A characterization of ordinary abelian varieties by the Frobenius push-forward of the structure sheaf

Abstract

For an ordinary abelian variety X, $F^e_*\mathcal {O}_X$ is decomposed into line bundles for every positive integer e. Conversely, if a smooth projective variety X satisfies this property and the Kodaira dimension of X is non-negative, then X is an ordinary abelian variety.

1 Introduction

Let k be an algebraically closed field of characteristic $p>0$. Let X be a smooth proper variety over k. When does X satisfy the following property $(*)$?

$(*)$ :

$F_*\mathcal {O}_X\simeq \bigoplus _j M_j$ where $F:X \rightarrow X$ is the absolute Frobenius morphism and each $M_j$ is a line bundle.

For example, an arbitrary smooth proper toric variety satisfies this property $(*)$ (cf. [1, 19]). Thus there are many varieties which satisfy $(*)$. But every toric variety has negative Kodaira dimension. On the other hand, we show that ordinary abelian varieties satisfy $(*)$. The main theorem of this paper is the following inverse result.

Theorem 1.1

(Theorem 4.7) Let k be an algebraically closed field of characteristic $p>0$. Let X be a smooth projective variety over k. Assume the following conditions.

• For infinitely many $e\in \mathbb Z_{>0}$, $F^e_*\mathcal {O}_X\simeq \bigoplus _j M_j^{(e)}$ where each $M_j^{(e)}$ is an invertible sheaf.

• $K_X$ is pseudo-effective (e.g. the Kodaira dimension of X is non-negative).

Then X is an ordinary abelian variety.

On the other hand, how about the opposite problem? More precisely, when does X satisfy the following property $(**)$?

$(**)$ :

$F_*\mathcal {O}_X$ is indecomposable, that is, if $F_*\mathcal {O}_X=E_1\oplus E_2$ holds for some coherent sheaves $E_1$ and $E_2$, then $E_1=0$ or $E_2=0$.

We study this problem for abelian varieties and curves.

Theorem 1.2

(Theorem 5.3) Let k be an algebraically closed field of characteristic $p>0$. Let X be an abelian variety over k. Set $r_X$ to be the p-rank of X. Then, for every $e\in \mathbb Z_{>0}$,

$$\begin{aligned} F_*^e\mathcal {O}_X \simeq E_1 \oplus \cdots \oplus E_{p^{er_X}} \end{aligned}$$

where each $E_i$ is an indecomposable locally free sheaf of rank $p^{e(\dim X-r_X)}$. In particular, $F_*^e\mathcal {O}_X$ is indecomposable if and only if $r_X=0$.

Theorem 1.3

(Theorem 5.5) Let k be an algebraically closed field of characteristic $p>0$. Let X be a smooth projective curve of genus g. Fix an arbitrary integer $e\in \mathbb Z_{>0}$. Then the following assertions hold.

(0):

If $g=0$, then $F^e_*\mathcal {O}_X \simeq \bigoplus L_j$ where every $L_j$ is a line bundle.

$(1\mathrm{or})$ :

If $g=1$ and X is an ordinary elliptic curve, then $F^e_*\mathcal {O}_X \simeq \bigoplus L_j$ where every $L_j$ is a line bundle.

$(1{ \mathrm ss})$ :

If $g=1$ and X is a supersingular elliptic curve, then $F^e_*\mathcal {O}_X$ is indecomposable.

(2):

If $g\ge 2$, then $F^e_*\mathcal {O}_X$ is indecomposable.

By Theorem 1.3(2), it is natural to ask whether $F_*\mathcal {O}_X$ is indecomposable for every smooth projective variety of general type X. If we drop the assumption that X is smooth, then the following theorem gives a negative answer to this question.

Theorem 1.4

Let k be an algebraically closed field of characteristic $p>0$. Then, there exists a projective normal surface X over k which satisfies the following properties.

• (1) The singularities of X are at worst canonical.

• (2) $K_X$ is ample.

• (3) $F_*\mathcal {O}_X$ is not indecomposable.

Remark 1.5

By [17], if X is a smooth projective curve of genus $g\ge 2$, then $F_*E$ is a stable vector bundle whenever so is E. Theorem 1.4 shows that there exists a projective normal canonical surface of general type X such that $F_*\mathcal {O}_X$ is not a stable vector bundle with respect to an arbitrary ample invertible sheaf H on X.

Proof of Theorem 1.1:

We overview the proof of Theorem 1.1. First of all, we can show that X is F-split, that is, $\mathcal {O}_X \rightarrow F_*\mathcal {O}_X$ splits as an $\mathcal {O}_X$-module homomorphism. This implies

$$\begin{aligned} H^0(X, -(p-1)K_X)\ne 0. \end{aligned}$$

Since $K_X$ is pseudo-effective, we obtain $(p-1)K_X \sim 0$. Then, by [6], we see that the Albanese map $\alpha :X \rightarrow {{\text {Alb}}}(X)$ is surjective. There are two main difficulties as follows.

• (1) To show that $\alpha$ is generically finite.

• (2) To treat the case where $\alpha$ is a finite surjective inseparable morphism.

