# Du Bois complexes of cones over singular varieties, local cohomological
dimension, and $K$-groups

Mihnea Popa  Department of Mathematics, Harvard University, 1 Oxford Street,
Cambridge, MA 02138, USA<EMAIL_ADDRESS>and  Wanchun Shen  Department
of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA
<EMAIL_ADDRESS>Dedicated to the memory of Lucian Bădescu, with
gratitude.

###### Abstract.

We compute the Du Bois complexes of abstract cones over singular varieties,
and use this to describe the local cohomological dimension and the non-
positive $K$-groups of such cones.

MP was partially supported by NSF grant DMS-2040378.

##  A. Introduction

As an undergraduate, the first author was introduced by Lucian Bădescu to the
beautiful world of algebraic geometry. One of his favorite topics was that of
Lefschetz-type theorems, in particular Barth’s celebrated result for arbitrary
smooth subvarieties in projective space; he also often brought up Hartshorne
and Ogus’ work on local cohomological dimension and its connections to the
topology of projective varieties. We dedicate this note on a related circle of
ideas to him, in the belief that he would have enjoyed it!

Concretely, in this paper we compute the Du Bois complexes of cones over
arbitrary subvarieties in projective space; the case of smooth varieties can
be found in [SVV] and the references therein. We use this for several
applications. The most important is not a new result, at least in the setting
of classical cones, but we believe it nicely illustrates how such Du Bois
complexes can be used in an almost algorithmic fashion: using the birational
characterization of local cohomological dimension provided in [MP1], expressed
in an equivalent form in terms of the depth of Du Bois complexes, we give an
alternative approach to Ogus’ [Og] interpretation of the local cohomological
dimension at the vertex of the cone over a projective variety $X$ in terms of
the topology of $X$. As a new result, we extend this characterization to all
abstract cones over $X$. In a different direction, feeding our computation
into the main technical result of [CHWW2], we describe the non-positive
$K$-groups of abstract cones over a projective variety in terms of its Du Bois
complexes, extending the result in _loc. cit._ to singular complex varieties.

Let $X\subseteq\mathbb{P}^{N}$ be a complex projective variety of dimension
$n$ and codimension $r=N-n$, and let $Z=C(X)\subseteq\mathbb{A}^{N+1}$ be the
affine cone over $X$. We denote by $\operatorname{lcd}(Z,\mathbb{A}^{N+1})$
the local cohomological dimension of $Z$ in $\mathbb{A}^{N+1}$, and call

${\rm lcdef}(Z):=\operatorname{lcd}(Z,\mathbb{A}^{N+1})-r$

the _local cohomological defect_ of $Z$. This invariant depends only on $Z$,
and not on the embedding in $\mathbb{A}^{N+1}$; see Section 2 for details. The
following is a slight reformulation of Ogus’ result [Og, Theorem 4.4] (also
recovered more recently in [HP] in the smooth case, and in [RSW] in general):

######  Theorem A.

For an integer $c\geq 0$, the following are equivalent:

  1. (1)

${\rm lcdef}(Z)\leq c$

  2. (2)

${\rm lcdef}(X)\leq c$ and the restriction maps
$H^{i}(\mathbb{P}^{N},\mathbb{C})\to H^{i}(X,\mathbb{C})$ are isomorphisms for
$i\leq n-1-c$.

In particular:

${\rm lcdef}(Z)=0\iff{\rm lcdef}(X)=0~{}{\rm
and}~{}H^{i}(\mathbb{P}^{N},\mathbb{C})\overset{\simeq}{\longrightarrow}H^{i}(X,\mathbb{C})\,\,{\rm
for~{}all}\,\,i\leq n-1.$

Note that in [Og], part (1) is phrased in terms of the cohomological dimension
${\rm cd}(\mathbb{P}^{N}\smallsetminus X)$, but it is well known via a
standard argument that $\operatorname{lcd}(Z,\mathbb{A}^{N+1})={\rm
cd}(\mathbb{P}^{N}\smallsetminus X)+1$.

The interest in the last equivalence in Theorem A stems from the fact that
both sides are satisfied when $X$ is a smooth complete intersection in
$\mathbb{P}^{N}$. It is therefore intimately related to Hartshorne’s
conjecture on small codimension subvarieties.

_Note._ Recall that Barth’s theorem (see [La, Scn.3.2] for a survey) states
that if $X$ is smooth, then we always have

$H^{i}(\mathbb{P}^{N},\mathbb{C})\overset{\simeq}{\longrightarrow}H^{i}(X,\mathbb{C})\,\,\,\,\,\,{\rm
for~{}all}\,\,\,\,i\leq n-r.$

According to Theorem A, this is equivalent to the universal bound ${\rm
lcdef}(Z)\leq r-1$.

We provide in fact an extension of Ogus’ result to any abstract cone over $X$,
defined by the choice of ample line bundle $L$, in terms of the action of
$c_{1}(L)$ on the singular cohomology of $X$; see Theorem 6.1 for the precise
statement. In our approach, this result, hence also Theorem A, follows from
two main points. One is a reformulation of the characterization of ${\rm
lcdef}(Z)$ in [MP1, Theorem E], for any complex variety $Z$, in terms of the
Du Bois complexes of $Z$:

${\rm lcdef}(Z)=\dim Z-\underset{k\geq
0}{\min}~{}\\{\operatorname{depth}\underline{\Omega}_{Z}^{k}+k\\}.$

See Section 2 for details. The second, and the main technical result here, is
the computation of the Du Bois complexes of (abstract) cones over arbitrary
subvarieties in $\mathbb{P}^{N}$, and of their depth.

######  Theorem B.

Let $X$ be a projective variety, endowed with an ample line bundle $L$. Let

$Z=C(X,L)=\operatorname{Spec}\big{(}\bigoplus_{m\geq 0}H^{0}(X,L^{m})\big{)}$

be the affine cone over $X$ associated to $L$, with cone point $x\in Z$. Then
the Du Bois complexes $\underline{\Omega}_{Z}^{k}$ can be computed explicitly
in terms of the Du Bois-Hodge theory of $X$ and $L$; see Theorem 3.1(1). The
same holds for $\operatorname{depth}_{x}\underline{\Omega}_{Z}^{k}$; see
Theorem 3.1(2).

On a different note, to prove Theorem 6.1 we also need a general result of
independent interest, namely a “dual” Nakano-type vanishing theorem for Du
Bois complexes. We establish this in Section 5, as a simple application of the
Kodaira-Saito vanishing theorem for mixed Hodge modules.

######  Theorem C.

Let $X$ be a complex projective variety of dimension $n$, and $L$ an ample
line bundle on $X$. Then

$\mathbb{H}^{q}(X,\underline{\Omega}_{X}^{p}\otimes L^{-1})=0\,\,\,\,\,\,{\rm
for~{}all}\,\,\,\,p+q<n-{\rm lcdef}(X).$

Finally, in a different direction, Theorem B is used in Section 8 to compute
the non-positive $K$-groups of cones over a projective variety in terms of its
Du Bois complexes, extending for complex varieties the result in [CHWW2] to
abstract cones over singular varieties; this is simply an appendix to _loc.
cit._ , as we are still using the main technical result of that paper.

######  Corollary D.

Let $X$ be an $n$-dimensional complex projective variety endowed with an ample
line bundle $L$, and let $Z=C(X,L)$ be the affine cone over $X$ associated to
$L$. Then:

  1. (1)

$K_{0}(Z)\simeq\mathbb{Z}\oplus\bigoplus_{i=1}^{n}\bigoplus_{m\geq
1}\mathbb{H}^{i}(X,\underline{\Omega}_{X/\mathbb{Q}}^{i}\otimes L^{m})$;

  2. (2)

$K_{-\ell}(Z)\simeq\bigoplus_{i=0}^{n-\ell}\bigoplus_{m\geq
1}\mathbb{H}^{\ell+i}(X,\underline{\Omega}_{X/\mathbb{Q}}^{i}\otimes L^{m})$
for all $\ell\geq 1$.

Here $\underline{\Omega}_{X/\mathbb{Q}}^{i}$ are the Du Bois complexes over
$\mathbb{Q}$ of the complex variety $X$; see the beginning of Section 8. When
$X$ is embedded in $\mathbb{P}^{N}$, with $L=\mathcal{O}_{X}(1)$, the method
applies to the classical cone $C(X)$ as well. For $l\geq 1$ we have
$K_{-l}(C(X))=K_{-l}(C(X,L))$, while for $l=0$ we have

$K_{0}(C(X))\simeq\mathbb{Z}\oplus\operatorname{Pic}(C(X))\oplus\bigoplus_{i=1}^{n}\bigoplus_{m\geq
1}\mathbb{H}^{i}(X,\underline{\Omega}_{X/\mathbb{Q}}^{i}\otimes L^{m}).$

Here $\operatorname{Pic}(C(X))\simeq R^{+}/R$, where $R$ is the homogeneous
coordinate ring of $X$, and

$R^{+}\simeq\bigoplus_{m\geq
0}\mathbb{H}^{0}(X,\underline{\Omega}_{X}^{0}\otimes L^{m})$

is its seminormalization. A few things can also be said about positive
$K$-groups, but a precise description can only be achieved in special cases.

Acknowledgements. We thank Elden Elmanto, Shiyue Li, Mircea Mustaţă, Sung Gi
Park, Anda Tenie and Duc Vo for helpful conversations.

##  B. Main results

###  1\. Preliminaries

We start by collecting the main technical definitions and facts used in this
paper.

Du Bois complexes. Let $X$ be a complex algebraic variety. The _filtered de
Rham complex_ $(\underline{\Omega}_{X}^{\bullet},F)$ is an object in the
bounded derived category of filtered differential complexes on $X$, introduced
by Du Bois in [DB], following the ideas of Deligne, and intended as a
replacement for the standard de Rham complex on smooth varieties. For each
$k\geq 0$, the (shifted) associated graded quotient

$\underline{\Omega}^{k}_{X}:=\operatorname{Gr}^{k}_{F}\underline{\Omega}^{\bullet}_{X}[k],$

is an object in $D^{b}_{\rm coh}(X)$, called the _$k$ -th Du Bois complex_ of
$X$. It follows from definition that

$\underline{\Omega}_{X}^{k}\simeq{\bf
R}\epsilon_{\bullet*}\Omega_{X_{\bullet}}^{k},$

where $\epsilon_{\bullet}\colon X_{\bullet}\to X$ is a hyperresolution of $X$.

Besides [DB], we refer for instance to [GNPP, Chapter V] or to [PS, Chapter
7.3] for the construction of hyperresolutions, and for a detailed treatment of
Du Bois complexes. Here we only collect the main properties that will be
needed in this paper; these will be used freely in the most obvious instances.

######  Lemma 1.1.

(1) We have $\underline{\Omega}_{X}^{k}|_{U}\simeq\underline{\Omega}_{U}^{k}$
for any open subset $U\subset X$, and any morphism $f:X\to Y$ induces a
canonical morphism $\underline{\Omega}_{Y}^{k}\to{\bf
R}f_{*}\underline{\Omega}_{X}^{k}$; see [DB, Section 3] and [PS, Chapter 7.3].

(2) For each $k\geq 0$, there is a canonical morphism
$\Omega_{X}^{k}\to\underline{\Omega}_{X}^{k}$, which is an isomorphism if $X$
is smooth; here $\Omega_{X}^{k}$ are the sheaves of Kähler differentials on
$X$; see [DB, Section 4.1] or [PS, p.175].

(3) If $X$ is projective, there exists a Hodge-to-de Rham spectral sequence

$E^{p,q}_{1}=\mathbb{H}^{q}(X,\underline{\Omega}_{X}^{p})\implies
H^{p+q}(X,\mathbb{C}),$

which degenerates at $E_{1}$; see [DB, Theorem 4.5(iii)] or [PS, Proposition
7.24].

(4) Given a discriminant square, i.e. a Cartesian square

${E}$${Y}$${Z}$${X}$$\scriptstyle{g}$$\scriptstyle{f}$

such that the horizontal maps are inclusions and $f:Y\smallsetminus E\to
X\smallsetminus Z$ is an isomorphism, then for each $k\geq 0$, we have an
exact triangle

$\underline{\Omega}_{X}^{k}\to\underline{\Omega}_{Z}^{k}\oplus{\bf
R}f_{*}\underline{\Omega}_{Y}^{k}\to{\bf
R}g_{*}\underline{\Omega}_{E}^{k}\xrightarrow{+1}.$

See [DB, Proposition 4.11] or [PS, Example 7.25].

