\mathscr{O}_{\mathbb{P}^{1}}(k)}=\mathscr{O}_{\mathbb{P}^{n}}^{\oplus
k+1}\oplus\mathscr{O}_{\mathbb{P}^{n}}(-1)^{\oplus n-1-k}.$

###### Proof.

Since $\sigma\colon\mathbb{P}^{n-1}\times\mathbb{P}^{1}\to\mathbb{P}^{n}$ is a
finite map, we have

$H^{i}(\mathbb{P}^{n},E_{n,\mathscr{O}_{\mathbb{P}^{1}}(k)}(m))=H^{i}(\mathbb{P}^{n-1}\times\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{n-1}}(m)\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(m+k)).$

for any $i\geq 0$ and $m\in\mathbb{Z}$. The Künneth formula shows that

$\begin{array}[]{l}H^{i}(\mathbb{P}^{n-1}\times\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{n-1}}(m)\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(m+k))\\\\[5.0pt]
=\big{(}H^{i-1}(\mathbb{P}^{n-1},\mathscr{O}_{\mathbb{P}^{n-1}}(m))\otimes
H^{1}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(m+k))\big{)}\oplus\big{(}H^{i}(\mathbb{P}^{n-1},\mathscr{O}_{\mathbb{P}^{n-1}}(m))\otimes
H^{0}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(m+k))\big{)}.\end{array}$

Then we see that

$\begin{array}[]{l}H^{i}(\mathbb{P}^{n},E_{n,\mathscr{O}_{\mathbb{P}^{1}}(k)}(m))=0~{}~{}\text{
for $2\leq i\leq n-2$ and $m\in\mathbb{Z}$},\\\\[5.0pt]
H^{1}(\mathbb{P}^{n},E_{n,\mathscr{O}_{\mathbb{P}^{1}}(k)}(m))=0~{}\text{for
$m\in\mathbb{Z}$}~{}\Longleftrightarrow~{}k\geq-1,\\\\[5.0pt]
H^{n-1}(\mathbb{P}^{n},E_{n,\mathscr{O}_{\mathbb{P}^{1}}(k)}(m))=0~{}\text{for
$m\in\mathbb{Z}$}~{}\Longleftrightarrow~{}k\leq n-1.\end{array}$

By the Horrocks criterion, the first assertion of the lemma follows. Now, we
observe that
$h^{0}(\mathbb{P}^{n},E_{n,\mathscr{O}_{\mathbb{P}^{1}}(k)})=k+1$, and
$h^{0}(\mathbb{P}^{n},E_{n,\mathscr{O}_{\mathbb{P}^{1}}(k)}(1))=n(k+2)=(n+1)(k+1)+n-1-k$.
This implies the second assertion of the lemma. ∎

######  Remark 2.4.

The following alternative approach to Lemma 2.3 was suggested by Lawrence Ein.
Let $D_{n}$ be the image of the injective map
$\mathbb{P}^{n-1}\times\mathbb{P}^{1}\to\mathbb{P}^{n}\times\mathbb{P}^{1}$
given by $(\xi,z)\mapsto(\xi+z,z)$. Note that
$\mathscr{O}_{\mathbb{P}^{n}\times\mathbb{P}^{1}}(-D_{n})=\mathscr{O}_{\mathbb{P}^{n}}(-1)\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(-n)$.
For any integer $k$, we have a short exact sequence on
$\mathbb{P}^{n}\times\mathbb{P}^{1}$:

$0\longrightarrow\mathscr{O}_{\mathbb{P}^{n}}(-1)\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(k-n)\longrightarrow\mathscr{O}_{\mathbb{P}^{n}}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(k)\longrightarrow\mathscr{O}_{\mathbb{P}^{n-1}}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(k)\longrightarrow
0.$

Let $p\colon\mathbb{P}^{n}\times\mathbb{P}^{1}\to\mathbb{P}^{n}$ be the
projection to the first component. When $-1\leq k\leq n-1$, by applying
$p_{*}$, we obtain a short exact sequence on $\mathbb{P}^{n}$:

$0\longrightarrow
H^{0}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(k))\otimes\mathscr{O}_{\mathbb{P}^{n}}\longrightarrow
E_{n,\mathscr{O}_{\mathbb{P}^{1}}(k)}\longrightarrow
H^{1}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(k-n))\otimes\mathscr{O}_{\mathbb{P}^{n}}(-1)\longrightarrow
0.$

Thus
$E_{n,\mathscr{O}_{\mathbb{P}^{1}}(k)}=(H^{0}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(k))\otimes\mathscr{O}_{\mathbb{P}^{n}})\oplus(H^{1}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(k-n))\otimes\mathscr{O}_{\mathbb{P}^{n}}(-1))$.
When $k\leq-2$ or $k\geq n$, it is easy to check that
$E_{n,\mathscr{O}_{\mathbb{P}^{1}}(k)}$ is not splitting.

######  Lemma 2.5.

Let $Y$ be a projective variety, and $\sigma\colon
Y\times\mathbb{P}^{n-1}\times\mathbb{P}^{1}\to Y\times\mathbb{P}^{n}$ be the
finite map given by $(y,\xi,z)\mapsto(y,\xi+z)$. If $M$ is a vector bundle on
$Y\times\mathbb{P}^{n}$, then

$H^{q}(Y\times\mathbb{P}^{n},M)=0~{}\Longleftrightarrow~{}H^{q}(Y\times\mathbb{P}^{n-1}\times\mathbb{P}^{1},\sigma^{*}M\otimes(\mathscr{O}_{Y\times\mathbb{P}^{n-1}}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(n-1)))=0$

for any $q\geq 0$.

###### Proof.

By Lemma 2.3 and the projection formula,
$\sigma_{*}\big{(}\sigma^{*}M\otimes(\mathscr{O}_{Y\times\mathbb{P}^{n-1}}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(n-1))\big{)}=M^{\oplus
n}$. Since $\sigma$ is a finite map, the assertion immediately follows. ∎

##  3\. Asymptotic Vanishing Theorem

The aim of this section is to prove Theorem 1.1. First, we construct a short
exact sequence of vector bundles, which allows us to give a quick proof of
Theorem 1.2 by induction on dimension. We then explain how one can deduce
Theorem 1.1 from Theorem 1.2.

###  3.1. Short Exact Sequence

Let $C$ be a smooth projective curve, and $L_{C}$ be a line bundle on $C$. Let
$Y$ be a smooth projective variety, and $L_{Y}$ be a line bundle on $Y$. Fix
an integer $k\geq 0$. Assume that the line bundle $L_{Y}\boxtimes
T_{k+1}(L_{C})$ on $Y\times C_{k+1}$ is globally generated. Notice that

$H^{0}(Y\times C_{k+1},L_{Y}\boxtimes T_{k+1}(L_{C}))=H^{0}(Y,L_{Y})\otimes
D^{k+1}H^{0}(C,L_{C}).$

We have a short exact sequence on $Y\times C_{k+1}$:

$0\longrightarrow M_{L_{Y}\boxtimes T_{k+1}(L_{C})}\longrightarrow
H^{0}(Y,L_{Y})\otimes D^{k+1}H^{0}(C,L_{C})\otimes\mathscr{O}_{Y\times
C_{k+1}}\longrightarrow L_{Y}\boxtimes T_{k+1}(L_{C})\longrightarrow 0.$

We can view $C_{k+1}=\\{\text{effective divisors of degree $k+1$ on $C$}\\}$
as the Hilbert scheme of $k+1$ points on $C$. Let

$\sigma\colon Y\times C_{k}\times C\longrightarrow Y\times C_{k+1}$

be the finite morphism given by $(y,\xi,z)\mapsto(y,\xi+z)$, and $p\colon
Y\times C_{k}\times C\rightarrow C$ be the projection to the last component.
By taking $\sigma^{*}$ of the above exact sequence on $Y\times C_{k+1}$, we
get a short exact sequence on $Y\times C_{k}\times C$:

$0\to\sigma^{*}M_{L_{Y}\boxtimes T_{k+1}(L_{C})}\to H^{0}(Y,L_{Y})\otimes
D^{k+1}H^{0}(C,L_{C})\otimes\mathscr{O}_{Y\times C_{k}\times C}\to
L_{Y}\boxtimes T_{k}(L_{C})\boxtimes L_{C}\to 0,$

By taking $p_{*}$ and considering (2.4), we get a short exact sequence on $C$:

$0\to H^{0}(Y,L_{Y})\otimes D^{k+1}M_{L_{C}}\to H^{0}(Y,L_{Y})\otimes
D^{k+1}H^{0}(C,L_{C})\otimes\mathscr{O}_{C}\to H^{0}(Y,L_{Y})\otimes
D^{k}H^{0}(C,L_{C})\otimes L_{C}\to 0.$

Then we obtain the following commutative diagram with exact sequences on
$Y\times C_{k}\times C$:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{L_{Y}\boxtimes
T_{k}(L_{C})}\boxtimes
L_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{p^{*}(H^{0}(Y,L_{Y})\otimes
D^{k+1}M_{L_{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{p^{*}(H^{0}(Y,L_{Y})\otimes
D^{k+1}H^{0}(C,L_{C})\otimes\mathscr{O}_{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{p^{*}(H^{0}(Y,L_{Y})\otimes
D^{k}H^{0}(C,L_{C})\otimes
L_{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\sigma^{*}M_{L_{Y}\boxtimes
T_{k+1}(L_{C})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(Y,L_{Y})\otimes
D^{k+1}H^{0}(C,L_{C})\otimes\mathscr{O}_{Y\times C_{k}\times
C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{Y}\boxtimes
T_{k}(L_{C})\boxtimes
L_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{M_{L_{Y}\boxtimes
T_{k}(L_{C})}\boxtimes
L_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$

Now, assume that $C=\mathbb{P}^{1}$ and
$L_{C}=\mathscr{O}_{\mathbb{P}^{1}}(d)$ with $d\geq 1$. For an integer $n\geq
1$, we have

$(\mathbb{P}^{1})_{n}=\mathbb{P}^{n},~{}~{}T_{n}(\mathscr{O}_{\mathbb{P}^{1}}(d))=\mathscr{O}_{\mathbb{P}^{n}}(d),~{}~{}M_{L_{Y}\boxtimes
T_{n}(\mathscr{O}_{\mathbb{P}^{1}}(d))}=M_{L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n}}(d)}.$

Since
$M_{\mathscr{O}_{\mathbb{P}^{1}}(d)}=\bigoplus\mathscr{O}_{\mathbb{P}^{1}}(-1)$,
it follows that

$D^{n}M_{\mathscr{O}_{\mathbb{P}^{1}}(d)}=\big{(}S^{n}M_{\mathscr{O}_{\mathbb{P}^{1}}(d)}^{*}\big{)}^{*}=\bigoplus\mathscr{O}_{\mathbb{P}^{1}}(-n).$

Then the left vertical short exact sequence in the above commutative diagram
gives a short exact sequence on $Y\times\mathbb{P}^{n-1}\times\mathbb{P}^{1}$:

(3.1)
$0\longrightarrow\bigoplus\mathscr{O}_{Y\times\mathbb{P}^{n-1}}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(-n)\longrightarrow\sigma^{*}M_{L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n}}(d)}\longrightarrow
M_{L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n-1}}(d)}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(d)\longrightarrow
0.$

When $Y$ is a point, the exact sequence (3.1) is

$0\longrightarrow\bigoplus\mathscr{O}_{\mathbb{P}^{n-1}}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(-n)\longrightarrow\sigma^{*}M_{\mathscr{O}_{\mathbb{P}^{n}}(d)}\longrightarrow
M_{\mathscr{O}_{\mathbb{P}^{n-1}}(d)}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(d)\longrightarrow
0.$

When $n=1$, the finite map $\sigma$ is an isomorphism and the exact sequence
(3.1) is

$0\longrightarrow\bigoplus\mathscr{O}_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(-1)\longrightarrow
M_{L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(d)}\longrightarrow
M_{L_{Y}}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(d)\longrightarrow 0.$

###  3.2. Case of Product of Projective Spaces

In this subsection, we prove Theorem 1.2. Recall that

$\begin{array}[]{l}X=\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{k}},~{}B=\mathscr{O}_{\mathbb{P}^{n_{1}}}(b_{1})\boxtimes\cdots\boxtimes\mathscr{O}_{\mathbb{P}^{n_{k}}}(b_{k}),~{}L=\mathscr{O}_{\mathbb{P}^{n_{1}}}(d_{1})\boxtimes\cdots\boxtimes\mathscr{O}_{\mathbb{P}^{n_{k}}}(d_{k}),\end{array}$

and $b=\min\\{b_{1},\ldots,b_{k}\\},~{}d=\min\\{d_{1},\ldots,d_{k}\\}$. Fix
$2\leq q\leq n_{1}+\cdots+n_{k}+1$. Our aim is to show that if $d+b\geq 0$,
then

$K_{p,q}(X,B;L)=0~{}~{}\text{ for $0\leq p\leq(1/n_{1}!\cdots
n_{k}!)(d^{q-1}+bd^{q-2})$.}$

We put

$Y:=\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{k-1}},~{}B_{Y}:=\mathscr{O}_{\mathbb{P}^{n_{1}}}(b_{1})\boxtimes\cdots\boxtimes\mathscr{O}_{\mathbb{P}^{n_{k-1}}}(b_{k-1}),~{}L_{Y}:=\mathscr{O}_{\mathbb{P}^{n_{1}}}(d_{1})\boxtimes\cdots\boxtimes\mathscr{O}_{\mathbb{P}^{n_{k-1}}}(d_{k-1}),$

and $n:=n_{k}$. Then

$X=Y\times\mathbb{P}^{n},~{}B=B_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n}}(b_{k}),~{}L=L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n}}(d_{k}).$

As $md+b\geq 0$ for any $m>0$, we have

$H^{i}(X,B\otimes L^{m})=0~{}~{}\text{ for $i>0$ and $m>0$}.$

By Proposition 2.1, for $p\geq 0$, we have

$K_{p,q}(X,B;L)=H^{q-1}\big{(}Y\times\mathbb{P}^{n},\wedge^{p+q-1}M_{L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n}}(d_{k})}\otimes((L_{Y}+B_{Y})\boxtimes\mathscr{O}_{\mathbb{P}^{n}}(d_{k}+b_{k}))\big{)}.$

We proceed by induction on $n_{1}+\cdots+n_{k}$. If $n_{1}+\cdots+n_{k}=1$,
then $q=2$ and the problem is to check the cohomology vanishing

$H^{1}\big{(}\mathbb{P}^{1},\wedge^{p+1}M_{\mathscr{O}_{\mathbb{P}^{1}}(d)}\otimes\mathscr{O}_{\mathbb{P}^{1}}(d+b)\big{)}=0~{}~{}\text{
for $0\leq p\leq d+b$.}$

As
$\wedge^{p+1}M_{\mathscr{O}_{\mathbb{P}^{1}}(d)}\otimes\mathscr{O}_{\mathbb{P}^{1}}(d+b)=\bigoplus\mathscr{O}_{\mathbb{P}^{1}}(d+b-p-1)$
and $d+b-p-1\geq-1$, the desired cohomology vanishing immediately follows.

Assume that $n_{1}+\cdots+n_{k}\geq 2$. Fix $0\leq p\leq(1/n_{1}!\cdots
n_{k}!)(d^{q-1}+bd^{q-2})$. By Lemma 2.5, it is sufficient to show the
cohomology vanishing on $Y\times\mathbb{P}^{n-1}\times\mathbb{P}^{1}$:

$H^{q-1}\big{(}\wedge^{p+q-1}\sigma^{*}M_{L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n}}(d_{k})}\otimes((L_{Y}+B_{Y})\boxtimes\mathscr{O}_{\mathbb{P}^{n-1}}(d_{k}+b_{k})\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(d_{k}+b_{k}+n-1))\big{)}=0,$

where $\sigma\colon Y\times\mathbb{P}^{n-1}\times\mathbb{P}^{1}\to
Y\times\mathbb{P}^{n}$ is the finite map given by $(y,\xi,z)\mapsto(y,\xi+z)$.
By considering the short exact sequence (3.1) and applying Lemma 2.2 to
$\wedge^{p+q-1}\sigma^{*}M_{L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n}}(d_{k})}$,
we can reduce the problem to proving the following:

$H^{q-1}\big{(}Y\times\mathbb{P}^{n-1}\times\mathbb{P}^{1},(\wedge^{i}M_{L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n-1}}(d_{k})}\otimes((L_{Y}+B_{Y})\boxtimes\mathscr{O}_{\mathbb{P}^{n-1}}(d_{k}+b_{k})))\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(a_{i})\big{)}=0$

for $0\leq i\leq p+q-1~{}(\leq(1/n_{1}!\cdots n_{k}!)(d^{q-1}+bd^{q-2})+q-1)$,
where

$a_{i}:=id_{k}-(p+q-1-i)n+d_{k}+b_{k}+n-1=i(d_{k}+n)+d_{k}+b_{k}+2n-qn-1-pn.$

By the Künneth formula, it is equivalent to showing that

(3.2)
$H^{q-1}\big{(}Y\times\mathbb{P}^{n-1},\wedge^{i}M_{L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n-1}}(d_{k})}\otimes((L_{Y}+B_{Y})\boxtimes\mathscr{O}_{\mathbb{P}^{n-1}}(d_{k}+b_{k}))\big{)}\otimes
H^{0}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(a_{i}))=0$

and

(3.3)
$H^{q-2}\big{(}Y\times\mathbb{P}^{n-1},\wedge^{i}M_{L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n-1}}(d_{k})}\otimes((L_{Y}+B_{Y}))\boxtimes\mathscr{O}_{\mathbb{P}^{n-1}}(d_{k}+b_{k})\big{)}\otimes
H^{1}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(a_{i}))=0$

for $0\leq i\leq p+q-1$.