(1) Let us overview how to show that $\alpha$ is generically finite. Set $r_X$ to be the p-rank of ${{\text {Alb}}}(X)$. It suffices to show $\dim X=r_X$. Note that $\alpha :X \rightarrow {{\text {Alb}}}(X)$ induces the following bijective group homomorphism:

$$\begin{aligned} \alpha ^*:{{\text {Pic}}}^0({{\text {Alb}}}(X)) \overset{\simeq }{\rightarrow }{{\text {Pic}}}^0(X),\,\,\, L\mapsto \alpha ^*L. \end{aligned}$$

Roughly speaking, since ${{\text {Pic}}}^{\tau }(X)/{{\text {Pic}}}^0(X)$ is a finite group, $r_X$ can be calculated by the asymptotic behavior of the number of $p^e$-torsion line bundles on X. Thus, we count the number of $p^e$-torsion line bundles on X. More precisely, we prove that the number of $p^e$-torsion line bundles on X is $p^{e\dim X}$ for infinitely many e.

Now we have

$$\begin{aligned} F^e_*\mathcal {O}_X=\bigoplus _{1\le j\le p^{e\dim X}} M_j \end{aligned}$$

where each $M_j$ is a line bundle. In our situation, we can show that every $p^e$-torsion line bundle L is isomorphic to some $M_j$ (cf. Lemma 3.3). Therefore, it suffices to prove that each $M_j$ is $p^e$-torsion. Tensor $M_j^{-1}$ with the above equation and take $H^0$. Then, we obtain $H^0(X, M_j^{-p^e})\ne 0$. If we have $H^0(X, M_j^{p^e})\ne 0$, then we are done. For this, we take the duality, that is, apply $\mathcal Hom_{\mathcal {O}_X}(-, \omega _X)$ to the above direct summand decomposition. Then we can also show $H^0(X, M^{p^e})\ne 0$. For more details on this argument, see Lemma 4.4.

(2) We overview how to treat the inseparable case. To clarify the idea, we assume that $\alpha$ is a finite surjective purely inseparable morphism of degree p. Then, Frobenius map $F_A$ of A factors through $\alpha$:

$$\begin{aligned} F_A:A \rightarrow X \overset{\alpha }{\rightarrow }A. \end{aligned}$$

By using the fact that X is F-split, we can show that

$$\begin{aligned} (F_A)_*\mathcal {O}_A \simeq \alpha _*\mathcal {O}_X\oplus E \end{aligned}$$

for some coherent sheaf E. Since $(F_A)_*\mathcal {O}_A$ is the direct sum of the p-torsion line bundles, we obtain

$$\begin{aligned} \alpha _*\mathcal {O}_X\simeq \bigoplus _{j=1}^p M_j \end{aligned}$$

where $M_1, \ldots , M_p$ are mutually distinct p-torsion line bundles. One of them, say $M_1$, satisfies $\alpha ^*M_1 \not \simeq \mathcal {O}_X$. By tensoring $M_1^{-1}$, we obtain

$$\begin{aligned} \alpha _*(\alpha ^*M_1^{-1})\simeq \mathcal {O}_A \oplus \bigoplus _{j=2}^p (M_j\otimes _{\mathcal {O}_A}M_1^{-1}) \end{aligned}$$

which induces the following contradiction:

$$\begin{aligned} 0=H^0(X, \alpha ^*M_1^{-1}) \simeq H^0(A, \mathcal {O}_A)\oplus H^0\left( A, \bigoplus _{j=2}^p \left( M_j\otimes _{\mathcal {O}_A}M_1^{-1}\right) \right) \ne 0. \end{aligned}$$

In the proof of Theorem 1.1, there appear other technical difficulties. For more details on the inseparable case, see Step 5 of the proof of Theorem 4.7.

Related results:

• (1) In [6], Hacon and Patakfalvi give a characterization of the varieties birational to ordinary abelian varieties.

• (2) Achinger [1] gives a characterization of smooth projective toric varieties as follows. For a smooth projective variety X in positive characteristic, X is toric if and only if $F_*L$ splits into line bundles for every line bundle L.

2 Preliminaries

2.1 Notation

We will not distinguish the notations line bundles, invertible sheaves and Cartier divisors. For example, we will write $L+M$ for line bundles L and M.

Throughout this paper, we work over an algebraically closed field k of characteristic $p>0$. For example, a projective scheme means a scheme which is projective over k.

Let X be a noetherian scheme. For a coherent sheaf F on X and a line bundle L on X, we define $F(L):=F\otimes _{\mathcal {O}_X} L$.

In this paper, a variety means an integral scheme which is separated and of finite type over k. A curve or a surface means a variety whose dimension is one or two, respectively.