(5) Let $X_{\bullet}\to X$ be a hyperresolution. Then there is a spectral
sequence

$E_{1}^{ij}=H^{j}(X_{i},\Omega_{X_{i}}^{k})\implies\mathbb{H}^{i+j}(X,\underline{\Omega}_{X}^{k}).$

See [GNPP, Chapter V, Lemma 3.1].

Depth of objects in the derived category. The notion of depth of a module has
a natural extension to complexes. Let $(R,\mathfrak{m})$ be a noetherian local
ring endowed with a dualizing complex $\omega_{R}^{\bullet}$ (for us this will
always be the local ring of $X$ at a closed point), and let $C$ be an element
of the bounded derived category of finitely generated $R$-modules. Then one
defines $\operatorname{depth}(C)$ in any of the following equivalent ways:

  1. (1)

$\min~{}\\{i~{}|~{}\operatorname{Ext}^{-i}_{R}(C,\omega_{R}^{\bullet})\neq
0\\}$;

  2. (2)

$\min~{}\\{i~{}|~{}\operatorname{Ext}^{i}_{R}(R/\mathfrak{m},C)\neq 0\\}$;

  3. (3)

$\min~{}\\{i~{}|~{}H^{i}_{\mathfrak{m}}(C)\neq 0\\}$,

with the convention that the depth is $-\infty$ if $C=0$. The last two
properties are studied and shown to be equivalent in [FY], while the
equivalence with the first follows from local duality.

If $X$ is a variety and $\mathcal{C}$ is an element in $D^{b}_{\rm coh}(X)$,
we will especially use the first and third interpretation, in the sense that
if $x\in X$ is a closed point, we have

(1.2)
$\operatorname{depth}_{x}(\mathcal{C})=\min~{}\\{i~{}|~{}\mathcal{E}xt^{-i}_{\mathcal{O}_{X,x}}(\mathcal{C}_{x},\omega_{X,x}^{\bullet})\neq
0\\}=\min~{}\\{i~{}|~{}\mathbb{H}^{i}_{x}(\mathcal{C})\neq 0\\}.$

We also set

$\operatorname{depth}(\mathcal{C}):=\min_{x\in
X}~{}\operatorname{depth}(\mathcal{C}_{x}),$

where the minimum is taken over the closed points of $X$.

###  2\. Local cohomological dimension via Du Bois complexes

Let $X$ be a complex variety. A cohomological invariant that is now understood
to figure prominently in the study of the Du Bois complexes of $X$ is the
local cohomological dimension. If $Y$ is a smooth variety containing $X$
(locally), this can be seen as

$\operatorname{lcd}(X,Y):=\max~{}\\{q~{}|~{}\mathcal{H}^{q}_{X}\mathcal{O}_{Y}\neq
0\\},$

where the sheaf in the parenthesis is the $q$-th local cohomology sheaf of
$\mathcal{O}_{Y}$ along $X$; see [MP1] for an extensive discussion. It is in
fact known that the sheaves $\mathcal{H}^{q}_{X}\mathcal{O}_{Y}$ are non-zero
precisely in the range $\operatorname{codim}_{Y}X\leq
q\leq\operatorname{lcd}(X,Y)$.

It follows from the main results in [Og] or [MP1] that $\dim
Y-\operatorname{lcd}(X,Y)$ depends only on $X$, and not the choice of $Y$; for
instance, in the language of [Og, Theorem 2.13], it is equal to the de Rham
depth of $X$. This provides a convenient invariant for our present purposes.

######  Definition 2.1.

The _local cohomological defect_ ${\rm lcdef}(X)$ of $X$ is

${\rm lcdef}(X):=\operatorname{lcd}(X,Y)-\operatorname{codim}_{Y}X.$

It measures how far $X$ is “numerically” from being a local complete
intersection, and again depends only on $X$, and not on the choice of $Y$;
indeed, we have

${\rm lcdef}(X)=\dim X-\big{(}\dim Y-\operatorname{lcd}(X,Y)\big{)}.$

This also shows that $\dim X\geq{\rm lcdef}(X)\geq 0$.

The key result we use here is [MP1, Theorem E], characterizing the local
cohomological dimension in terms of the vanishing of higher direct images of
forms with log poles on a log resolution. Via results of Steenbrink, this was
equivalently phrased in terms of Du Bois complexes as follows:

######  Theorem 2.2 ([MP1, Corollary 12.6]).

Let $X$ be a subvariety of a smooth variety $Y$. For any integer $c$ we have

$\operatorname{lcd}(X,Y)\leq
c\iff\mathcal{E}xt^{j+k+1}_{\mathcal{O}_{Y}}(\underline{\Omega}_{X}^{k},\omega_{Y})=0\,\,\,\,{\rm
for~{}all~{}}j\geq c\,\,\,\,{\rm and}\,\,\,\,k\geq 0.$

or equivalently

${\rm lcdef}(X)\leq
c\iff\mathcal{E}xt^{j+k+1}_{\mathcal{O}_{X}}(\underline{\Omega}_{X}^{k},\omega_{X}^{\bullet})=0\,\,\,\,{\rm
for~{}all~{}}j\geq c-\dim X\,\,\,\,{\rm and}\,\,\,\,k\geq 0.$

The second equivalent statement is simply obtained by translating into the
language introduced here, and using Grothendieck duality for the inclusion
$X\hookrightarrow Y$. The fundamental consequence is the following:

######  Corollary 2.3.

We have the identity

${\rm lcdef}(X)=\dim X-\underset{k\geq
0}{\min}~{}\\{\operatorname{depth}\underline{\Omega}_{X}^{k}+k\\}.$

###### Proof.

Using Theorem 2.2, the identity follows from (1.2). ∎

This formula opens the door to the study of ${\rm lcdef}(X)$ based on objects
in the bounded derived category of coherent sheaves, providing an alternative
to the topological approach in [Og] (or in [RSW] and [BBLSZ]).

######  Example 2.4 (Varieties with ${\rm lcdef}(X)=0$).

An important point is to recognize those varieties that behave numerically
like local complete intersections, without necessarily being so. Here this is
encoded in the condition ${\rm lcdef}(X)=0$, or equivalently
$\operatorname{lcd}(X,Y)=\operatorname{codim}_{Y}X$ in any embedding. Besides
local complete intersections, this is known to hold for instance when $X$ has
quotient singularities [MP1, Corollary 11.22], when $X$ is affine with Cohen-
Macaulay Stanley-Reisner coordinate algebra [MP1, Corollary 11.26], when $X$
is an arbitrary Cohen-Macaulay surface or threefold by [Og, Remark p.338-339]
and [DT, Corollary 2.8] respectively, and when $X$ is a Cohen-Macaulay
fourfold whose local analytic Picard groups are torsion [DT, Theorem 1.3].

###  3\. Du Bois complexes of cones over singular varieties

The set-up for this entire section and the next is the following: $X$ is a
projective variety of dimension $n$, endowed with an ample line bundle $L$,
and

$Z=C(X,L)=\operatorname{Spec}\big{(}\bigoplus_{m\geq 0}H^{0}(X,L^{m})\big{)}$

is the affine cone over $X$ associated to $L$, with cone point $x\in Z$.

Our first goal is to describe the Du Bois complexes
$\underline{\Omega}_{Z}^{k}$, generalizing the result obtained in [SVV,
Appendix A] (and earlier in different language in [CHWW2]) when $X$ is smooth.
More precisely, noting that $Z$ is affine, we describe the global sections of
their cohomology sheaves. Our second goal is to use this description in order
to compute their depth at the cone point. We combine all of this in the
following:

######  Theorem 3.1.

With the notation above:

(1) The Du Bois complexes $\underline{\Omega}_{Z}^{k}$ are given by

$\Gamma(Z,\mathcal{H}^{0}\underline{\Omega}_{Z}^{0})\simeq\mathbb{C}\oplus\bigoplus_{m\geq
1}\mathbb{H}^{0}(X,\underline{\Omega}_{X}^{0}\otimes
L^{m})\simeq\mathbb{C}\oplus\bigoplus_{m\geq
1}H^{0}(X,\mathcal{H}^{0}\underline{\Omega}_{X}^{0}\otimes L^{m}),$
$\Gamma(Z,\mathcal{H}^{i}\underline{\Omega}_{Z}^{0})\simeq\bigoplus_{m\geq
1}\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{0}\otimes L^{m})\text{ for $i\geq
1$},$

and for $k\geq 1$,

$\Gamma(Z,\mathcal{H}^{i}\underline{\Omega}_{Z}^{k})\cong\bigoplus_{m\geq
1}\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k}\otimes
L^{m})\oplus\bigoplus_{m\geq
1}\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k-1}\otimes L^{m}).$

(2) We have $\operatorname{depth}_{x}\underline{\Omega}_{Z}^{k}\geq 1$ for
every $k\geq 0$. Moreover, if $d\geq 1$ is an integer:

(i) $\operatorname{depth}_{x}\underline{\Omega}_{Z}^{0}>d$ if and only if

  * •

$\mathbb{H}^{0}(X,\underline{\Omega}_{X}^{0}\otimes L^{m})=0$ for all
$m\leq-1$;

  * •

$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{0}\otimes L^{m})=0$ for all $m\leq
0$ and $1\leq i\leq d-1$.

(ii) If $k\geq 1$, then $\operatorname{depth}_{x}\underline{\Omega}_{Z}^{k}>d$
if and only if

  * •

$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k}\otimes
L^{m})=\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k-1}\otimes L^{m})=0$ for
$m\leq-1$ and $0\leq i\leq d-1$;

  * •

$\mathbb{H}^{0}(X,\underline{\Omega}_{X}^{k})=0$;

  * •

The map
$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k-1})\to\mathbb{H}^{i+1}(X,\underline{\Omega}_{X}^{k})$
is an isomorphism for $0\leq i\leq d-2$ and injective for $i=d-1$.

In this section we focus on the proof of Theorem 3.1(1), while (2) will be
proved in the next. To this end, we consider the blow-up square

${E\simeq X}$${{\widetilde{Z}}=\mathbf{Spec}_{X}\big{(}\bigoplus_{m\geq
0}L^{m}\big{)}}$${x}$${Z}$$\scriptstyle{i}$$\scriptstyle{f}$$\scriptstyle{f}$

obtained by blowing up $Z$ at the cone point. We denote

$U:=Z\smallsetminus\\{x\\}=\widetilde{Z}\smallsetminus E.$

Using Lemma 1.1(4), we have the following exact triangles:

$\underline{\Omega}_{Z}^{0}\to{\bf
R}f_{*}\underline{\Omega}_{\widetilde{Z}}^{0}\oplus\mathcal{O}_{x}\to{\bf
R}f_{*}\underline{\Omega}_{E}^{0}\xrightarrow{+1}$

and

$\underline{\Omega}_{Z}^{k}\to{\bf
R}f_{*}\underline{\Omega}_{\widetilde{Z}}^{k}\to{\bf
R}f_{*}\underline{\Omega}_{E}^{k}\xrightarrow{+1}$

for $k\geq 1$. Since $\widetilde{Z}$ is an $\mathbb{A}^{1}$-bundle over
$E\simeq X$, these exact triangles are split:

######  Lemma 3.2.

With the notation above, for each $k\geq 0$ there exists a “pullback” map

${\bf R}f_{*}\underline{\Omega}_{E}^{k}\to{\bf
R}f_{*}\underline{\Omega}^{k}_{\widetilde{Z}}$

inverse to the natural restriction map ${\bf
R}f_{*}\underline{\Omega}^{k}_{\widetilde{Z}}\to{\bf
R}f_{*}\underline{\Omega}_{E}^{k}$.

As a consequence, after passing to hypercohomology, the exact triangles above
lead to direct sum decompositions:

(3.3)
$\mathbb{H}^{0}(\widetilde{Z},\underline{\Omega}^{0}_{\widetilde{Z}})\oplus\mathbb{C}\simeq\mathbb{H}^{0}(Z,\underline{\Omega}_{Z}^{0})\oplus\mathbb{H}^{0}(E,\underline{\Omega}_{E}^{0})$

and

(3.4)
$\mathbb{H}^{i}(\widetilde{Z},\underline{\Omega}^{k}_{\widetilde{Z}})\simeq\mathbb{H}^{i}(Z,\underline{\Omega}_{Z}^{k})\oplus\mathbb{H}^{i}(E,\underline{\Omega}_{E}^{k})$

for $i\geq 0$ and $k\geq 1$. Moreover, the summands
$\mathbb{H}^{i}(E,\underline{\Omega}_{E}^{k})$ can be identified with
$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k})$, via the zero section of $\pi$.
We next compute $\pi_{*}\underline{\Omega}^{k}_{\widetilde{Z}}$, via the
projection $\pi\colon\widetilde{Z}\to X$.