By induction and Proposition 2.1, we can assume that

(3.4)
$H^{q-1}\big{(}Y\times\mathbb{P}^{n-1},\wedge^{i}M_{L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n-1}}(d_{k})}\otimes((L_{Y}+B_{Y})\boxtimes\mathscr{O}_{\mathbb{P}^{n-1}}(d_{k}+b_{k}))\big{)}=0$

for $0\leq i\leq(1/n_{1}!\cdots n_{k-1}!(n_{k}-1)!)(d^{q-1}+bd^{q-2})+q-1$
when $q\geq 2$, and

(3.5)
$H^{q-2}\big{(}Y\times\mathbb{P}^{n-1},\wedge^{i}M_{L_{Y}\boxtimes\mathscr{O}_{\mathbb{P}^{n-1}}(d_{k})}\otimes((L_{Y}+B_{Y})\boxtimes\mathscr{O}_{\mathbb{P}^{n-1}}(d_{k}+b_{k}))\big{)}=0$

for $0\leq i\leq(1/n_{1}!\cdots n_{k-1}!(n_{k}-1)!)(d^{q-2}+bd^{q-3})+q-2$
when $q\geq 3$. Observe that (3.2) immediately follows from (3.4). It only
remains to check (3.3). If $q=2$, then

$a_{i}=i(d_{k}+n)+d_{k}+b_{k}-1-np\geq d_{k}+b_{k}-1-np\geq d+b-1-np\geq-1$

since $p\leq(1/n_{1}!\cdots n_{k}!)(d+b)\leq(1/n)(d+b)$. Thus

$H^{1}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(a_{i}))=0,$

so the cohomology vanishing (3.3) holds for $q=2$. Next, we consider the case
that $q\geq 3$. If $0\leq i\leq(1/n_{1}!\cdots
n_{k-1}!(n_{k}-1)!)(d^{q-2}+bd^{q-3})+q-2$, then (3.3) immediately follows
from (3.5). If $i\geq(1/n_{1}!\cdots
n_{k-1}!(n_{k}-1)!)(d^{q-2}+bd^{q-3})+q-1$, then

$\begin{array}[]{rcl}a_{i}&=&id_{k}+\underbrace{(d_{k}+b_{k}+2n+in-qn)}_{\geq
0}-1-np\\\ &\geq&id-1-np\\\ &\geq&(1/n_{1}!\cdots
n_{k-1}!(n-1)!)(d^{q-1}+bd^{q-2})-1-np\\\ &\geq&-1\end{array}$

since $p\leq(1/n_{1}!\cdots n_{k-1}!n!)(d^{q-1}+bd^{q-2})$. Thus

$H^{1}(\mathbb{P}^{1},\mathscr{O}_{\mathbb{P}^{1}}(a_{i}))=0,$

so the cohomology vanishing (3.3) holds in this case as well. We have shown
(3.2) and (3.3), and they imply $K_{p,q}(X,B;L)=0$ as desired. We complete the
proof of Theorem 1.2.

###  3.3. General Case

In this subsection, we prove Theorem 1.1. As we mentioned in the introduction,
Raicu proved that Theorem 1.1 can be deduced from Theorem 1.2 for the case
$k=3$ (see [28, Corollary A.5]). Here we reproduce his proof for the
completeness. Recall that $X$ is an $n$-dimensional projective variety, $B$ is
coherent sheaf, $A$ is an ample divisor, $P$ is an arbitrary divisor on $X$,
and $L_{d}:=\mathscr{O}_{X}(dA+P)$ for an integer $d\geq 1$. Our aim is to
show that for each $2\leq q\leq n+1$ (the case that $q=1$ is trivial), there
is a constant $C>0$ depending on $X,A,B,P$ such that if $d$ is sufficiently
large, then

$K_{p,q}(X,B;L_{d})=0~{}~{}\text{ for $0\leq p\leq Cd^{q-1}$}.$

We can choose integers $a_{1},a_{2},a_{3}\geq 1$ with
$\text{gcd}(a_{1},a_{2}+a_{3})=1$ such that

$A_{1}:=a_{1}A,~{}A_{2}:=a_{2}A+P,~{}A_{3}:=a_{3}A-P$

are very ample and the natural maps

(3.6) $S^{m_{1}}H^{0}(X,A_{1})\otimes S^{m_{2}}H^{0}(X,A_{2})\otimes
S^{m_{3}}H^{0}(X,A_{3})\longrightarrow
H^{0}(X,m_{1}A_{1}+m_{2}A_{2}+m_{3}A_{3})$

are surjective for all $m_{1},m_{2},m_{3}>0$. We may assume that $a_{1}\gg
a_{2}+a_{3}$. Note that $a_{1},a_{2},a_{3}$ are depending only on $X,A,P$. As
$d$ is sufficiently large, we can find integers $d_{1},d_{2}\geq 1$ such that
$d_{1}\approx d/2a_{1},d_{2}\approx d/2(a_{2}+a_{3})$ and
$d+a_{3}=a_{1}d_{1}+(a_{2}+a_{3})d_{2}$. Let $d_{3}:=d_{2}-1$. Then
$d=a_{1}d_{1}+a_{2}d_{2}+a_{3}d_{3}$, and
$L_{d}=\mathscr{O}_{X}(d_{1}A_{1}+d_{2}A_{2}+d_{3}A_{3})$. Note that
$d_{1}<d_{3}<d_{2}$.

Next, consider the commutative diagrams

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}H^{0}(X,A_{1})\times\mathbb{P}H^{0}(X,A_{2})\times\mathbb{P}H^{0}(X,A_{3})=:\mathbb{P}^{n_{1}}\times\mathbb{P}^{n_{2}}\times\mathbb{P}^{n_{3}}=Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{r}:=\mathbb{P}H^{0}(X,L_{d})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}H^{0}(Y,\mathscr{O}_{\mathbb{P}^{n_{1}}}(d_{1})\boxtimes\mathscr{O}_{\mathbb{P}^{n_{2}}}(d_{2})\boxtimes\mathscr{O}_{\mathbb{P}^{n_{3}}}(d_{3}))=:\mathbb{P}^{N}.}$

Clearly, $n_{1},n_{2},n_{3}$ are depending only on $X,A,P$. Notice that
$\mathbb{P}^{r}$ is a linear subspace of $\mathbb{P}^{N}$ by the surjectivity
of (3.6). We can regard $B$ as a coherent sheaf on $Y,~{}\mathbb{P}^{r},$ and
$\mathbb{P}^{N}$. The syzygies of $B$ on $\mathbb{P}^{N}$ are the syzygies of
$B$ on $\mathbb{P}^{r}$ tensoring with a Koszul complex of linear forms. By
letting

$L_{Y}:=\mathscr{O}_{\mathbb{P}^{n_{1}}}(d_{1})\boxtimes\mathscr{O}_{\mathbb{P}^{n_{2}}}(d_{2})\boxtimes\mathscr{O}_{\mathbb{P}^{n_{3}}}(d_{3}),$

we see that

(3.7) $\min\\{p\mid K_{p,q}(X,B;L_{d})\neq 0\\}=\min\\{p\mid
K_{p,q}(Y,B;L_{Y})\neq 0\\}.$