######  Lemma 3.5.

With the notation above, we have

$\pi_{*}\underline{\Omega}^{0}_{\widetilde{Z}}\simeq\bigoplus_{m\geq
0}\underline{\Omega}_{X}^{0}\otimes L^{m},$

and for each $k\geq 1$, a split exact triangle

$\bigoplus_{m\geq 0}\underline{\Omega}_{X}^{k}\otimes
L^{m}\to\pi_{*}\underline{\Omega}^{k}_{\widetilde{Z}}\to\bigoplus_{m\geq
1}\underline{\Omega}_{X}^{k-1}\otimes L^{m}\xrightarrow{+1}.$

###### Proof.

Recall that $\widetilde{Z}=\mathbf{Spec}_{X}(\operatorname{Sym}L)$. Given a
hyperresolution $\epsilon_{\bullet}:X_{\bullet}\to X$, it induces a
hyperresolution
$\widetilde{Z}_{\bullet}=\mathbf{Spec}_{X_{\bullet}}(\operatorname{Sym}\epsilon_{\bullet}^{*}L)$
of $\widetilde{Z}$. For each $\widetilde{Z_{i}}\to X_{i}$, we have

$\pi_{*}\mathcal{O}_{\widetilde{Z_{i}}}\simeq\bigoplus_{m\geq 0}L^{m}$

and for $k\geq 1$, a split exact sequence

$0\to\bigoplus_{m\geq
0}\Omega_{X_{i}}^{p}\otimes\epsilon_{i}^{*}L^{m}\to\pi_{*}\Omega^{p}_{\widetilde{Z}_{i}}\to\bigoplus_{m\geq
1}\Omega_{X_{i}}^{p-1}\otimes\epsilon_{i}^{*}L^{m}\to 0.$

Since the splitting given by $d$ is compatible with the maps in the
hyperresolution, we get the result by pushing these forward to $X$, and using
the projection formula. ∎

We are now ready to deduce the result stated at the beginning of the section:

###### Proof of Theorem 3.1(1).

Putting together (3.3), (3.4), and Lemma 3.5, we obtain

$\mathbb{H}^{0}(Z,\underline{\Omega}_{Z}^{0})\simeq\mathbb{C}\oplus\bigoplus_{m\geq
1}\mathbb{H}^{0}(X,\underline{\Omega}_{X}^{0}\otimes L^{m})$
$\mathbb{H}^{i}(Z,\underline{\Omega}_{Z}^{0})\simeq\bigoplus_{m\geq
1}\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{0}\otimes L^{m})\,\,\,\,{\rm
for}\,\,\,\,i\geq 1,$

and for each $i$ and $k\geq 1$, a split exact sequence

$0\to\bigoplus_{m\geq 1}\mathbb{H}^{i}(\underline{\Omega}_{X}^{k}\otimes
L^{m})\to\mathbb{H}^{i}(Z,\underline{\Omega}^{k}_{Z})\to\bigoplus_{m\geq
1}\mathbb{H}^{i}(\underline{\Omega}_{X}^{k-1}\otimes L^{m})\to 0.$

Since $Z$ is affine, we have
$\Gamma(Z,\mathcal{H}^{i}\underline{\Omega}_{Z}^{k})=\mathbb{H}^{i}(Z,\underline{\Omega}_{Z}^{k})$,
from which the result is immediate. ∎

It is also worth noting the following quick consequence of Theorem 3.1(1),
presumably known to experts:

######  Corollary 3.6.

The cone $Z=C(X,L)$ is seminormal if and only if $X$ is seminormal.

###### Proof.

It is well known that a variety $X$ is seminormal if and only if the natural
map $\mathcal{O}_{X}\to\mathcal{H}^{0}\underline{\Omega}^{0}_{X}$ is an
isomorphism.

In our case, the cone $Z$ is seminormal if and only if

$\varphi:\Gamma(Z,\mathcal{O}_{Z})=\bigoplus_{m\geq
0}H^{0}(X,L^{m})\to\Gamma(Z,\mathcal{H}^{0}\underline{\Omega}_{Z}^{0})=\mathbb{C}\oplus\bigoplus_{m\geq
1}H^{0}(X,\mathcal{H}^{0}\underline{\Omega}_{X}^{0}\otimes L^{m})$

is an isomorphism. Hence it is clear that if $X$ is seminormal, then so is
$Z$.

Conversely, suppose $Z$ is seminormal. Since the map $\varphi$ preserves the
grading given by $L$, we have isomorphisms

$H^{0}(X,L^{m})\to H^{0}(X,\mathcal{H}^{0}\underline{\Omega}_{X}^{0}\otimes
L^{m})$

for all $m\geq 1$, induced by the inclusion of sheaves
$\mathcal{O}_{X}\hookrightarrow\mathcal{H}^{0}\underline{\Omega}_{X}^{0}$;
using Serre’s theorems, it is then straightforward to check that this
inclusion has to be an isomorphism. ∎

Finally, when $L$ is very ample, one can also consider the classical cone
$C(X)$ over the corresponding embedding in projective space. For future use,
we record the following result that relates $C(X,L)$ to $C(X)$.

######  Lemma 3.7.

Suppose $L=\mathcal{O}_{X}(1)$ is very ample, defining an embedding
$X\subset\mathbb{P}^{n}$. Then for each $k$,

$\underline{\Omega}^{k}_{C(X)}\to{\bf
R}\pi_{*}\underline{\Omega}^{k}_{C(X,L)}$

is a quasi-isomorphism, where $\pi$ is the natural map $C(X,L)\to C(X)$.
Consequently, $\operatorname{lcdef}(C(X))=\operatorname{lcdef}(C(X,L))$.

###### Proof.

The isomorphism $\underline{\Omega}_{C(X)}^{k}\to{\bf
R}\pi_{*}\underline{\Omega}^{k}_{C(X,L)}$ follows from Lemma 1.1(4), given
that $\pi$ is an isomorphism away from the vertex points. The last assertion
then follows from Corollary 2.3.

∎

###  4\. Depth of Du Bois complexes of cones

In this section we focus on the proof of Theorem 3.1(2). In other words, we
characterize the depth of $\underline{\Omega}_{Z}^{k}$, where $Z$ is the cone
over a possibly singular projective variety $X$, in terms of cohomological
data of the Du Bois complexes of $X$.

To do so, still using the notation of the previous section, we must first also
understand the Du Bois complexes of the complement $U=Z\smallsetminus\\{x\\}$
of the vertex of the cone. Note first that a completely similar argument as in
Lemma 3.5 gives:

######  Lemma 4.1.

For $U=\widetilde{Z}\smallsetminus E=Z\smallsetminus\\{x\\}$ we have

$\pi_{*}\underline{\Omega}^{0}_{U}\simeq\bigoplus_{m\in\mathbb{Z}}\underline{\Omega}_{X}^{0}\otimes
L^{m},$

and for each $k\geq 1$, a (not necessarily split) exact triangle

$\bigoplus_{m\in\mathbb{Z}}\underline{\Omega}_{X}^{k}\otimes
L^{m}\to\pi_{*}\underline{\Omega}^{k}_{U}\to\bigoplus_{m\in\mathbb{Z}}\underline{\Omega}_{X}^{k-1}\otimes
L^{m}\xrightarrow{+1}.$

Passing to hypercohomology, we obtain

$\mathbb{H}^{i}(U,\underline{\Omega}^{0}_{U})\simeq\bigoplus_{m\in\mathbb{Z}}\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{0}\otimes
L^{m})$

for $i\geq 0$, and for $k\geq 1$, a long exact sequence

$\cdots\to\bigoplus_{m\in\mathbb{Z}}\mathbb{H}^{i}(\underline{\Omega}_{X}^{k}\otimes
L^{m})\to\mathbb{H}^{i}(U,\underline{\Omega}^{k}_{U})\to\bigoplus_{m\in\mathbb{Z}}\mathbb{H}^{i}(\underline{\Omega}_{X}^{k-1}\otimes
L^{m})\xrightarrow{d_{i}}\bigoplus_{m\in\mathbb{Z}}\mathbb{H}^{i+1}(\underline{\Omega}_{X}^{k}\otimes
L^{m})\to\cdots$

######  Lemma 4.2.

The differentials $d_{i}$ in the long exact sequence above are zero, except on
the $m=0$ summand, where

$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k-1})\xrightarrow{d_{i}}\mathbb{H}^{i+1}(X,\underline{\Omega}_{X}^{k})$

is induced by cup product with $c_{1}(\epsilon_{j}^{*}L)$ on each term $X_{j}$
in a hyperresolution of $X$.

###### Proof.

For each $k$ and $m$, completely analogously to Lemma 1.1(5), we have a
spectral sequence

$E_{1}^{q,p}=H^{p}(X_{q},\Omega_{X_{q}}^{k}\otimes\epsilon_{q}^{*}L^{m})\implies\mathbb{H}^{p+q}(X,\underline{\Omega}_{X}^{k}\otimes
L^{m}),$

by means of which the differentials $d_{i}$, with $i=p+q$, are induced by the
corresponding differentials

$H^{p}(X_{q},\Omega_{X_{q}}^{k-1}\otimes\epsilon_{q}^{*}L^{m})\xrightarrow{d_{p}}H^{p+1}(X_{q},\Omega_{X_{q}}^{k}\otimes\epsilon_{q}^{*}L^{m})$

on each term in the hyperresolution.

It suffices therefore to show the statement of the Lemma when $X$ is smooth,
meaning that in this case

$d_{p}\colon H^{p}(X,\Omega_{X}^{k-1}\otimes L^{m})\to
H^{p+1}(X,\Omega_{X}^{k}\otimes L^{m})$

is $0$ on the terms with $m\neq 0$, and cup product with $c_{1}(L)$ on the
terms with $m=0$.

To this end, let $\omega\in H^{p}(X,\Omega_{X}^{k-1}\otimes L^{m})$ be
represented by the Čech $p$-cocycle

$\\{\omega_{i_{0}i_{1}\cdots i_{p}}t_{i_{0}i_{1}\cdots
i_{p}}^{m}\\}_{i_{0}i_{1}\cdots
i_{p}}\in\check{C}^{p}(X,\Omega^{k-1}_{X}\otimes L^{m}).$

where we fix an open cover $\\{U_{i}=\operatorname{Spec}A_{i}\\}_{i}$ over
which $L=t_{i}A_{i}$ is locally free, and we denote by $t_{i_{0}i_{1}\cdots
i_{p}}:=t_{i_{0}}|_{U_{i_{0}}\cap\cdots\cap U_{i_{p}}}$. To simplify notation,
we will denote this restriction by $t_{i_{0}}$ as well.

If $m\neq 0$, then $\omega$ lifts to a cocycle
$\\{\frac{1}{m}d(\omega_{i_{0}i_{1}\cdots
i_{p}}t_{i_{0}}^{m})\\}_{i_{0}i_{1}\cdots
i_{p}}\in\check{C}^{p}(\Omega^{k}_{\tilde{Z}\smallsetminus E})$, thus its
image in $\check{C}^{p+1}(\Omega^{k}_{\tilde{Z}\smallsetminus E})$ is zero.

If $m=0$, then $\omega$ lifts to $\\{\frac{1}{t_{i_{0}i_{1}\cdots
i_{p}}}d(\omega_{i_{0}i_{1}\cdots i_{p}}t_{i_{0}})\\}_{i_{0}i_{1}\cdots
i_{p}}\in\check{C}^{p}(\Omega^{k}_{\tilde{Z}\smallsetminus E})$. Using the
cocycle condition $\sum_{j}(-1)^{j}\omega_{i_{0}i_{1}\cdots\hat{i_{j}}\cdots
i_{p}i_{p+1}}=0$, one computes that

$\displaystyle\varphi_{p}(\omega)$ $\displaystyle=\\{\omega_{i_{1}i_{2}\cdots
i_{p+1}}\wedge\frac{dt_{i_{1}}}{t_{i_{1}}}+\sum_{j>0}(-1)^{j}\omega_{i_{0}i_{1}\cdots\hat{i_{j}}\cdots
i_{p}i_{p+1}}\wedge\frac{dt_{i_{0}}}{t_{i_{0}}}\\}_{i_{0}i_{1}\cdots
i_{p}i_{p+1}}$ $\displaystyle=\\{\omega_{i_{1}i_{2}\cdots
i_{p+1}}\wedge(\frac{dt_{i_{1}}}{t_{i_{1}}}-\frac{dt_{i_{0}}}{t_{i_{0}}})\\}_{i_{0}i_{1}\cdots
i_{p}i_{p+1}},$

which (up to sign) represents the class $\omega\cup\\{d\log g_{ij}\\}_{i,j}$,
where $g_{ij}$ are the transition maps of $L$, satisfying $t_{i}=g_{ij}t_{j}$.
Since the class of $d\log g_{ij}$ can be identified with $c_{1}(L)\in
H^{2}(X,\mathbb{Z})\cap H^{1,1}(X)$ (see e.g. [GH, p.141]), we obtain the
conclusion. ∎

######  Remark 4.3.

It is well known that for line bundle $L$ on a (possibly singular) projective
variety $X$, we have

$c_{1}(L)\in
F^{1}H^{2}(X,\mathbb{C})=H^{2}(X,F^{1}\underline{\Omega}_{X}^{\bullet}),$

and therefore it defines an element in
$\mathbb{H}^{1}(X,\underline{\Omega}_{X}^{1})$; see e.g. [AK] for a
discussion. The differentials in Lemma 4.2 can be seen as cup product with
this element; equivalently, they arise from the action of $c_{1}(L)$ on
singular cohomology

$\cdot\cup c_{1}(L)\colon H^{i+k-1}(X,\mathbb{C})\to H^{i+k+1}(X,\mathbb{C}),$

via Lemma 1.1(3). However, Lemma 4.2 can also be taken to be a definition of
this action.

When in addition $L$ is very ample, giving an embedding
$\iota:X\hookrightarrow\mathbb{P}^{N}$, there are induced morphisms of mixed
Hodge structures $\iota^{*}_{i}:H^{i}(\mathbb{P}^{N},\mathbb{C})\to
H^{i}(X,\mathbb{C})$, commuting with cup product, and such that
$\iota^{*}_{2}(c_{1}(\mathcal{O}(1)))=c_{1}(L)$. Thus the Lefschetz maps on
$\mathbb{P}^{N}$ and $X$ are compatible, in the sense that for each $i$ there
is a commutative diagram

${H^{i}(\mathbb{P}^{N},\mathbb{C})}$${H^{i+2}(\mathbb{P}^{N},\mathbb{C})}$${H^{i}(X,\mathbb{C})}$${H^{i+2}(X,\mathbb{C}).}$$\scriptstyle{c_{1}(\mathcal{O}(1))}$$\scriptstyle{c_{1}(L)}$

As a consequence of the calculations above and of the results of the previous
section, we obtain:

######  Corollary 4.4.

With notation as above, we have:

(i) The maps
$\mathbb{H}^{i}(Z,\underline{\Omega}_{Z}^{k})\to\mathbb{H}^{i}(U,\underline{\Omega}_{U}^{k})$
induced by restriction are injective for all $i,k$.

(ii) The map
$\mathbb{H}^{0}(Z,\underline{\Omega}_{Z}^{0})\to\mathbb{H}^{0}(U,\underline{\Omega}_{U}^{0})$
is an isomorphism if and only if

$\mathbb{H}^{0}(X,\underline{\Omega}_{X}^{0}\otimes L^{m})=0\,\,\,\,\text{ for
all }\,\,m\leq-1.$

For $i\geq 1$, the map
$\mathbb{H}^{i}(Z,\underline{\Omega}_{Z}^{0})\to\mathbb{H}^{i}(U,\underline{\Omega}_{U}^{0})$
is an isomorphism if and only if

$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{0}\otimes L^{m})=0\,\,\,\,\text{ for
all }\,\,m\leq 0.$

(iii) For $k\geq 1$, the map
$\mathbb{H}^{i}(Z,\underline{\Omega}_{Z}^{k})\to\mathbb{H}^{i}(U,\underline{\Omega}_{U}^{k})$
is an isomorphism if and only if

  * •

$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k}\otimes
L^{m})=\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k-1}\otimes L^{m})=0$ for
$m\leq-1$;

  * •

The differential

$\mathbb{H}^{i-1}(X,\underline{\Omega}_{X}^{k-1})\xrightarrow{d_{i}}\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k})\text{
is surjective}$

and the differential

$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k-1})\xrightarrow{d_{i}}\mathbb{H}^{i+1}(X,\underline{\Omega}_{X}^{k})\text{
is injective}.$

We are finally ready to compute the depth at the Du Bois complexes of $Z$,
which is crucial for the proof of Theorem 6.1, but may also be of independent
interest.

###### Proof of Theorem 3.1(2).

Recall that we denote $U=Z\smallsetminus\\{x\\}$. For each $k$, there is a
long exact sequence

$\cdots\to\mathbb{H}^{i}_{x}(Z,\underline{\Omega}_{Z}^{k})\to\mathbb{H}^{i}(Z,\underline{\Omega}_{Z}^{k})=H^{0}(Z,\mathcal{H}^{i}\underline{\Omega}_{Z}^{k})\to\mathbb{H}^{i}(U,\underline{\Omega}_{U}^{k})\to\cdots.$

Thus, the condition

$\operatorname{depth}_{x}\underline{\Omega}_{Z}^{k}>d\iff\mathbb{H}_{x}^{i}(Z,\underline{\Omega}_{Z}^{k})=0\,\,\,\,\text{
for all }\,\,\,\,i\leq d$

is equivalent to the following two conditions:

  1. (1)

$\mathbb{H}^{i}(Z,\underline{\Omega}_{Z}^{k})\to\mathbb{H}^{i}(U,\underline{\Omega}_{U}^{k})$
is an isomorphism for $0\leq i\leq d-1$;

  2. (2)

$\mathbb{H}^{d}(Z,\underline{\Omega}_{Z}^{k})\to\mathbb{H}^{d}(U,\underline{\Omega}_{U}^{k})$
is injective.

The assertion is then an immediate consequence of Corollary 4.4. ∎

######  Remark 4.5 (The case when $X$ is smooth).

We make some remarks about the content of Theorem 3.1(2) when $X$ is smooth,
as a blueprint for what we should aim for in the singular case. With this
assumption, the statement translates into the fact, say when $k\geq 1$, that
$\operatorname{depth}_{x}\underline{\Omega}_{Z}^{k}>d$ is equivalent to the
following two conditions being satisfied simultaneously:

(i) $H^{i}(X,\Omega_{X}^{k}\otimes L^{\otimes
m})=H^{i}(X,\Omega_{X}^{k-1}\otimes L^{\otimes m})=0$ for all $m\leq-1$ and
$0\leq i\leq d-1$.

(ii) The maps $H^{k-1,i}(X)\to H^{k,i+1}(X)$ induced by cup product with
$c_{1}(L)$ are isomorphisms for $-1\leq i\leq d-2$, and injective for $i=d-1$.

If $d\leq n-k+2$, then $i\leq d-1$ implies $i+k<n=\dim X$, and therefore
condition (i) holds automatically thanks to Nakano vanishing. The injectivity
of all the maps in (ii) is also automatic, because of the Hard Lefschetz
theorem. These observations and similar ones for $k=0$, in conjunction with
Corollary 2.3 and Remark 4.3, essentially finish the proof of Theorem A when
$X$ is smooth.

Thus when $X$ is singular we will need in particular a suitable replacement
for the Nakano vanishing theorem. This is of independent interest, and is the
topic of the next section.

###  5\. Vanishing for dual Du Bois complexes

The Du Bois complexes of a projective variety $X$ of dimension $n$ are known
by [GNPP, Theorem V.5.1] to satisfy the analogue of the Kodaira-Akizuki-Nakano
vanishing theorem, in the sense that

$\mathbb{H}^{q}(X,\underline{\Omega}_{X}^{p}\otimes L)=0\,\,\,\,\,\,{\rm
for~{}all}\,\,\,\,p+q>n,$

for any ample line bundle $L$ on $X$. The Serre dual statement for twists by
$L^{-1}$ does not however hold automatically, as in general we only have
morphisms
$\underline{\Omega}_{X}^{n-p}\to\mathbb{D}(\underline{\Omega}_{X}^{p})$ that
are not necessarily isomorphisms, where

$\mathbb{D}(-)={\bf R}\mathcal{H}om(-,\omega_{X}^{\bullet})[-n]$

is the (shifted) Grothendieck duality functor. Instead, the correct statement
turns out to be the following, which is an expanded version of Theorem C in
the Introduction.

######  Theorem 5.1.

Let $X$ be a projective variety of dimension $n$, and $L$ an ample line bundle
on $X$. Then

$\mathbb{H}^{q}(X,\mathbb{D}(\underline{\Omega}_{X}^{p})\otimes
L)=0\quad\text{ for }\,\,\,\,q-p>\operatorname{lcdef}(X).$

Equivalently,

$\mathbb{H}^{q}(X,\underline{\Omega}_{X}^{p}\otimes L^{-1})=0\quad\text{ for
}\,\,\,\,p+q<n-\operatorname{lcdef}(X).$

###### Proof.

The second statement follows from the first by Grothendieck-Serre duality.

To check the first statement, fix an embedding
$X\hookrightarrow\mathbb{P}^{N}$ given by a sufficiently high power of $L$.
Let $\mathbb{Q}_{\mathbb{P}^{N}}^{H}[N]$ denote the trivial Hodge module on
$\mathbb{P}^{N}$. Then, as in [MP1, Proposition 13.1], the Du Bois complexes
satisfies

$\underline{\Omega}_{X}^{p}\simeq{\bf
R}\mathcal{H}om_{\mathcal{O}_{Y}}(\operatorname{Gr}^{F}_{p-N}\operatorname{DR}_{\mathbb{P}^{N}}i_{*}i^{!}\mathbb{Q}_{\mathbb{P}^{N}}^{H}[N],\omega_{\mathbb{P}^{N}}^{\bullet})[p-N]$

Dualizing, we obtain

$\displaystyle\mathbb{D}(\underline{\Omega}_{X}^{p})$ $\displaystyle={\bf
R}\mathcal{H}om_{\mathcal{O}_{X}}(\underline{\Omega}_{X}^{p},\omega_{X}^{\bullet})[-n]$
$\displaystyle\cong{\bf
R}\mathcal{H}om_{\mathcal{O}_{\mathbb{P}^{N}}}(\underline{\Omega}_{X}^{p},\omega_{\mathbb{P}^{N}}^{\bullet})[-n]$
$\displaystyle\cong(\operatorname{Gr}^{F}_{p-N}\operatorname{DR}_{\mathbb{P}^{N}}i_{*}i^{!}\mathbb{Q}_{\mathbb{P}^{N}}^{H}[N])[N-p-n].$

Consider now the spectral sequence

$E_{2}^{i,j}=\mathbb{H}^{i}(X,\operatorname{Gr}^{F}_{p-N}\operatorname{DR}_{\mathbb{P}^{N}}\mathcal{H}^{j}(i_{*}i^{!}\mathbb{Q}_{\mathbb{P}^{N}}^{H}[N])\otimes
L)\implies$
$\implies\mathbb{H}^{i+j}(X,\operatorname{Gr}^{F}_{p-N}\operatorname{DR}_{\mathbb{P}^{N}}(i_{*}i^{!}\mathbb{Q}_{Y}^{H}[N])\otimes
L).$

We have that

  * •

$\mathcal{H}^{j}(i_{*}i^{!}\mathbb{Q}_{\mathbb{P}^{N}}^{H}[N])=\mathcal{H}^{j}_{X}\mathcal{O}_{\mathbb{P}^{N}}=0$
for $j>\operatorname{lcd}(X,\mathbb{P}^{N})$;

  * •

$\mathbb{H}^{i}(X,\operatorname{Gr}^{F}_{p-N}\operatorname{DR}_{\mathbb{P}^{N}}\mathcal{H}^{j}(i_{*}i^{!}\mathbb{Q}_{\mathbb{P}^{N}}^{H}[N])\otimes
L)$=0 for $i>0$, by Saito’s vanishing theorem for mixed Hodge modules [Sa,
Scn.2.g].

It follows that $E_{2}^{i,j}=0$ if $i+j>\operatorname{lcd}(X,\mathbb{P}^{N})$.
We conclude that

$\mathbb{H}^{q}(X,\mathbb{D}(\underline{\Omega}_{X}^{p})\otimes L)=0$

when $q-p+N-n>\operatorname{lcd}(X,\mathbb{P}^{N})$, or equivalently $q-p>{\rm
lcdef}(X)$, as required. ∎

###  6\. Proof of the main result

The following statement is the more general version of Theorem A promised in
the Introduction, that applies to any abstract cone over $X$.

######  Theorem 6.1.

Let $X$ be an $n$-dimensional projective variety endowed with an ample line
bundle $L$, and let $Z=C(X,L)$ be the associated cone over $X$. Then the
following are equivalent:

  1. (1)

$\operatorname{lcdef}(Z)\leq c$

  2. (2)

$\operatorname{lcdef}(X)\leq c$, and the “Lefschetz” maps

$H^{i}(X,\mathbb{C})\to H^{i+2}(X,\mathbb{C})$

given by cup product with $c_{1}(L)$ are isomorphisms for $-1\leq i\leq n-3-c$
and injective for $i\leq n-2-c$, with the convention that
$H^{-1}(X,\mathbb{C})=0$.111In particular, all $H^{i}(X,\mathbb{C})$ in this
range are $0$ for $i$ odd, and $1$-dimensional for $i$ even.