Let $S:=S(H^{0}(X,A_{1})\oplus H^{0}(X,A_{2})\oplus H^{0}(X,A_{3}))$ be the
total coordinate ring of
$Y=\mathbb{P}^{n_{1}}\times\mathbb{P}^{n_{2}}\times\mathbb{P}^{n_{3}}$ with
the usual $\mathbb{Z}^{3}$-grading. Then

$M:=\bigoplus_{(a_{1},a_{2},a_{3})\in\mathbb{Z}^{3}_{\geq
0}}H^{0}(Y,B\otimes\mathscr{O}_{\mathbb{P}^{n_{1}}}(a_{1})\boxtimes\mathscr{O}_{\mathbb{P}^{n_{2}}}(a_{2})\boxtimes\mathscr{O}_{\mathbb{P}^{n_{3}}}(a_{3}))$

is a finitely generated graded $S$-module. Consider the minimal free
resolution of $M$:

$\textstyle{0}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
E_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
0}$

where

$E_{i}=\bigoplus_{b^{j}=(b_{1}^{j},b_{2}^{j},b_{3}^{j})\in
S_{i}}F_{i,b^{j}}\otimes S(b^{j})$

for some finite dimensional vector space $F_{i,b^{j}}$ over $\Bbbk$ and finite
subsets $S_{i}\subseteq\mathbb{Z}^{3}$. Let

$B_{Y}^{j}:=\mathscr{O}_{\mathbb{P}^{n_{1}}}(b_{1}^{j})\boxtimes\mathscr{O}_{\mathbb{P}^{n_{2}}}(b_{2}^{j})\boxtimes\mathscr{O}_{\mathbb{P}^{n_{3}}}(b_{3}^{j})~{}~{}\text{
and }~{}~{}b:=\min\\{b_{1}^{j},b_{2}^{j},b_{3}^{j}\mid b^{j}\in S_{i},0\leq
i\leq m\\}.$

Note that $b$ is depending only on $X,A,B,P$. Now, fix $2\leq q\leq
n+1~{}(\leq n_{1}+n_{2}+n_{3}+1)$. By Theorem 1.2, for $0\leq i\leq m$ and
$b^{j}\in S_{i}$, we have

(3.8) $K_{p-i,q+i}(Y,B_{Y}^{j};L_{Y})=0~{}~{}\text{ for $0\leq
p\leq(1/n_{1}!n_{2}!n_{3}!)(d_{1}^{q+i-1}+bd_{1}^{q+i-2})+i$}.$

Then (3.7) and [28, Theorem A.1] (cf. [21, Proposition 1.d.3]) show that

(3.9) $K_{p,q}(X,B;L_{d})=K_{p,q}(Y,B;L_{Y})=0~{}~{}\text{ for $0\leq
p\leq(1/n_{1}!n_{2}!n_{3}!)(d_{1}^{q-1}+bd_{1}^{q-2})$}.$

Recall that the numbers $a_{1},a_{2},a_{3},n_{1},n_{2},n_{3},b$ are depending
only on $X,A,B,P$ but not on $d$. Since $d_{1}\approx d/2a_{1}$ grows linearly
in $d$ and $d$ is sufficiently large, Theorem 1.1 follows.

######  Remark 3.1.

Instead of applying [28, Theorem A.1], one can alternatively prove (3.9) as
follows. From the minimal free resolution of $M$, we have an exact sequence on
$Y$:

$\textstyle{0}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{E}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{E}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathscr{E}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
0}$

where

$\mathscr{E}_{i}=\bigoplus_{b^{j}\in S_{i}}F_{i,b^{j}}\otimes B_{Y}^{j}.$

By using Proposition 2.1 and chasing through the above exact sequence, we see
that

$K_{p,q}(Y,B;L_{Y})=H^{q-1}(Y,\wedge^{p+q-1}M_{L_{Y}}\otimes B\otimes
L_{Y})=0$

is implied by

$K_{p-i,q+i}(Y,B_{Y}^{j};L_{Y})=H^{q+i-1}(Y,\wedge^{(p-i)+(q+i)-1}M_{L_{Y}}\otimes
B_{Y}^{j}\otimes L_{Y})=0$

for all $0\leq i\leq m$ and $b^{j}\in S_{i}$. Thus (3.7) and (3.8) imply
(3.9).

##  4\. Open Problems

In this section, we discuss some open problems and conjectures. Let $X$ be a
smooth projective variety of dimension $n$, and $B$ be a coherent sheaf on
$X$. Fix an ample divisor $A$ and an arbitrary divisor $P$ on $X$, and put
$L_{d}:=\mathscr{O}_{X}(dA+P)$ for an integer $d\geq 1$.

For each $2\leq q\leq n+1$, it would be extremely interesting to find an
explicit constant $c>0$ in terms of $X,A,B,P$, and $q,d$ such that if $d$ is
sufficiently large, then

$K_{p,q}(X,B;L_{d})=0$ for $0\leq p\leq c$ and $K_{c+1,q}(X,B;L_{d})\neq 0$.

However, this problem is already very difficult for $q=2$. A generalization of
Mukai’s conjecture (cf. [11, Conjecture 4.2]) asks whether the property
$N_{d}$ holds for $K_{X}+(n+2+d)A$ when $X$ is a smooth projective complex
variety. But it is widely open even when $n=2$ and $d=0$. Moreover, Fujita’s
conjecture, which predicts that $K_{X}+(n+2+d)A$ is very ample for $d\geq 0$,
is unknown when $n\geq 3$. However, when $A$ is very ample, Ein–Lazarsfeld
established in [11, Theorem 1] that

$K_{p,q}(X,K_{X}+(n+1+d)A)=0~{}~{}\text{ for $0\leq p\leq d$ and $q\geq 2$}.$

It is reasonable to expect extending this result for $q\geq 3$.

######  Problem 4.1.

Let $X$ be a smooth projective complex variety of dimension $n$, and $A$ be a
very ample divisor on $X$. For each $2\leq q\leq n+1$ and $d\geq 0$, find an
explicit polynomial $P(x)$ of degree $q-1$ such that

$K_{p,q}(X,K_{X}+(n+1+d)A)=0~{}~{}\text{ for $0\leq p\leq P(d)$}$

One can also consider the effective asymptotic vanishing problem for the
syzygies of products of projective spaces.

######  Problem 4.2.

Let $k\geq 1$ be an integer, $n_{1},\ldots,n_{k},d_{1},\ldots,d_{k}$ be
positive integers, and $b_{1},\ldots,b_{k}$ be integers. Set

$\begin{array}[]{l}X:=\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{k}},~{}B:=\mathscr{O}_{\mathbb{P}^{n_{1}}}(b_{1})\boxtimes\cdots\boxtimes\mathscr{O}_{\mathbb{P}^{n_{k}}}(b_{k}),~{}L:=\mathscr{O}_{\mathbb{P}^{n_{1}}}(d_{1})\boxtimes\cdots\boxtimes\mathscr{O}_{\mathbb{P}^{n_{k}}}(d_{k}),\end{array}$

For each $2\leq q\leq n_{1}+\cdots+n_{k}+1$, find a constant $c>0$ in terms of
$b_{1},\ldots,b_{k},d_{1},\ldots,d_{k}$, and $q$ such that if
$d_{1},\ldots,d_{k}$ are sufficiently large, then

(4.1) $K_{p,q}(X,B;L)=0$ for $0\leq p\leq c$ and $K_{c+1,q}(X,B;L)\neq 0$.

In Theorem 1.2, we prove that if $d+b\geq 0$, then

$K_{p,q}(X,B;L)=0~{}~{}\text{ for $0\leq p\leq(1/n_{1}!\cdots
n_{k}!)(d^{q-1}+bd^{q-2})$},$

where $b:=\min\\{b_{1},\ldots,b_{k}\\},~{}d:=\min\\{d_{1},\ldots,d_{k}\\}$. By
a small modification of the proof, we can improve the bound on $p$, but our
method only gives a bound on $p$ depending on dimensions $n_{1},\ldots,n_{k}$.
It is expected that the constant $c$ in Problem 4.2 is independent of
$n_{1},\ldots,n_{k}$. It would be also interesting to find a constant $d>0$ in
terms of $b_{1},\ldots,b_{k},n_{1},\ldots,n_{k}$, and $q$ such that if
$d_{1},\ldots,d_{k}\geq d$, then (4.1) holds. When $k=1$, there is a precise
prediction on $c$ and $d$ (see [12, Conjecture 7.5], [14, Conjecture 2.3]).