###### Proof.

First we note the inequality

(6.2) $\operatorname{lcdef}(X)\leq\operatorname{lcdef}(Z).$

Indeed, we clearly have
$\operatorname{lcdef}(Z\smallsetminus\\{x\\})\leq\operatorname{lcdef}(Z)$,
where $x\in Z$ is the cone point, while
$\operatorname{lcdef}(Z\smallsetminus\\{x\\})=\operatorname{lcdef}(X)$, since
$Z\smallsetminus\\{x\\}$ is a locally trivial $\mathbb{C}^{*}$-bundle over
$X$.

Note now that by Corollary 2.3, we have

${\rm lcdef}(Z)=\dim Z-\min_{k\geq
0}~{}\\{\operatorname{depth}\underline{\Omega}_{Z}^{k}+k\\}.$

As a consequence, we have an equivalence between the following two conditions:

  1. (1)

$\operatorname{lcdef}(Z)\leq c$

  2. (2)

$\operatorname{lcdef}(X)\leq c$, and

$\operatorname{depth}_{x}\underline{\Omega}_{Z}^{k}\geq n+1-c-k$

holds for all $k\geq 0$.

Hence we focus on understanding when this last condition holds. By Theorem
3.1(2), applied with $d=n-c-k$,222Note that the Proposition applies when
$d\geq 1$, or equivalently $k\leq n-c-1$. it is in turn equivalent to the
following conditions being satisfies simultaneously:

  1. (a)

$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{0})=0$ for $1\leq i\leq n-c-1$;

  2. (b)

$\mathbb{H}^{0}(X,\underline{\Omega}_{X}^{k})=0$ for $1\leq k\leq n-c-1$;

  3. (c)

$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k}\otimes L^{m})=0$ for all
$m\leq-1$ and $i+k\leq n-c-1$;

  4. (d)

For $1\leq k\leq n-c-2$, the map
$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k-1})\to\mathbb{H}^{i+1}(X,\underline{\Omega}_{X}^{k})$
is isomorphism when $0\leq i\leq n-c-k-2$, and injective when $i=n-c-k-1$.

The main point is now to note that by Theorem C, condition (c) is guaranteed
to hold precisely when $\operatorname{lcdef}(X)\leq c$.

On the other hand, condition (a), (b) and (d) taken together are easily seen
to be equivalent to

  1. (d′)

For all $i,k\geq 0$, the map

$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{k})\to\mathbb{H}^{i+1}(X,\underline{\Omega}_{X}^{k+1})$

is an isomorphism when $-1\leq i+k\leq n-c-3$ and injective when $i+k=n-c-2$.

This in turn is equivalent to the condition on the Lefschetz maps in (2) in
the statement of the Theorem, using Lemma 1.1(3) and the fact that these maps
are compatible with the differentials in Lemma 4.2; see also Remark 4.3. ∎

###### Proof of Theorem A.

In the classical setting, we have in addition that $X$ is embedded in some
$\mathbb{P}^{N}$, and $L=\mathcal{O}_{X}(1)$. By Lemma 3.7, it suffices to
prove Theorem A for $Z=C(X,L)$. The point is simply to observe that the Hodge-
theoretic conditions in (2) in Theorem 6.1 are in this case equivalent to the
condition that the restriction maps are isomorphisms

(6.3)
$H^{i}(\mathbb{P}^{N},\mathbb{C})\overset{\simeq}{\longrightarrow}H^{i}(X,\mathbb{C})\,\,\,\,\,\,{\rm
for~{}all}\,\,\,\,i\leq n-1-c.$

Note that the Lefschetz maps on $\mathbb{P}^{N}$ and $X$ are compatible, as in
the diagram in Remark 4.3. Recall that the second condition in (2) in Theorem
6.1 says that the maps

$H^{i}(X,\mathbb{C})\to H^{i+2}(X,\mathbb{C}),$

given by cup product with $c_{1}(L)$, are isomorphisms for all $-1\leq i\leq
n-3-c$, and injective for $i\leq n-2-c$. This condition certainly holds if
($\ref{eqn:WL})$ is satisfied. (Injectivity holds for $i=n-2-c$, since in this
case $H^{i}(X,\mathbb{C})$ must be $0$ or $1$-dimensional, while the Lefschetz
map is nontrivial if $H^{i}(\mathbb{P}^{N},\mathbb{C})$ is so.)

Conversely, since $H^{1}(X,\mathbb{C})=0$, assuming that the bottom horizontal
maps in the diagram are isomorphisms for $i\leq n-3-c$, it is clear that (6.3)
holds. ∎

##  C. Other applications

Here we describe further consequences of our calculation of the Du Bois
complexes of cones over singular varieties.

###  7\. Higher Du Bois singularities and Bott vanishing

A natural strengthening of the notion of Du Bois singularities has been
considered in [MOPW], [JKSY], [MP1] and [MP2] in the case of local complete
intersections. The weaker condition of _pre- $p$-Du Bois_ singularities was
defined in general in [SVV]; it means that up to degree $p$ the Du Bois
complexes have no higher cohomologies:

$\underline{\Omega}_{X}^{j}\simeq\mathcal{H}^{0}\underline{\Omega}_{X}^{j}\,\,\,\,\,\,{\rm
for~{}all}\,\,\,\,j\leq p.$

Using Theorem 3.1(1), we obtain a criterion for cones over singular varieties
to be pre-$p$-Du Bois, extending [SVV, Corollary 7.4].

######  Corollary 7.1.

Let $X$ be a projective variety endowed with an ample line bundle $L$, and let
$Z=C(X,L)$ be the associated cone over $X$. Then $Z$ has pre-$p$-Du Bois
singularities if and only if

$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{j}\otimes L^{m})=0\quad\text{ for
all }\,\,i,m\geq 1,~{}j\leq p.$

Following [KT, Section 2.3], we say that a projective variety $X$ _satisfies
Bott vanishing_ if

$H^{i}(X,\Omega_{X}^{[j]}\otimes A)=0$

for all $j\geq 0$ and $i>0$, and any ample line bundle $A$ on $X$. Here
$\Omega_{X}^{[j]}$ denotes the reflexive differentials
$(\Omega_{X}^{j})^{\vee\vee}$.

######  Corollary 7.2.

Let $X$ be a projective variety with rational singularities. Then the
following are equivalent:

  1. (1)

$Z=C(X,L)$ has pre-$p$-Du Bois singularities for all ample line bundles $L$
and all $p$.

  2. (2)

$X$ has pre-$p$-Du Bois singularities for all $p$ and satisfies Bott
vanishing.

###### Proof.

Due to the main result of [KS], when $X$ has rational singularities we have
$\mathcal{H}^{0}\underline{\Omega}_{X}^{j}\simeq\Omega_{X}^{[j]}$ for all $j$.
Thus (1) follows from (2) by Corollary 7.1. Conversely, suppose first that $X$
does not have pre-$p$-Du Bois singularities, meaning
$\mathcal{H}^{i}\underline{\Omega}_{X}^{j}\neq 0$ for some $i>0$ and $j\leq
p$. By Serre’s theorems, we can choose $L$ sufficiently positive, so that
$H^{s}(X,\mathcal{H}^{t}\underline{\Omega}_{X}^{j}\otimes L)=0$ for all $s>0$
and $t\geq 0$, and $H^{0}(X,\mathcal{H}^{i}\underline{\Omega}_{X}^{j}\otimes
L)\neq 0$. Then, the spectral sequence

$E_{2}^{s,t}=H^{s}(X,\mathcal{H}^{t}\underline{\Omega}_{X}^{j}\otimes
L)\implies\mathbb{H}^{s+t}(X,\underline{\Omega}_{X}^{j}\otimes L)$

degenerates, and gives

$\mathbb{H}^{i}(X,\underline{\Omega}_{X}^{j}\otimes L)\simeq
H^{0}(X,\mathcal{H}^{i}\underline{\Omega}_{X}^{j}\otimes L)\neq 0.$

This is a contradiction, again by Corollary 7.1, which also implies that $X$
satisfies Bott vanishing once we know that it is pre-$p$-Du Bois. ∎

######  Remark 7.3.

For varieties whose singularities are not rational, it may in fact make sense
to define the Bott vanishing condition in terms of
$\mathcal{H}^{0}\underline{\Omega}_{X}^{j}$ rather than $\Omega_{X}^{[j]}$.
The statement of the Corollary, minus the rational singularities assumption,
would remain unchanged.

For example, one can show along the lines of [KT] that if $X$ is a variety
with int-amplified endomorphisms, then for every ample line bundle $A$ on $X$,
we have

$H^{i}(X,\mathcal{H}^{j}\underline{\Omega}_{X}^{k}\otimes A)=0$

for $i>0$ and $j,k\geq 0$.

###  8\. $K$-groups of cones

Another consequence of Theorem 3.1 is the computation of the non-positive
$K$-groups of abstract cones over a projective variety in terms of its Du Bois
complexes, following the approach in [CHWW2] for classical cones over smooth
varieties, and using the main technical result of that paper. Please see the
introduction of _loc. cit._ for a discussion of the problem, and of the
important techniques introduced by the authors in a series of papers dedicated
to the study of $K$-groups.

As in [CHWW1], for any subfield $F\subset\mathbb{C}$ and complex variety $X$,
one can define the $p$-th Du Bois complex of $X$ over $F$ as

$\underline{\Omega}_{X/F}^{p}={\bf
R}\epsilon_{\bullet*}\Omega^{p}_{X_{\bullet}/F}$

where $\epsilon_{\bullet}:X_{\bullet}\to X$ is a hyperresolution of $X$. The
formula for the Du Bois complexes of cones in Theorem 3.1 remains valid in
this setting, with the same proof where $\underline{\Omega}_{X}^{p}$ is
replaced by $\underline{\Omega}_{X/F}^{p}$. Moreover, one checks exactly as in
the slightly more restrictive [CHWW1, Lemma 2.1] that there is an isomorphism

$\underline{\Omega}_{X/F}^{p}\simeq\Omega^{p}_{cdh,X/F},$

where $\Omega^{p}_{cdh,X/F}:={\bf R}a_{*}a^{*}\Omega^{p}_{X/F}$ is the
sheafification of $\Omega_{X/F}^{p}$ in the cdh-topology, the map
$a:X_{cdh}\to X_{zar}$ being the natural change-of-topology morphism.

We are now ready to describe the calculation of non-positive $K$-groups of
cones.

###### Proof of Corollary D.

By [CHWW2, Thereom 1.2], for the affine cone $Z=C(X,L)$ over the
$n$-dimensional projective variety $X$, we have

  * •

$K_{0}(Z)\simeq\mathbb{Z}\oplus\bigoplus_{i=1}^{\dim
Z-1}\Gamma(\mathcal{H}^{i}\underline{\Omega}_{Z/\mathbb{Q}}^{i})/d\big{(}\Gamma(\mathcal{H}^{i}\underline{\Omega}_{Z/\mathbb{Q}}^{i-1})\big{)}$;

  * •

$K_{-\ell}(Z)\simeq\Gamma(Z,\mathcal{H}^{\ell}\underline{\Omega}_{Z/\mathbb{Q}}^{0})\oplus\bigoplus_{i=1}^{n-\ell}\Gamma(Z,\mathcal{H}^{\ell+i}\underline{\Omega}_{Z/\mathbb{Q}}^{i})/d\big{(}\Gamma(Z,\mathcal{H}^{\ell+i}\underline{\Omega}_{Z/\mathbb{Q}}^{i-1})\big{)}$,
$\ell\geq 1$.

Here $d$ is the action on global sections of the differential
$\mathcal{H}^{\ell+i}\underline{\Omega}_{Z/\mathbb{Q}}^{i-1}\to\mathcal{H}^{\ell+i}\underline{\Omega}_{Z/\mathbb{Q}}^{i}$.

Using the calculation of Du Bois complexes of abstract cones in Theorem 3.1(1)
(in place of [CHWW2, Corollary 2.10]) then yields

  * •

$K_{0}(Z)\simeq\mathbb{Z}\oplus\bigoplus_{i=1}^{n}\bigoplus_{m\geq
1}\mathbb{H}^{i}(X,\underline{\Omega}_{X/\mathbb{Q}}^{i}\otimes L^{m})$;

  * •

$K_{-\ell}(Z)\simeq\bigoplus_{i=0}^{n-\ell}\bigoplus_{m\geq
1}\mathbb{H}^{\ell+i}(X,\underline{\Omega}_{X/\mathbb{Q}}^{i}\otimes L^{m})$,
$\ell\geq 1$.