######  Conjecture 4.3 (Ein–Lazarsfeld).

Fix $n\geq 1$, $b\geq 0$, and $0\leq q\leq n$. If $d\geq b+q+1$, then

$K_{p,q}(\mathbb{P}^{n},\mathscr{O}_{\mathbb{P}^{n}}(b);\mathscr{O}_{\mathbb{P}^{n}}(d))=0~{}~{}\text{
for }~{}0\leq p\leq{d+q\choose q}-{d-b-1\choose q}-q-1.$

Notice that the conjecture gives the precise vanishing range because
Ein–Erman–Lazarsfeld [10, Theorem 2.1] (see also [14, Theorem 2.1]) proved
that
$K_{p,q}(\mathbb{P}^{n},\mathscr{O}_{\mathbb{P}^{n}}(b);\mathscr{O}_{\mathbb{P}^{n}}(d))\neq
0$ for all

${d+q\choose q}-{d-b-1\choose q}-q\leq p\leq{d+n\choose n}+{d+n-q\choose
n-q}-{n+b\choose q+b}-q-1,$

In [26], Ottaviani–Paoletti conjectured that if $n\geq 3,d\geq 3$, then
$\mathscr{O}_{\mathbb{P}^{n}}(d)$ satisfies the property $N_{3d-3}$. They also
consider the cases that $n\leq 2$ or $d\leq 2$, but these cases are already
settled. By [26, Theorem 1.6], the property $N_{3d-3}$ for
$\mathscr{O}_{\mathbb{P}^{n}}(d)$ is implied by that
$K_{p,2}(\mathbb{P}^{n},\mathscr{O}_{\mathbb{P}^{n}}(d))=0$ for $0\leq p\leq
3d-3$. Thus Conjecture 4.3 for $b=0$ and $q=2$ is equivalent to
Ottaviani–Paoletti’s conjecture. At this moment, we only know that
$\mathscr{O}_{\mathbb{P}^{n}}(d)$ satisfies the property $N_{d+1}$ by
Bruns–Conca–Römer [7], and a small change of the proof of Theorem 1.2 yields
that $\mathscr{O}_{\mathbb{P}^{3}}(d)$ satisfies the property $N_{d+2}$. A new
idea might be needed to solve Conjecture 4.3.

It is also a fascinating problem to study the asymptotic behavior of the
_Betti numbers_

$k_{p,q}(X,B;L_{d}):=\dim K_{p,q}(X,B;L_{d})$

when $d$ is sufficiently large (see [12, Problem 7.3]). In this direction,
Ein–Erman–Lazarsfeld conjectured that for each $1\leq q\leq n$, the Betti
numbers $k_{p,q}(X,L_{d})$ converge to a normal distribution (see [9,
Conjecture B], [14, Conjecture 3.2]). This normal distribution conjecture has
not been verified even for $\mathbb{P}^{2}$ and
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, and it seems that the conjecture is
already very challenging for Veronese embeddings (cf. [5, 6]).

######  Conjecture 4.4 (Ein–Erman–Lazarsfeld).

Fix $n\geq 1$ and $1\leq q\leq n$. Then there is a normalizing function
$F_{q}(d)$ such that

$F_{q}(d)\cdot
k_{p_{d},q}(\mathbb{P}^{n},\mathscr{O}_{\mathbb{P}^{n}}(d))\longrightarrow
e^{-a^{2}/2}$

as $d\to\infty$ and $p_{d}\to r_{d}/2+a\sqrt{r_{d}}/2$.

Notice that

$k_{p,q}(\mathbb{P}^{n},\mathscr{O}_{\mathbb{P}^{n}}(d))=\frac{1}{n}\cdot
h^{q}\big{(}\mathbb{P}^{n-1}\times\mathbb{P}^{1},\wedge^{p+q}\sigma^{*}M_{\mathscr{O}_{\mathbb{P}^{n}}(d)}\otimes(\mathscr{O}_{\mathbb{P}^{n-1}}\boxtimes\mathscr{O}_{\mathbb{P}^{1}}(n-1))\big{)}$

for $p\geq 0$ and $1\leq q\leq n$. It is tempting to wonder if there is a
clever way to compute this Betti number.

We refer to [6] and [12, 14] for more problems and conjectures on syzygies of
Veronese embeddings and asymptotic syzygies of algebraic varieties,
respectively.

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# Wrapping dynamics and full uptake conditions for nonspherical active
nanoparticles

Ke Xiao<EMAIL_ADDRESS>Wenzhou Institute, University of Chinese Academy of
Sciences, Wenzhou 325016, People’s Republic of China  Department of Physics,
College of Physical Science and Technology, Xiamen University, Xiamen 361005,
People’s Republic of China  Rui Ma  Fujian Provincial Key Lab for Soft
Functional Materials Research, Research Institute for Biomimetics and Soft
Matter, Department of Physics, College of Physical Science and Technology,
Xiamen University, Xiamen 361005, People’s Republic of China  Chen-Xu Wu
<EMAIL_ADDRESS>Fujian Provincial Key Lab for Soft Functional Materials
Research, Research Institute for Biomimetics and Soft Matter, Department of
Physics, College of Physical Science and Technology, Xiamen University, Xiamen
361005, People’s Republic of China

(August 27, 2024)

###### Abstract

The cellular uptake of self-propelled nanoparticles (NPs) or viruses, usually
nonspherical, by cell membrane is crucial in may biological processes. In this
study, using Onsager variational principle, we obtain a general wrapping
equation for nonspherical self-propelled nanoparticles. Two analytical
critical conditions are theoretically derived, one for the continuous full
uptake of prolate particles and the other for snapthrough full wrapping of
oblate particles. They capture considerably well the full uptake critical
boundaries in the phase diagrams constructed in terms of active force, aspect
ratio, adhesion energy density, and membrane tension based on numerical
calculations. It is found that enhancing activity (active force), reducing
effective dynamic viscosity, increasing adhesion energy density, and
decreasing membrane tension, can significantly improve the wrapping efficiency
for the self-propelled particles. These results elucidate some of the previous
specific investigations conclusively and may offer novel possibilities for
designing an effective active NP-based vehicle for controlled drug delivery.

The lipid bilayer plasma membrane, a physical barrier defining organelles of
cells and plenty of their surrounding environment, plays a crucial role for a
spectrum of biological processes B.Alberts2005 . Examples range from the
transduction of biochemical signal and the intake of nutrients F.Frey2019PRE
to budding and fission G.V.Meer2004 ; J.T.Groves2010 , and endocytosis of
viruses, pathogens, and particles G.Bao2005 ; S.Zhang2015 . The engulfing of a
particle or virus (pathogen) by a plasma membrane is a widely encountered
phenomenon in endocytosis processes, including inter- and intracellular
transport S.Behzadi2017 , delivering therapeutic agents enveloped by
nanoparticles into tumor cells D.Peer2007 ; W.Rao2015 ; Y.Min2015 , and virus
infection N.Kol2007 ; R.F.Bruinsma2021 ; C.B.Jackson2022 . Especially,
cellular uptake, which involves the interaction between cell membrane and
nanoparticles (NPs) or viruses, is an essential step for a wide range of
healthy and disease-related processes R.F.Bruinsma2021 .

Over the past two decades, considerable efforts using experiment, theoretical
modeling, and numerical simulation have been devoted to characterizing how the
physical parameters, including the particle size Deserno2002 ; Deserno2003 ;
M.Deserno2004 ; S.Zhang2009 ; B.D.Chithrani2006 ; J.Agudo2015 ; C.Contini2020
, shape F.Frey2019 ; K.Yang2010 ; Z.Shen2019 , elastic properties of the
particle X.Yi2011 ; J.C.Shillcock2005 ; X.Yi2014 ; A.Verma2010 ; X.Ma2021 ,
ligand/receptor density H.Yuan2010PRL ; H.Yuan2010 ; T.Wiegand2020 , as well
as the mechanical properties of the membrane J.Agudo-Canalejo2015 ;
H.T.Spanke2020 , affect the invading behaviors. Though the cellular uptake of
passive particles via endocytic process has been studied extensively, little
work has been done on the active entry of self-propelled bacterial pathogens.
To name a few examples, it has been found that some cytosolic bacteria such as
Rickettsia rickettsii are able to produce active force to facilitate their
mobility by forming actin tails P.M.Colonne2016 , and Listeria monocytogenes
can generate active force to push out a tube-like protuberance from the plasma
membrane by hijacking the actin polymerization-depolymerization apparatus of
their host J.A.Theriot1992 ; J.R.Robbins1999 ; T.Chakraborty1999 ;
F.E.Ortega2019 ; G.C.Dowd2020 . How the active force of these self-propelled
agents affects the engulfing dynamics at the cell membrane remains to be
elucidated.