Note that if $X$ is embedded in $\mathbb{P}^{N}$ by $L=\mathcal{O}_{X}(1)$,
one can consider the classical cone $C(X)$, and the method in loc. cit.
applies to this as well. Thus for $\ell\geq 1$ we have

$K_{-\ell}(C(X))=K_{-\ell}(C(X,L))$

thanks to Lemma 3.7. For $\ell=0$, we have

$K_{0}(C(X))\simeq\mathbb{Z}\oplus\operatorname{Pic}(C(X))\oplus\bigoplus_{i=1}^{n}\bigoplus_{m\geq
1}\mathbb{H}^{i}(X,\underline{\Omega}_{X/\mathbb{Q}}^{i}\otimes L^{m})$

where $\operatorname{Pic}(C(X))\simeq R^{+}/R$; here $R$ is the homogeneous
coordinate ring of $X$, and

$R^{+}\simeq\bigoplus_{m\geq
0}\mathbb{H}^{0}(X,\underline{\Omega}_{X}^{0}\otimes L^{m})$

is its seminormalization. The last statement follows from Theorem 3.1(1) and
the well-known fact that the structure sheaf $\mathcal{O}_{Z^{\rm sn}}$ of the
seminormalization of $Z$ can be identified with
$\mathcal{H}^{0}\underline{\Omega}_{Z}^{0}$. For the statement about
$\operatorname{Pic}(C(X))$ we are using [CHWW2, Proposition 1.5]. ∎

######  Remark 8.1.

As a consequence, $K_{-\ell}(Z)=0$ for $l>n+1$, where as usual $n=\dim X$.
This vanishing result holds in greater generality, see [CHSW, Theorem 6.2] and
[KST, Theorem B]. Moreover, if $X$ is a complex algebraic variety with Du Bois
singularities, then

$K_{-n-1}(Z)\simeq\bigoplus_{m=1}^{\infty}H^{n}(X,L^{m}),$

as in the case when $X$ is smooth. More generally, if the complex algebraic
variety $X$ has pre-$p$-Du Bois singularities (see the previous section), the
formula for $K_{-\ell}(Z)$ essentially coincides with that in the smooth case
for $\ell\geq n-p$, since in this case we have that
$\mathcal{H}^{0}\underline{\Omega}_{X/\mathbb{Q}}^{k}\simeq\underline{\Omega}_{X/\mathbb{Q}}^{k}$
for $k\leq p$; indeed, this is the case by definition for
$\underline{\Omega}_{X}^{k}=\underline{\Omega}_{X/\mathbb{C}}^{k}$, and the
statement over $\mathbb{Q}$ follows inductively from this via exact triangles
of the type

$\Omega_{\mathbb{C}/\mathbb{Q}}^{1}\otimes_{\mathbb{Q}}\underline{\Omega}_{X/\mathbb{Q}}^{p-1}\to\underline{\Omega}_{X/\mathbb{Q}}^{p}\to\underline{\Omega}^{p}_{X/\mathbb{C}}\xrightarrow{+1}$

obtained from the corresponding exact sequences at the level of Kähler
differentials on a hyperresolution. Note that when $X$ has rational
singularities we have
$\mathcal{H}^{0}\underline{\Omega}_{X}^{k}\simeq\Omega_{X}^{[k]}$ for all $k$,
while if $X$ is a $p$-Du Bois local complete intersection, then
$\mathcal{H}^{0}\underline{\Omega}_{X}^{k}\simeq\Omega_{X}^{k}$ for $k\leq p$;
see [MP1] for a general discussion.

######  Remark 8.2 (Higher $K$-groups).

With the same notation as above, according to [CHWW2, Theorem 1.13], for
$l\geq 1$ the group $K_{l}(Z)$ breaks up into a direct sum of eigenspaces of
the Adams operator, given by:

  * •

$K_{l}^{(i)}(Z)\cong HC_{l-1}^{(i-1)}(R)$ for $0<i<l$;

  * •

$K_{l}^{(l)}(Z)\cong
K_{l}^{(l)}(\mathbb{C})\oplus\operatorname{tors}\Omega_{Z/\mathbb{Q}}^{l-1}/d\operatorname{tors}\Omega_{Z/\mathbb{Q}}^{l-2}$;

  * •

$K_{l}^{(l+1)}(Z)\cong\operatorname{coker}(\Gamma(Z,\mathcal{H}^{0}\underline{\Omega}_{Z/\mathbb{Q}}^{l-1})\xrightarrow{d}\Gamma(\mathcal{H}^{0}\underline{\Omega}_{Z/\mathbb{Q}}^{l}/\Omega_{Z/\mathbb{Q}}^{l}))$;

  * •

$K_{l}^{(i)}(Z)\cong\operatorname{coker}(\Gamma(\mathcal{H}^{i-l-1}\underline{\Omega}_{Z/\mathbb{Q}}^{i-2})\xrightarrow{d}\Gamma(\mathcal{H}^{i-l-1}\underline{\Omega}_{Z/\mathbb{Q}}^{i-1}))$
for $i\geq l+2$.

For $i\geq l+2$, Theorem 3.1(1) and a similar argument as in the computation
of non-positive $K$-groups gives

$K_{l}^{(i)}(Z)\simeq\bigoplus_{m\geq
1}\mathbb{H}^{i-l-1}(X,\underline{\Omega}_{X/\mathbb{Q}}^{i-1}\otimes L^{m}).$

Still using Theorem 3.1, further explicit calculations can be performed for
special choices of $X$; these will appear later.

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# Sum of elements in finite Sidon sets II

Yuchen Ding  (Yuchen Ding) School of Mathematical Science, Yangzhou
University, Yangzhou 225002, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

A set $S\subset\\{1,2,...,n\\}$ is called a Sidon set if all the sums
$a+b~{}~{}(a,b\in S)$ are different. Let $S_{n}$ be the largest cardinality of
the Sidon sets in $\\{1,2,...,n\\}$. In a former article, the author proved
the following asymptotic formula

$\sum_{a\in S,~{}|S|=S_{n}}a=\frac{1}{2}n^{3/2}+O(n^{111/80+\varepsilon}),$

where $\varepsilon>0$ is an arbitrary small constant. In this note, we give an
extension of the above formula. We show that

$\sum_{a\in
S,~{}|S|=S_{n}}a^{\ell}=\frac{1}{\ell+1}n^{\ell+1/2}+O\left(n^{\ell+61/160}\right)$

for any positive integers $\ell$. Besides, we also consider the asymptotic
formulae of other type summations involving Sidon sets. The proofs are
established in a more general setting, namely we obtain the asymptotic
formulae of the Sidon sets with $t$ elements when $t$ is near the magnitude
$n^{1/2}$.

###### Key words and phrases:

Sidon sets; asymptotic formula

###### 2010 Mathematics Subject Classification:

Primary 11B75; Secondary 11B83.

##  1\. Introduction

The notion of the Sidon set dates back to a paper of Sidon [23] where he
investigated the coefficients of the Fourier series. Since then, the
researches of Sidon sets became rich after the past ninety years. Let $n$ be a
positive integer and $S$ be a subset of the first $n$ numbers
$\\{1,2,...,n\\}$. We call $S$ a Sidon set if all the sums $a+b~{}~{}(a,b\in
S)$ are different. Denoting by $S_{n}$ the largest cardinality of the Sidon
sets in $\\{1,2,...,n\\}$, i.e.,

$S_{n}=\max_{S\subset\\{1,2,...,n\\},~{}S~{}\text{Sidon}}|S|.$

It is a natural question to ask the magnitude of the number $S_{n}$. And there
are a few literatures involving this magnitude. In 1941, Erdős and Turán [12]
proved that

$\frac{1}{\sqrt{2}}\leqslant\liminf_{n\rightarrow\infty}\frac{S_{n}}{\sqrt{n}}\leqslant\limsup_{n\rightarrow\infty}\frac{S_{n}}{\sqrt{n}}\leqslant
1.$

Actually, they gave more explicit upper bound $S_{n}\leqslant
n^{1/2}+O(n^{1/4})$. This upper bound was sharpen to $S_{n}\leqslant
n^{1/2}+n^{1/4}+1$ by Lindström [17]. Later, Cilleruelo [6] improved slightly
this bound by showing that $S_{n}\leqslant n^{1/2}+n^{1/4}+1/2$. In about
2017, Professor Yong–Gao Chen in Nanjing Normal University showed me that the
original method of Erdős and Turán with some more elaborate calculations can
already lead to the bound $S_{n}\leqslant n^{1/2}+n^{1/4}+5/8$ (private
communications). Recently, Balogh, Füredi and Roy [3] proved that
$S_{n}\leqslant n^{1/2}+0.998n^{1/4}$ for sufficiently large $n$ by combining
the methods of Erdős–Turán and Lindström.

In the other direction, Chowla [5] and Erdős [9] independently found that the
constructions of Singer [24] and Bose [4] provided the evidences of
$S_{n}\geqslant(1+o(1))n^{1/2}$. Based on the results of gaps between primes
(up to the year of 1999), it has been remarked by Ruzsa [22] that
$S_{n}\geqslant n^{1/2}+O(n^{11/40+\varepsilon})$, where $\varepsilon>0$ is an
arbitrary small constant. This lower bound together with the upper bound give
us that

$S_{n}=n^{1/2}+O(n^{11/40+\varepsilon}).$ (1.1)

It was conjectured by Erdős [11] that for any $\varepsilon>0$, we have
$S_{n}=n^{1/2}+O(n^{\varepsilon})$. He offered one thousand dollars for the
first one to solve this conjecture.

In the famous book ‘Unsolved Problems in Number Theory’ of Guy [13, Problem
E28], it asked what is the superemum of the sum of reciprocal terms taken from
a Sidon set. It has been proved that

$\max_{\begin{subarray}{c}S\subseteq\\{1,2,..,n\\}\\\
\text{Sidon}\end{subarray}}\sum_{a\in S}\frac{1}{a}$

is near $2$ for sufficiently large $n$. For results of this topic, one can
refer to [1, 10, 15, 16, 18, 21, 25, 26, 27, 28].

For the collections of researches about Sidon sets, one can refer to the
excellent survey article [19] and P.hD. thesis [20] of O’Bryant.

In the Lindström argument for bounding the maximal size of a Sidon set, it
arises naturally the estimate of

$\sum_{\begin{subarray}{c}S=\\{a_{1}<a_{2}<\cdot\cdot\cdot<a_{t}\\}\\\
S\subset[1,n]~{}\text{Sidon}\end{subarray}}(n+1-i)a_{i}.$

In 2021, the author [8] considered the magnitude of the sum of elements in
Sidon sets. To be precise, let $S$ be a Sidon set in $\\{1,2,...,n\\}$ with
$|S|=S_{n}$, using the asymptotic formula of $S_{n}$ in equation (1.1) the
author proved for any $\varepsilon>0$

$\sum_{a\in S}a=\frac{1}{2}n^{3/2}+O(n^{111/80+\varepsilon}).$ (1.2)

In this subsequent note, we provide the asymptotic formulae of the following
two type summations:

$\sum_{\begin{subarray}{c}S=\\{a_{1}<a_{2}<\cdot\cdot\cdot<a_{t}\\}\\\
S\subset[1,n]~{}\text{Sidon}\end{subarray}}a_{i}^{\ell}~{}~{}~{}\qquad~{}~{}~{}\text{and}~{}~{}~{}\sum_{\begin{subarray}{c}S=\\{a_{1}<a_{2}<\cdot\cdot\cdot<a_{t}\\}\\\
S\subset[1,n]~{}\text{Sidon}\end{subarray}}ia_{i}^{\ell}.$

Using these estimates with $\ell=1$, we finally attain the asymptotic formula
of

$\sum_{\begin{subarray}{c}S=\\{a_{1}<a_{2}<\cdot\cdot\cdot<a_{t}\\}\\\
S\subset[1,n]~{}\text{Sidon}\end{subarray}}(n+1-i)a_{i}$

which arose in the text of Lindström. Although this formula can not effect the
upper bound of the maximal Sidon sets in Lindström’s argument, this is of
interest itself.

Our first theorem deals with the general case of equation (1.2).