Recently, many model systems by using lipid vesicles to encapsulate natural
swimmers (Escherichia coli bacteria, Bacillus subtilis bacteria, etc.) or
artificial microswimmers (synthetic Janus particles) have been developed to
study the active membrane behaviors in vitro H.R.Vutukuri2020 ;
S.C.Takatori2020 ; C.Wang2019 ; Y.Li2019 ; M.S.E.Peterson2021 ;
L.LeNagarda2022 . Such systems are out-of-equilibrium and hence give rise to
many intriguing behaviors, such as membrane fluctuations and large
deformations H.R.Vutukuri2020 ; S.C.Takatori2020 , shape transformations
C.Wang2019 ; Y.Li2019 ; M.S.E.Peterson2021 , and even deformation of lipid
vesicles into flagellated swimmers L.LeNagarda2022 . Therefore, in biology,
the specific interactions between vesicles and bacteria or artificial self-
propelled particles plays a key role in designing active matter systems
A.T.Brownet2016 . Besides, the wrapping dynamics of particles by cell
membranes, which is important for understanding the cellular uptake, has been
carried out experimentally, theoretically, and numerically. In the limit of
low membrane tension and weak reversible adhesion, Spanke et. al.
H.T.Spanke2022  experimentally investigated how the spontaneous wrapping
dynamics of micron-sized particles by giant unilamellar vesicles changes with
the adhesion energy. By combining computer simulations and theoretical
analysis, the cellular uptake of active particles in the absence of membrane
tension was studied P.Chen2020 , and the deterministic and stochastic uptake
dynamics of passive nanoparticles with different geometries were also reported
F.Frey2019 . Understanding such effect of forces and membrane properties
(adhesion energy density and membrane tension) on the dynamics of cellular
uptake are critical to designing efficient strategies for potential biomedical
applications, including drug/gene delivery Panyam2003 ; Xu2018 ; Wang2019 ,
cell operation and manipulation M.Medina2018 ; Xie2019 , and
bioimaging/sensing R.Weissleder2006 ; D.Peer2007 . In addition, in reality
many pathogens and viruses are nonspherical C.Hulo2011 , such as egg-shaped
malaria parasite S.Dasgupta2014  and cylindrical Listeria monocytogenes,
indicating the significance of probing the wrapping dynamics of nonspherical
particles for the entry of certain pathogens into cells.

To better understand the cellular uptake dynamics of a nonspherical self-
propelled nanoparticle by a plasma membrane, a detailed and comprehensive
investigation of how the wrapping time depends on the active force, the
particle’s aspect ratio, the viscosity, and the membrane properties (adhesion
energy density and membrane tension) is needed. Here in this paper, the
wrapping dynamics is studied by employing the Onsager variational principle
OnsagerI1931 ; OnsagerII1931  for out-of-equilibrium systems. Our results show
that active force, low effective dynamic viscosity, strong adhesion force, and
small membrane tension play a positive role in the wrapping time, indicating
that it can be manipulated by changing the activity and the aspect ratio of
the particles, the viscosity, and the properties of the membrane such as
adhesion energy density and membrane tension. The physical insights gained
from this work could clarify the mechanism of wrapping dynamics of
nonspherical active nanoparticles.

We model a self-propelled invader as an axis-symmetric ellipsoid (prolate or
oblate spheroid) with its principle rotational axis orthogonal to the flat
membrane, as shown in Fig. 1, where

Figure 1: (Color online) Schematic depicting wrapping phases of an active
particle from nonwrapping to partial wrapping and to full uptake.

$a$ and $b$ denote the semi-axes perpendicular to and along the principle
rotational axis, respectively. The geometry of the particle is parameterized
by the aspect ratio $e=b/a$, with $e>1$ for a prolate ellipsoid, and $e<1$ for
an oblate one. In practice, it is convenient to write the area element in
terms of polar angle: $dA=2\pi
a^{2}\sin\theta\sqrt{\cos^{2}\theta+e^{2}\sin^{2}\theta}\leavevmode\nobreak\
d\theta.$ To model the uptake dynamics of an active particle, we have to write
down the total free energy of the system. Following the classical Canham-
Helfrich continuum model Helfrich1973 ; F.Julicher1994 ; M.Deserno2004 , such
an energy is given by

$\displaystyle E_{\mathrm{tot}}=\int_{A_{\rm
mem}}\frac{\kappa}{2}(2H)^{2}dA+\sigma\Delta A-\int_{A_{\rm
ad}}\omega\leavevmode\nobreak\ dA-fZ,$ (1)

where the elastic energy of the membrane, the adhesion energy between the
particle and the membrane, and the work done by the active particle are taken
into account. Here the first term, with $\kappa$ the bending rigidity and $H$
the local mean curvature, denotes the bending energy of the membrane

$\displaystyle E_{\rm bend}=$ $\displaystyle\int_{0}^{\alpha}\pi\kappa
e^{2}\sin\theta\frac{[2+(e^{2}-1)\sin^{2}\theta]^{2}}{[1+(e^{2}-1)\sin^{2}\theta]^{3}}\times$
$\displaystyle\sqrt{\cos^{2}\theta+e^{2}\sin^{2}\theta}\leavevmode\nobreak\
d\theta,$ (2)

an integral over the contact area between the membrane and the particle. The
second term of Eq. (1) is contributed by the surface tension:

$\displaystyle E_{\rm ten}=$ $\displaystyle\int_{0}^{\alpha}2\pi\sigma
a^{2}\sin\theta\biggl{[}1-\frac{\cos\theta}{\sqrt{\cos^{2}\theta+e^{2}\sin^{2}\theta}}\biggr{]}\times$
$\displaystyle\sqrt{\cos^{2}\theta+e^{2}\sin^{2}\theta}\leavevmode\nobreak\
d\theta.$ (3)

The third term of Eq. (1) representing the gain in adhesive energy,
characterized by a negative adhesive energy $-\omega$ per unit area, can be
written as an integral over the wrapping area. The last term of Eq. (1) arises
from work done by the active force $f$ acting on the particle, which is
calculated as $E_{\rm f}=-fae(1-{\rm cos}\alpha)$.

As the particle is being engulfed by the membrane, it has been found that it
is the friction force near the membrane-particle contact line with its
circumference given by $L(\alpha)=2\pi a\sin\alpha$ that largely dissipates
the energy H.T.Spanke2022 . In the limit of low Reynolds number, the

Figure 2: (Color online) The engulfing angle $\alpha/\pi$ and the wrapping
velocity $v$ of an active particle with different aspect ratios (a) $e=0.75$,
(b) $e=1$, and (c) $e=1.5$, where the bending rigidity, the adhesion energy
density, the tension modulus, and the effective dynamic viscosity are set as
$25\leavevmode\nobreak\ k_{B}T$, $0.044\leavevmode\nobreak\ {\rm mJ/m^{2}}$,
$0.9\times 10^{-5}\leavevmode\nobreak\ {\rm N/m}$, and $1\leavevmode\nobreak\
{\rm Pa\cdot s}$, respectively.

dissipation function reads

$\displaystyle\Phi=\pi\eta
a^{3}\sin\alpha(\cos^{2}\theta+e^{2}\sin^{2}\theta)\dot{\alpha}^{2},$ (4)

where $\eta$ is the effective dynamic viscosity with a typical order of 1
Pa$\cdot$s. In order to obtain the equation governing the wrapping dynamics of
the active particle, first of all we construct a Rayleighian
$\mathcal{R}=\dot{E}_{\mathrm{tot}}+\Phi$, with $\dot{E}_{\mathrm{tot}}$ the
time derivative of the free energy of the system given by

$\displaystyle\dot{E}_{\mathrm{tot}}=$ $\displaystyle\biggl{\\{}\frac{\kappa
e^{2}}{a^{2}}\frac{[2+(e^{2}-1)\sin^{2}\alpha]^{2}}{[1+(e^{2}-1)\sin^{2}\alpha]^{3}}+$
$\displaystyle
2\sigma\biggl{(}1-\frac{\cos\alpha}{\sqrt{\cos^{2}\alpha+e^{2}\sin^{2}\alpha}}\biggr{)}-2\omega\biggr{\\}}\times$
$\displaystyle\pi
a^{2}\sin\alpha\sqrt{\cos^{2}\alpha+e^{2}\sin^{2}\alpha}\dot{\alpha}-fae\sin\alpha\dot{\alpha},$
(5)

and $\Phi$ the energy dissipation function. Minimizing $\mathcal{R}$ with
respect to $\dot{\alpha}$ following the Onsager variational principle, i.e.,
$\partial\mathcal{R}/\partial\dot{\alpha}=0$, we obtain the cellular uptake
dynamics equation

$\displaystyle\dot{\alpha}=$ $\displaystyle\frac{1}{\eta
a\sqrt{\cos^{2}\alpha+e^{2}\sin^{2}\alpha}}\biggl{\\{}\omega+\frac{fe}{2\pi
a\sqrt{\cos^{2}\alpha+e^{2}\sin^{2}\alpha}}$ $\displaystyle-\frac{\kappa
e^{2}}{2a^{2}}\frac{[2+(e^{2}-1)\sin^{2}\alpha]^{2}}{[1+(e^{2}-1)\sin^{2}\alpha]^{3}}$
$\displaystyle-\sigma\biggl{(}1-\frac{\cos\alpha}{\sqrt{\cos^{2}\alpha+e^{2}\sin^{2}\alpha}}\biggr{)}\biggl{\\}},$
(6)

for a nonspherical active particle. For spherical particles $a=b=R$, the above
equation reduces to Eq. (2) in Ref. F.Frey2019  if $f=0$.

A detailed theoretical analysis of Eq. (6) shows that there exist two types of
critical conditions for a full uptake to occur. One is governed by
$\dot{\alpha}\bigl{|}_{\alpha=\pi}=0$, or

$\displaystyle\omega+\frac{f}{2\pi a}e-\frac{2\kappa}{a^{2}}e^{2}-2\sigma=0,$
(7)

corresponding to a second-order wrapping transition for prolate particles and
spherical particles. For a spherical particle, Eq. (7) reduces to a critical
radius
$R_{c}=[-f+\sqrt{f^{2}+32\pi^{2}\kappa(\omega-2\sigma)}]/[4\pi(\omega-2\sigma)]$
when $\omega\neq 2\sigma$. The other is given by
$\dot{\alpha}\bigl{|}_{\alpha=\pi/2+B/(2C)}=0$ with $B=\sigma/e$, and

$\displaystyle C=\frac{1-e^{2}}{2e^{2}}\biggl{(}\sigma-\omega-\frac{f}{\pi
a}\biggr{)}+\frac{\kappa(3e^{2}+7)}{4a^{2}e^{6}},$

leading to a critical condition

$\displaystyle\omega+\frac{f}{2\pi
a}-\frac{\kappa(1+e^{2})^{2}}{2a^{2}e^{4}}-\sigma-\frac{B^{2}}{4C}=0,$ (8)

for a first-order wrapping transition from a partial wrapping to a full
uptake. If the surface tension satisfies
$(2e^{2}-1)\sigma\ll\kappa(3e^{2}+7)/(a^{2}e^{4})-2(1-e^{2})[\omega+f/(\pi
a)]$, the above condition reduces to

$\displaystyle\omega+\frac{f}{2\pi
a}-\frac{\kappa(1+e^{2})^{2}}{2a^{2}e^{4}}-\sigma=0.$ (9)

To investigate the wrapping dynamics of the active particle, we first solve
Eq. (6) numerically and analyze the evolution of the wrapping angle
$\alpha/\pi$. For an oblate spheroid, the engulfing angle (black curves) as
shown in Fig. 2(a) exhibits a plateau around $\alpha=\pi/2$, indicating that
there exists a stable partial wrapping state at this point. This can be
reflected by its corresponding low wrapping velocity (blue curve) as the
oblate particle has to cross over an energy barrier before it is fully
wrapped, corresponding to a first-order wrapping transition. Figures 2(b) and
2(c) show the similar wrapping behavior with a monotonic increase of wrapping
angle to a completely wrapped state ($\alpha/\pi=1$) for both the prolate
spheroid and the spherical particle. The wrapping velocity for the spherical
particle shows a significant and monotonic drop off after the particle is
initially internalized at a fast speed, until it slows down and is fully
wrapped by the fluidic film [see Fig. 2(b)]. While for the prolate spheroid,
the engulfing speed increases against time in the beginning, followed by a
dramatic decrease, and finally levels off to zero, terminating at a full
wrapping state ($\alpha/\pi=1$) [see Fig. 2(c)].

To systematically study how the active force and the aspect ratio of the
particle, the effective dynamic viscosity, and the membrane properties affect
the uptake dynamics of the active particle at cell membrane, we numerically
calculate the wrapping time by changing these parameter values. Figure 3 shows
the wrapping time as a function of particle size for different active forces
and dynamic viscosities.

Figure 3: (Color online) Wrapping times as functions of particle size for (a)
oblate particle, and (b) spherical particle with different active forces.
Uptake time as a function of particle size for (c) oblate particle, and (d)
spherical particle with different effective dynamic viscosities. Parameter
values (if not varied) are fixed at $\kappa=25\leavevmode\nobreak\ k_{B}T$,
$\omega=0.044\leavevmode\nobreak\ {\rm mJ/m^{2}}$, $\sigma=0.9\times
10^{-5}\leavevmode\nobreak\ {\rm N/m}$, and $f=0\leavevmode\nobreak\ {\rm
pN}$, $\eta=1\leavevmode\nobreak\ {\rm Pa\cdot s}$.

For fixed $f$ and $\eta$, the curves of uptake time share a similar feature
that with the increase of the particle size, $t_{\rm up}$ drops very quickly
before it bounces back mildly. Meanwhile, the full wrapping regimes in the
parameter space are also apparently broadened with the increase of the active
force and the effective dynamic viscosity. Figures 3(a) and 3(b) compare the
resulting wrapping times of the oblate and the spherical particle for
different values of active force. For the oblate particle, we adjust the
length of the symmetry axis $a$ by keeping the aspect ratio unchanged. Both
ellipsoid and spherical particles share a similar behavior that there exists a
critical particle size beyond which a full uptake occurs. Interestingly, the
calculation results demonstrate that particles with larger active forces are
uptaken faster than those with smaller forces, indicating that particle
activity facilitates the uptake process. This conclusion is in line with the
simulation predictions in Ref. P.Chen2020 , where the authors claimed that the
uptake efficiency can be enhanced with the increase of Péclet number
quantifying the strength of active force.

In order to probe how effectively the dynamic viscosity affect the wrapping
dynamics, we plot the wrapping time for an oblate particle and a spherical
particle against particle size with different effective dynamic viscosities,
as shown in Figs. 3(c) and 3(d), respectively. It is found that decreasing the
effective dynamic viscosity $\eta$ clearly decreases the threshold particle
size for the occurrence of the complete uptake. This is due to the fact that
large viscosity increases the friction, which correspondingly inhibits the
uptake process and hence leads to longer uptake time. Therefore, to achieve
faster (slower) wrapping process, enhancing (weakening) the particle activity
and reducing (raising) the effective dynamic viscosity might be an effective
option.

Pathogens and viruses come in many different shapes C.Hulo2011 , but the most
frequent occurrence is ellipsoid. Therefore, here we focus our discussion on
ellipsoid with its shape characterized by aspect ratio. Furthermore, to gain
more insight into the interrelated influence of the active force $f$ and the
particle aspect ratio $e$ on the wrapping time, we construct a phase diagram
for the wrapping time in the $f-e$ space under the condition of fixed particle
volume, as shown in Fig. 4.

Figure 4: (Color online) A two-dimensional phase diagram on the ($f-e$) plane
characterizes the interrelated effects of active force and particle aspect
ratio on the uptake time of wrapping process, where the bending rigidity, the
adhesion energy density, the membrane tension, and the effective dynamic
viscosity are set as $25\leavevmode\nobreak\ k_{B}T$,
$0.044\leavevmode\nobreak\ {\rm mJ/m^{2}}$, $0.9\times
10^{-5}\leavevmode\nobreak\ {\rm N/m}$, and $1\leavevmode\nobreak\ {\rm
Pa\cdot s}$, respectively. The volume of the particle is fixed at $V=4\pi
ea^{3}/3=4\pi R_{0}^{3}/3$ with $R_{0}=100\leavevmode\nobreak\ {\rm nm}$.