######  Theorem 1.1.

Let $S=\\{a_{1}<a_{2}<\cdot\cdot\cdot<a_{t}\\}$ be a Sidon set in
$\\{1,2,...,n\\}$. Then for any positive integer $\ell$, we have

$\sum_{i=1}^{t}a_{i}^{\ell}=\begin{cases}\frac{1}{\ell+1}tn^{\ell}+O_{\ell}\left(n^{\ell+3/8}\right),~{}&~{}\text{if~{}}t\geqslant
n^{1/2}-n^{1/4}\\\ \\\
\frac{1}{\ell+1}tn^{\ell}+O_{\ell}\left(n^{\ell+1/4}\sqrt{n^{1/2}-t}\right),~{}&\text{if~{}}t<n^{1/2}-n^{1/4}.\end{cases}$

Several corollaries can be deduced from Theorem 1.1. The asymptotic formula
mentioned in the abstract is proved by taking $t=S_{n}=n^{1/2}+O(n^{21/80})$
(see Lemma 2.10).

Taking $\ell=1$, we have the following corollary, which is another extension
of equation (1.2).

######  Corollary 1.2.

Let $S=\\{a_{1}<a_{2}<\cdot\cdot\cdot<a_{t}\\}$ be a Sidon set in
$\\{1,2,...,n\\}$. Then

$\sum_{i=1}^{t}a_{i}=\begin{cases}\frac{1}{2}nt+O(n^{11/8}),~{}&\text{if~{}}t\geqslant
n^{1/2}-n^{1/4}\\\ \\\
\frac{1}{2}nt+O\left(n^{5/4}\sqrt{n^{1/2}-t}\right),~{}&\text{if~{}}t<n^{1/2}-n^{1/4}.\end{cases}$

Taking $t=S_{n}$ and then Corollary 1.2 reduces to the following Corollary
1.3.

######  Corollary 1.3.

Let $S$ be a Sidon set in $\\{1,2,...,n\\}$ with $|S|=S_{n}$, then we have

$\sum_{a\in S}a=\frac{1}{2}n^{3/2}+O(n^{221/160}).$

It is certainly that $11/8<221/160<111/80$, so this is a small improvement of
the error term in equation (1.2). In the former article, the author pointed
out that one can improve the error term in equation (1.2) to $O(n^{11/8})$
under the assumption $S_{n}=n^{1/2}+O(n^{1/4})$. This assumption is still open
at present due to the barrier of the gaps between primes. By some applications
of the mean value estimate rather than the direct use of the size of Sidon
sets, we can prove the following stronger result comparing with Corollary 1.3.

######  Theorem 1.4.

Let $S$ be a Sidon set in $\\{1,2,...,n\\}$ with $|S|=S_{n}$, then for all
$n\leqslant N$ but at most $O(N/(\log N)^{7/19})$ exceptions, we have

$\sum_{a\in S}a=\frac{1}{2}n^{3/2}+O(n^{11/8}\log n).$

From Theorem 1.4, we immediately have the following corollary.

######  Corollary 1.5.

Let $S$ be a Sidon set in $\\{1,2,...,n\\}$ with $|S|=S_{n}$, then the
asymptotic formula $\sum_{a\in S}a=\frac{1}{2}n^{3/2}+O(n^{11/8}\log n)$ is
true for almost all integers $n$.

As an application of Theorem 1.1, we establish the last theorem in this note.

######  Theorem 1.6.

Let $S=\\{a_{1}<a_{2}<\cdot\cdot\cdot<a_{t}\\}$ be a Sidon set in
$\\{1,2,...,n\\}$. Then for any positive integer $\ell$ we have

$\sum_{i=1}^{t}ia_{i}^{\ell}=\begin{cases}\frac{1}{\ell+2}t^{2}n^{\ell}+O_{\ell}\left(n^{\ell+3/8}t\right),~{}&~{}\text{if~{}}t\geqslant
n^{1/2}-n^{1/4}\\\ \\\
\frac{1}{\ell+2}t^{2}n^{\ell}+O_{\ell}\left(n^{\ell+1/4}t\sqrt{n^{1/2}-t}\right),~{}&\text{if~{}}t<n^{1/2}-n^{1/4}.\end{cases}$

By Theorem 1.6, we finally offer the asymptotic formula of the summation
occurred in the Lindström article.

######  Corollary 1.7.

Let $S=\\{a_{1}<a_{2}<\cdot\cdot\cdot<a_{t}\\}$ be a Sidon set in
$\\{1,2,...,n\\}$. Then

$\sum_{i=1}^{t}(n+1-i)a_{i}=\begin{cases}\frac{1}{2}n^{2}t+O\left(n^{19/8}\right),~{}&~{}\text{if~{}}t\geqslant
n^{1/2}-n^{1/4}\\\ \\\
\frac{1}{2}n^{2}t+O\left(n^{9/4}\sqrt{n^{1/2}-t}\right),~{}&\text{if~{}}t<n^{1/2}-n^{1/4}.\end{cases}$

##  2\. Lemmas

In this section, we provide some lemmas which shall be used in the proofs of
our theorems.

######  Lemma 2.8 (Baker–Harman–Pintz).

[2] Let $p_{k}$ be the $k$–th prime, then we have

$p_{k+1}-p_{k}\ll p_{k}^{0.525}.$

######  Lemma 2.9 (Bose).

[4] Let $p$ be a prime, then there are at least $p$ elements between
$[1,p^{2}-1]$ such that all the sums of two of these elements are different
modulo $p^{2}-1$. In other words, we have $S_{p^{2}-1}\geqslant p$ for primes
$p$.

######  Lemma 2.10.

For $n\geqslant 2$, we have $S_{n}=n^{1/2}+O(n^{21/80})$.

This is mentioned in [19] without proof and we offer one here for complement.

###### Proof.

Let $p_{i}$ be the $i$–th prime. Suppose that $p_{t}$ is the largest prime
such that $p_{t}^{2}-1<n$, then we have

$p_{t}^{2}-1<n\leqslant p_{t+1}^{2}-1.$ (2.1)

By Lemmas 2.8 and 2.9, it follows that

$S_{n}\geqslant S_{p_{t}^{2}-1}\geqslant
p_{t}=p_{t+1}-(p_{t+1}-p_{t})\geqslant\sqrt{n+1}+O(p_{t}^{0.525}).$

From the equation (2.1), it is clear that $p_{t}^{0.525}\ll
n^{0.525/2}=n^{21/80}$. Now the lemma follows from the fact that
$S_{n}\leqslant n^{1/2}+O(n^{1/4})$. ∎

######  Lemma 2.11 (Cilleruelo).

[7] Given a Sidon set $A\subset\\{1,2,...,n\\}$ with $|A|=n^{1/2}-L$. Then
every subinterval $I\subset[1,n]$ with length $cn$ contains $c|A|+E_{I}$
elements of $A$, where

$|E_{I}|\leqslant
52n^{1/4}(1+c^{1/2}n^{1/8})(1+L_{+}^{1/2}n^{-1/8}),~{}~{}L_{+}=\max\\{0,L\\}.$

Let $B_{n}$ denote the $n$–th Bernoulli number, i.e., it is given by

$\frac{x}{e^{x}-1}=\sum_{n=0}^{\infty}\frac{B_{n}}{n!}x^{n}.$

It is well known that (see for example [14])

$B_{0}=1,~{}B_{1}=-\frac{1}{2},~{}B_{2}=\frac{1}{6},~{}B_{4}=-\frac{1}{30},~{}B_{6}=\frac{1}{42},~{}B_{8}=-\frac{1}{30},~{}B_{10}=\frac{5}{66},...$

and $B_{n}=0$ for odd integers $n\geqslant 3$. The Bernoulli polynomial
$B_{n}(x)~{}(n=0,1,2,3,...)$ is defined by

$\displaystyle B_{n}(x)=\sum_{i=0}^{n}\binom{n}{i}B_{i}x^{n-i},$ (2.2)

where $\binom{n}{i}=\frac{n!}{i!(n-i)!}$. We have the following identity.

######  Lemma 2.12.

[14, Page 93] For natural numbers $r$ and $x$, we have

$\sum_{n=0}^{x-1}n^{r-1}=\frac{1}{r}(B_{r}(x)-B_{r}).$

##  3\. Proofs

######  Proof of Theorem 1.1.

We start with calculations of the summation $\sum_{k=1}^{n}k^{\ell-1}S(k)$,
where $S(k)=|S\cap[1,k]|$ for any $1\leqslant k\leqslant n$. On one hand,

$\displaystyle\sum_{k=1}^{n}k^{\ell-1}S(k)$
$\displaystyle=\sum_{k=1}^{n}k^{\ell-1}\sum_{\begin{subarray}{c}a\in S\\\
1\leqslant a\leqslant k\end{subarray}}1=\sum_{a\in S}\sum_{a\leqslant
k\leqslant n}k^{\ell-1}.$ (3.1)

By Lemma 2.12, we have

$\displaystyle\sum_{a\leqslant k\leqslant
n}k^{\ell-1}=\frac{1}{\ell}B_{\ell}(n+1)-\frac{1}{\ell}B_{\ell}(a).$ (3.2)

Taking equation (3.2) into equation (3.1) gives

$\displaystyle\sum_{k=1}^{n}k^{\ell-1}S(k)=\frac{1}{\ell}B_{\ell}(n+1)t-\frac{1}{\ell}\sum_{a\in
S}B_{\ell}(a).$ (3.3)

On the other hand, by Lemma 2.11 with $A=S$ and $I=S\cap[1,k]$ and Lemma 2.12,

$\displaystyle\sum_{k=1}^{n}k^{\ell-1}S(k)$
$\displaystyle=\sum_{k=1}^{n}k^{\ell-1}|S\cap[1,k]|=\sum_{k=1}^{n}k^{\ell-1}\left(\frac{k}{n}t+E_{k}\right)$
$\displaystyle=\frac{t}{n(\ell+1)}\left(B_{\ell+1}(n+1)-B_{\ell+1}\right)+\sum_{k=1}^{n}k^{\ell-1}E_{k},$
(3.4)

where $|E_{k}|\leqslant
52n^{1/4}(1+(k/n)^{1/2}n^{1/8})(1+L_{+}^{1/2}n^{-1/8}),~{}~{}L_{+}=\max\\{0,n^{1/2}-t\\}.$
Hence, we get

$\displaystyle\sum_{a\in S}B_{\ell}(a)$
$\displaystyle=\left(B_{\ell}(n+1)-\frac{\ell}{\ell+1}\frac{B_{\ell+1}(n+1)}{n}\right)t+\frac{\ell
t}{(\ell+1)n}B_{\ell+1}-\sum_{k=1}^{n}k^{\ell-1}E_{k}.$ (3.5)

From the definition of the Bernoulli polynomial (equation (2.2)), we have

$\displaystyle
B_{\ell}(n+1)-\frac{\ell}{\ell+1}\frac{B_{\ell+1}(n+1)}{n}=\frac{1}{\ell+1}n^{\ell}+O_{\ell}\left(n^{\ell-1}\right)$
(3.6)

and

$\displaystyle B_{\ell}(a)=a^{\ell}+O_{\ell}\left(a^{\ell-1}\right).$ (3.7)

Therefore,

$\displaystyle\sum_{i=1}^{t}a_{i}^{\ell}=\sum_{a\in
S}a^{\ell}=\frac{1}{\ell+1}tn^{\ell}+O_{\ell}\left(tn^{\ell-1}\right)-\sum_{k=1}^{n}k^{\ell-1}E_{k}.$
(3.8)

It remains to bound the sum involving $E_{k}$. Since

$\displaystyle|E_{k}|$
$\displaystyle\ll\begin{cases}n^{1/4}\left(1+(k/n)^{1/2}n^{1/8}\right),~{}&\text{if~{}}t\geqslant
n^{1/2}-n^{1/4}\\\
n^{1/8}\left(1+(k/n)^{1/2}n^{1/8}\right)\sqrt{n^{1/2}-t},~{}&\text{if~{}}t<n^{1/2}-n^{1/4}\end{cases}$
$\displaystyle\ll\begin{cases}n^{1/4}+k^{1/2}n^{-1/8},~{}&\text{if~{}}t\geqslant
n^{1/2}-n^{1/4}\\\
\left(n^{1/8}+k^{1/2}n^{-1/4}\right)\sqrt{n^{1/2}-t},~{}&\text{if~{}}t<n^{1/2}-n^{1/4},\end{cases}$
(3.9)

it follows that

$\displaystyle\sum_{k=1}^{n}k^{\ell-1}E_{k}\ll\begin{cases}n^{\ell+3/8},~{}&~{}\text{if~{}}t\geqslant
n^{1/2}-n^{1/4}\\\
n^{\ell+1/4}\sqrt{n^{1/2}-t},~{}&\text{if~{}}t<n^{1/2}-n^{1/4}.\end{cases}$
(3.10)