On one hand, a comparison of the wrapping time between the prolate ellipsoidal
particle ($e>1$) and the spherical particle ($e=1$) shows that an prolate
ellipsoidal particle is taken up slower than a spherical one. However, upon
decreasing the aspect ratio, the uptake time for the active oblate ellipsoidal
particles ($e<1$) displays a nonmonotonic feature by decreasing to a minimum
value first and then bouncing back gradually, in stark contrast to the
monotonic dependence for active prolate particles with an aspect ratio of
$e>1$. On the other hand, Figure 4 also verifies that enhancing the activity
gives rise to the wrapping efficiency (with a decrease of $t_{\rm up}$).
Therefore, the wrapping efficiency can be regulated by tuning the aspect ratio
and the activity of particles. To determine the boundaries separating the full
uptake and no uptake regimes, we plotted two boundary curves based on Eq. (7)
[see the dash curve in Fig. 4] and Eq. (9) [see the dot curve in Fig. 4],
respectively, which match the numerical calculations very well. The
consistence once again indicates that there exist two ways for a nonspherical
particle to reach full wrapping, i.e. a continuous uptake for prolate
particles and a snapthrough uptake for oblate particles. Such a conclusion is
in agreement with the simulations done by Khosravanizadeh et. al
Khosravanizadeh2022 , who demonstrated that the oblate ellipsoidal particles
exhibit discontinuous wrapping phase transition from partial wrapping to full
wrapping during the uptake process, while the prolate ellipsoidal particles
show a continuous wrapping transition behavior.

In order to gain more insights into the effects of the membrane properties on
the wrapping time, we explore the wrapping dynamics of an active oblate
particle, a prolate ellipsoidal one, and a spherical one with different
adhesion energy densities and membrane tensions, as shown in Fig. 5.

Figure 5: (Color online) Wrapping time $t_{\rm up}$ as a function of $f$ and
$\omega$ for (a) oblate ellipsoidal, (b) spherical, and (c) prolate
ellipsoidal particles at equal particle volume. Uptake times as a function of
active force $f$ and membrane tension $\sigma$ for (d) oblate ellipsoidal, (e)
spherical, and (f) prolate ellipsoidal particles at equal particle volume.
Parameter values (if not varied) are fixed at $\kappa=25\leavevmode\nobreak\
k_{B}T$, $\omega=0.044\leavevmode\nobreak\ {\rm mJ/m^{2}}$, $\sigma=0.9\times
10^{-5}\leavevmode\nobreak\ {\rm N/m}$, and $\eta=1\leavevmode\nobreak\ {\rm
Pa\cdot s}$. The volume of the particle is fixed at $V=4\pi ea^{3}/3=4\pi
R_{0}^{3}/3$ with $R_{0}=100\leavevmode\nobreak\ {\rm nm}$.

The colored contour maps of $t_{\rm up}$ on the $f-\omega$ [see Figs. 5(a),
5(b), and 5(c)] and $f-\sigma$ [see Figs. 5(d), 5(e), and 5(f)] planes show
that given an aspect ratio and a fixed active force, a higher (lower) wrapping
efficiency can be achieved under a stronger (weaker) adhesion force or a lower
(higher) membrane tension, i.e. higher adhesion and looser membrane leads to
faster wrapping. Such a conclusion can be supported by experimental
observations H.T.Spanke2022  reported recently that higher adhesion leads to
faster wrapping, and simulation results in Ref. F.Frey2019  that the uptake
time is strongly decreased for higher adhesion and looser membrane. Here it is
important to note that the boundaries of the full uptake (read lines) can be
very well captured by the critical conditions Eqs (7), (8) and (9)
theoretically derived using our model.

Finally we discuss the influence of the membrane properties and the aspect
ratio of particle to the wrapping process. Figures 6(a) and 6(b) show the two-
dimensional phase diagrams of wrapping time in the parametric planes of
$\omega-e$ and $\sigma-e$.

Figure 6: (Color online) Two-dimensional wrapping time phase diagrams in the
projection planes of (a) $\omega$ and $e$ and (b) $\sigma$ and $e$, where the
bending rigidity, the adhesion energy density, the membrane tension, the
active force and the effective dynamic viscosity (if not varied) are set as
$25\leavevmode\nobreak\ k_{B}T$, $0.044\leavevmode\nobreak\ {\rm mJ/m^{2}}$,
$0.9\times 10^{-5}\leavevmode\nobreak\ {\rm N/m}$, $f=0\leavevmode\nobreak\
{\rm pN}$ and $1\leavevmode\nobreak\ {\rm Pa\cdot s}$, respectively. The
volume of the particle is fixed at $V=4\pi ea^{3}/3=4\pi R_{0}^{3}/3$ with
$R_{0}=100\leavevmode\nobreak\ {\rm nm}$.

What is striking is that, for the oblate ellipsoidal particles, the uptake
time nonmonotonically depends on $e$ within the scale of $\omega$ or $\sigma$,
in comparison to the monotonic dependence of the prolate ellipsoidal
particles. Tuning the particle’s aspect ratio to an optimal value enables one
to obtain a minimal wrapping time. In addition, Figure 6 also confirms that
the uptake efficiency can be improved (abated) by increasing (decreasing) the
adhesion energy density, and reducing (raising) the membrane tension. The dash
and solid curves based analytical results Eq. (7) and Eq. (8) coincide with
the boundaries obtained from numerical calculations. It has been shown, by
providing a stochastic model to study the kinetics of particle wrapping by a
vesicle, that increasing the attraction strength between the particle and
vesicle causes the improvement of uptake rate S.Mirigian2013 . This is again a
result in agreement with our present conclusions.

Consequently, we argue that the wrapping time decreases with the increase of
the active force; the higher the viscosity, the longer the wrapping time;
strong adhesion and low tension improve the wrapping efficiency. The reason
stems from that the wrapping process is largely controlled by the competition
among the three types of energy: the elastic energy (consisting of bending
energy and tension energy), the adhesive energy, and the work done by the
active force. Reducing wrapping time can be realized based on the condition
that the adhesion energy and the work done by the active force is sufficient
to overcome the energy barrier, namely, the sum of the elastic energy and the
viscous dissipation. In the presence of active force, the work done by the
active force reduces the free energy and as a result change the uptake force.
The positive correlation between active force and uptake force gives rise to
the positive dependence of wrapping time on the active force. According to Eq.
(6), the adhesion energy driving the wrapping process is positively
proportional to the adhesion energy density and negatively proportional to the
membrane tension. As a result, increasing the adhesion energy density
corresponds to increasing the driving force for wrapping, decreasing the
membrane tension decreases the energy penalty for uptake, indicating a
decreases of the wrapping time. As for the effective dynamic viscosity, an
increase of it means that the induced energy dissipation requires more
adhesion energy and work done by the active force to compensate, which leads
to an increase of wrapping time.

In summary, we propose a theoretical model to investigate the wrapping
dynamics of a nonspherical active particle by a lipid plasma membrane, by
taking into account the influence of the active force, the particle shape, the
effective dynamic viscosity, and the membrane properties (including the
adhesion energy density and the membrane tension). The wrapping equation for
the active particle, which quantitatively couples the elastic deformation of
the membrane, the work done by the active particle, and the energy
dissipation, is derived by using Onsager variational principle. Two critical
conditions, one for the continuous full uptake of prolate particles and the
other for snapthrough full wrapping of oblate particles, are obtained
theoretically. Our results reveal that enhancing activity (active force),
reducing effective dynamic viscosity, increasing adhesion energy density, and
decreasing membrane tension, can significantly improve the wrapping efficiency
for the self-propelled particles. Intriguingly, with an increase of aspect
ratio, the wrapping time for oblate ellipsoidal particles exhibits a
nonmonotonic dependence, in stark contrast to the monotonic dependence for
prolate ellipsoidal particles. Therefore, the wrapping time can be manipulated
by changing the activity and the aspect ratio of the particles, the effective
dynamic viscosity, and the properties of the membrane, such as adhesion energy
density and membrane tension. All these findings may not only shed light on
the influence of the activity and the aspect ratio of particles, the
viscosity, and the properties of membrane on the dynamic behaviors of wrapping
process, but also provide guidelines to improve the efficiency of active
particle-based drug delivery systems.

##  I ACKNOWLEDGMENTS

We acknowledge financial support from National Natural Science Foundation of
China under Grant Nos.12147142, 11974292, 12174323, and 1200040838, and 111
project B16029.

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