Now the theorem follows from the trivial estimates

$tn^{\ell-1}\ll n^{\ell+1/8}$

for $t<n^{1/2}+O(n^{1/4})$ and

$tn^{\ell-1}\ll n^{\ell+1/4}\sqrt{n^{1/2}-t}$

provided that $t<n^{1/2}-n^{1/4}$. ∎

######  Proof of Theorem 1.4.

The proof is an application of the mean value idea. For any $1\leqslant
n\leqslant N$, let $\mathscr{S}_{n}$ be a Sidon set in $\\{1,2,...,n\\}$ with
$|\mathscr{S}_{n}|=S_{n}$. By Lemma 2.10 and equation (3.5) with $\ell=1$ and
$S=\mathscr{S}_{n}$, we know that

$\displaystyle\sum_{a\in\mathscr{S}_{n}}a$
$\displaystyle=\left(B_{1}(n+1)-\frac{1}{2}\frac{B_{2}(n+1)}{n}\right)S_{n}+O\left(\frac{1}{\sqrt{n}}\right)-\sum_{k=1}^{n}E_{n,k}$
$\displaystyle=\frac{1}{2}n^{3/2}+O(n^{101/80})-\sum_{k=1}^{n}E_{n,k},$ (3.11)

where

$\displaystyle|E_{n,k}|\leqslant
52n^{1/4}(1+(k/n)^{1/2}n^{1/8})(1+L_{n}^{1/2}n^{-1/8}),~{}~{}L_{n}=\max\\{0,n^{1/2}-S_{n}\\}.$
(3.12)

In another word, we shall have

$\displaystyle\left|\sum_{a\in\mathscr{S}_{n}}a-\frac{1}{2}n^{3/2}\right|\ll
n^{101/80}+\sum_{k=1}^{n}|E_{n,k}|.$ (3.13)

By equation (3.13), we immediately obtain that

$\displaystyle\sum_{n=1}^{N}\left|\sum_{a\in\mathscr{S}_{n}}a-\frac{1}{2}n^{3/2}\right|\ll
N^{181/80}+\sum_{n=1}^{N}\sum_{k=1}^{n}|E_{n,k}|.$ (3.14)

We now treat the summation involving $E_{n,k}$. From equation (3.12), we know
that

$\displaystyle\sum_{k=1}^{n}|E_{n,k}|\ll(n^{1/4}+L_{n}^{1/2}n^{1/8})\sum_{k=1}^{n}(1+k^{1/2}n^{-3/8})\ll
n^{11/8}+L_{n}^{1/2}n^{5/4}.$ (3.15)

From which it can be deduced that

$\displaystyle\sum_{n=1}^{N}\sum_{k=1}^{n}|E_{n,k}|\ll\sum_{n=1}^{N}(n^{11/8}+L_{n}^{1/2}n^{5/4})\ll
N^{19/8}+\sum_{n=1}^{N}L_{n}^{1/2}n^{5/4}.$ (3.16)

Suppose that $p_{\ell}$ is the least prime such that $N\leqslant
p_{\ell}^{2}-1$, where $p_{i}$ is the $i$–th prime and we define $p_{0}$ to be
$1$. Then we have

$\displaystyle\sum_{n=1}^{N}L_{n}^{1/2}n^{5/4}$
$\displaystyle\leqslant\sum_{m=0}^{\ell-1}\sum_{\begin{subarray}{c}p_{m}^{2}-1<n\leqslant
p_{m+1}^{2}-1\end{subarray}}L_{n}^{1/2}n^{5/4}.$ (3.17)

Recall that $S_{p^{2}-1}\geqslant p$ from Lemma 2.9, thus by the definition of
$L_{n}$ in equation (3.12), we get

$\displaystyle\sum_{\begin{subarray}{c}p_{m}^{2}-1<n\leqslant
p_{m+1}^{2}-1\end{subarray}}L_{n}^{1/2}n^{5/4}$
$\displaystyle\leqslant\sum_{\begin{subarray}{c}p_{m}^{2}-1<n\leqslant
p_{m+1}^{2}-1\end{subarray}}n^{5/4}(n^{1/2}-p_{m})^{1/2}$
$\displaystyle<\sum_{\begin{subarray}{c}p_{m}^{2}<n\leqslant
p_{m+1}^{2}\end{subarray}}n^{5/4}(n^{1/2}-p_{m})^{1/2}$ $\displaystyle\ll
p_{m+1}^{5/4}\sum_{\begin{subarray}{c}p_{m}^{2}<n\leqslant
p_{m+1}^{2}\end{subarray}}(n^{1/2}-p_{m})^{1/2}$ $\displaystyle\ll
p_{m+1}^{5/4}\int_{p_{m}^{2}}^{p_{m+1}^{2}}(x^{1/2}-p_{m})^{1/2}dx$
$\displaystyle\xlongequal[x=(t+p_{m})^{2}]{t=x^{1/2}-p_{m}}p_{m+1}^{5/4}\int_{0}^{p_{m+1}-p_{m}}2t^{1/2}(t+p_{m})dt$
$\displaystyle\ll
p_{m+1}^{5/4}(p_{m+1}-p_{m})^{5/2}+p_{m+1}^{5/4}p_{m}(p_{m+1}-p_{m})^{3/2}$
$\displaystyle\ll p_{m+1}^{9/4}(p_{m+1}-p_{m})^{3/2}.$ (3.18)

Taking this into equation (3.17), we get

$\displaystyle\sum_{n=1}^{N}L_{n}^{1/2}n^{5/4}$
$\displaystyle\ll\sum_{m=0}^{\ell-1}p_{m+1}^{9/4}(p_{m+1}-p_{m})^{3/2}$
$\displaystyle\leqslant\sum_{m=0}^{\ell-1}p_{m+1}^{11/4}(p_{m+1}-p_{m})$
$\displaystyle\leqslant p_{\ell}^{11/4}\sum_{m=0}^{\ell-1}(p_{m+1}-p_{m})$
$\displaystyle\leqslant p_{\ell}^{15/4}.$ (3.19)

Recall that $p_{\ell-1}^{2}-1<N\leqslant p_{\ell}^{2}-1$ from the definition
of $\ell$, so we have

$\displaystyle
p_{\ell}=p_{\ell-1}+(p_{\ell}-p_{\ell-1})\leqslant\sqrt{N+1}+O(p_{\ell-1}^{0.525})\ll
N^{1/2}.$ (3.20)

It follows that

$\displaystyle\sum_{n=1}^{N}L_{n}^{1/2}n^{5/4}\ll N^{15/8}.$ (3.21)

Thus, we conclude that

$\displaystyle\sum_{n=1}^{N}\left|\sum_{a\in\mathscr{S}_{n}}a-\frac{1}{2}n^{3/2}\right|\ll
N^{19/8}$ (3.22)

from equations (3.14), (3.16) and (3.21). Now let $\mathscr{A}$ be the set of
integers $n\leqslant N$ such that

$\displaystyle\left|\sum_{\begin{subarray}{c}a\in\mathscr{S}_{n}\end{subarray}}a-\frac{1}{2}n^{3/2}\right|>n^{11/8}\log
n.$ (3.23)

If there are more than $N/(\log N)^{7/19}$ integers in $\mathscr{A}$, we shall
have

$\displaystyle\sum_{n\in\mathscr{A}}\left|\sum_{\begin{subarray}{c}a\in\mathscr{S}_{n}\end{subarray}}a-\frac{1}{2}n^{3/2}\right|>\sum_{n\leqslant
N/(\log N)^{7/19}}n^{11/8}\log n\gg N^{19/8}(\log N)^{1/8}.$ (3.24)

This a contradiction with equation (3.22) for sufficiently large $N$, which
means that we should have $|\mathscr{A}|<N/(\log N)^{7/19}$ provided that $N$
is large enough. Therefore, for all integers $n\leqslant N$ but at most
$N/(\log N)^{7/19}$ exceptions, we have

$\displaystyle\left|\sum_{\begin{subarray}{c}a\in\mathscr{S}_{n}\end{subarray}}a-\frac{1}{2}n^{3/2}\right|\leqslant
n^{11/8}\log n.$ (3.25)

∎

######  Remark 3.1.

It is sure that the error term $O(n^{11/8}\log n)$ in Theorem 1.4 can be
improved to $O(n^{11/8}f(n))$ for the integers of density one provided that
$f(n)$ tends to infinity. It seems that this is the best possible which can be
achieved by the present method.

######  Proof of Theorem 1.6.

It is clear that

$\displaystyle\sum_{i=1}^{t}ia_{i}^{\ell}=\sum_{a\in
S}a^{\ell}\sum_{\begin{subarray}{c}b\in S\\\ b\leqslant
a\end{subarray}}1=\sum_{a\in
S}a^{\ell}\left(\frac{a}{n}t+E_{a}\right)=\frac{t}{n}\sum_{a\in
S}a^{\ell+1}+\sum_{a\in S}a^{\ell}E_{a}$ (3.26)

from Lemma 2.11 with $A=S$ and $I=S\cap[1,a]$, where

$\displaystyle|E_{a}|$ $\displaystyle\leqslant
52n^{1/4}(1+(a/n)^{1/2}n^{1/8})(1+L_{+}^{1/2}n^{-1/8})~{}~{}(\text{with}~{}~{}L_{+}=\max\\{0,n^{1/2}-t\\})$
$\displaystyle\ll\begin{cases}n^{1/4}+a^{1/2}n^{-1/8},~{}&\text{if~{}}t\geqslant
n^{1/2}-n^{1/4}\\\
\left(n^{1/8}+a^{1/2}n^{-1/4}\right)\sqrt{n^{1/2}-t},~{}&\text{if~{}}t<n^{1/2}-n^{1/4}.\end{cases}$
(3.27)

For $t\geqslant n^{1/2}-n^{1/4}$, we have

$\displaystyle\sum_{a\in S}a^{\ell}E_{a}$ $\displaystyle\ll n^{1/4}\sum_{a\in
S}a^{\ell}+n^{-1/8}\sum_{a\in S}a^{\ell+1/2}$ $\displaystyle\ll
n^{1/4}n^{\ell}t+n^{-1/8}n^{\ell+1/2}t$ $\displaystyle\ll n^{\ell+3/8}t.$
(3.28)

For $t<n^{1/2}-n^{1/4}$, we have

$\displaystyle\sum_{a\in S}a^{\ell}E_{a}$ $\displaystyle\ll
n^{1/8}\sqrt{n^{1/2}-t}\sum_{a\in
S}a^{\ell}+n^{-1/4}\sqrt{n^{1/2}-t}\sum_{a\in S}a^{\ell+1/2}$
$\displaystyle\ll
n^{1/8}n^{\ell}t\sqrt{n^{1/2}-t}+n^{-1/4}n^{\ell+1/2}t\sqrt{n^{1/2}-t}$
$\displaystyle\ll n^{\ell+1/4}t\sqrt{n^{1/2}-t}.$ (3.29)

Recall that

$\displaystyle\sum_{a\in
S}a^{\ell+1}=\begin{cases}\frac{1}{\ell+2}tn^{\ell+1}+O_{\ell}\left(n^{\ell+11/8}\right),~{}&~{}\text{if~{}}t\geqslant
n^{1/2}-n^{1/4}\\\
\frac{1}{\ell+2}tn^{\ell+1}+O_{\ell}\left(n^{\ell+5/4}\sqrt{n^{1/2}-t}\right),~{}&\text{if~{}}t<n^{1/2}-n^{1/4}.\end{cases}$
(3.30)

from Theorem 1.1, thus the theorem follows from equations (3.26), (3), (3) and
(3.30). ∎

###### Proof of Corollary 1.7.

From Corollary 1.2 and Theorem 1.6 with $\ell=1$, we have

$\displaystyle\sum_{i=1}^{t}(n+1-i)a_{i}$
$\displaystyle=(n+1)\sum_{i=1}^{t}a_{i}-\sum_{i=1}^{t}ia_{i}$
$\displaystyle=\begin{cases}\frac{1}{2}n^{2}t+O\left(n^{19/8}\right),~{}&~{}\text{if~{}}t\geqslant
n^{1/2}-n^{1/4}\\\
\frac{1}{2}n^{2}t+O\left(n^{9/4}\sqrt{n^{1/2}-t}\right),~{}&\text{if~{}}t<n^{1/2}-n^{1/4}.\end{cases}$
(3.31)

∎

## Acknowledgments

The author is supported by the Natural Science Foundation of Jiangsu Province
of China, Grant No. BK20210784. He is also supported by the foundations of the
projects “Jiangsu Provincial Double–Innovation Doctor Program”, Grant No.
JSSCBS20211023 and “Golden Phoenix of the Green City–Yang Zhou” to excellent
PhD, Grant No. YZLYJF2020PHD051.

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