For a proper scheme X, let ${{\text {Pic}}}(X)$ be the group of line bundles on X and let ${{\text {Pic}}}^0(X)$ (resp. ${{\text {Pic}}}^{\tau }(X)$) be the subgroup of ${{\text {Pic}}}(X)$ of line bundles which are algebraically (resp. numerically) equivalent to zero:

$$\begin{aligned} {{\text {Pic}}}^0(X) \subset {{\text {Pic}}}^{\tau }(X) \subset {{\text {Pic}}}(X). \end{aligned}$$

For a normal variety X and a coherent sheaf M on X, we say M is reflexive if the natural map $M \rightarrow \mathcal Hom_{\mathcal {O}_X}(\mathcal Hom_{\mathcal {O}_X}(M, \mathcal {O}_X), \mathcal {O}_X)$ is an isomorphism. We say M is divisorial if M is reflexive and $M|_{\mathcal {O}_{X, \xi }}$ is rank one where $\xi$ is the generic point. It is well-known that a divisorial sheaf M is isomorphic to the sheaf $\mathcal {O}_X(D)$ associated to a Weil divisor D on X.

Let X be a scheme of finite type over k. We say X is F-split if the absolute Frobenius

$$\begin{aligned} \mathcal {O}_X \rightarrow F_*\mathcal {O}_X,\,\,\, a\mapsto a^p \end{aligned}$$

splits as an $\mathcal {O}_X$-module homomorphism.

We say a coherent sheaf F is indecomposable if, for every isomorphism $F\simeq F_1\oplus F_2$ with coherent sheaves $F_1$ and $F_2$, we obtain $F_1=0$ or $F_2=0$.

We recall the definition of ordinary abelian varieties.

Definition-Proposition 2.1

Let X be an abelian variety. We say X is ordinary if one of the following conditions hold. Moreover, the following conditions are equivalent.

• (1) For some $e\in \mathbb Z_{>0}$, the number of $p^e$-torsion points is $p^{e\cdot \dim X}$.

• (2) For every $e\in \mathbb Z_{>0}$, the number of $p^e$-torsion points is $p^{e\cdot \dim X}$.

• (3) $F:H^1(X, \mathcal {O}_X) \rightarrow H^1(X, \mathcal {O}_X)$ is bijective.

• (4) $F:H^i(X, \mathcal {O}_X) \rightarrow H^i(X, \mathcal {O}_X)$ is bijective for every $i\ge 0$.

• (5) X is F-split.

Proof

(1) and (2) are equivalent by [13, Section 15, Thep-rank]. (2) and (3) are equivalent by [13, Section 15, Theorem 3]. (Note that, in older editions of [13], there are two Theorem 2 in Section 15.) (3) and (4) are equivalent by [14, Example 5.4]. (4) and (5) are equivalent by [12, Lemma 1.1]. $\square$

2.2 Albanese varieties

In this subsection, we recall the definition and fundamental properties of the Albanese varieties. For details, see [4, Section 9].

For a projective normal variety X and a closed point $x\in X$, there uniquely exists a morphism $\alpha _X:X \rightarrow {{\text {Alb}}}(X)$ to an abelian variety ${{\text {Alb}}}(X)$, called the Albanese variety of X, such that $\alpha _X(x)=0$ and that every morphism to an abelian variety $g:X \rightarrow B$, with $g(x)=0_B$, factors through $\alpha _X$ (cf. [4, Remark 9.5.25]). Note that ${{\text {Alb}}}(X) \simeq \underline{{{\text {Pic}}}}^0(\underline{{{\text {Pic}}}}^0(X)_{{{\text {red}}}})$, where $\underline{{{\text {Pic}}}}(X):=\mathbf {Pic}_{X/k}$ in the sense of [4, Section 9].

The Albanese morphism $\alpha _X:X \rightarrow {{\text {Alb}}}(X)$ induces a natural morphism

$$\begin{aligned} \alpha _X^*:\underline{{{\text {Pic}}}}^0({{\text {Alb}}}(X)) \rightarrow \underline{{{\text {Pic}}}}^0(X)_{{{\text {red}}}}. \end{aligned}$$

It is well-known that $\alpha _X^*$ is an isomorphism. In particular, the induced group homomorphism

$$\begin{aligned} \alpha _X^*:{{\text {Pic}}}^0({{\text {Alb}}}(X)) \rightarrow {{\text {Pic}}}^0(X) \end{aligned}$$

is bijective.

2.3 The number of $p^e$-torsion line bundles

The asymptotic behavior of the number of $p^e$-torsion line bundles is determined by the p-rank of the Picard variety ${{\text {Pic}}}^0(X)_{{{\text {red}}}}$.

Proposition 2.2

Let X be a projective normal variety. Then, the following assertions hold.

• (1) There exists the following exact sequence

  $$\begin{aligned} 0 \rightarrow {{\text {Pic}}}^0(X) \rightarrow {{\text {Pic}}}^{\tau }(X) \rightarrow G(X) \rightarrow 0 \end{aligned}$$

  where G(X) is a finite group.

• (2) If $r_X$ is the p-rank of $\underline{{{\text {Pic}}}}^0(X)_{{{\text {red}}}}$, then there exists $\xi \in \mathbb Z_{>0}$ such that

  $$\begin{aligned} p^{er_X} \le |{{\text {Pic}}}(X)[p^e]| \le p^{er_X}\times \xi \end{aligned}$$

  for every $e\in \mathbb Z_{>0}$ where ${{\text {Pic}}}(X)[p^e]$ is the group of $p^e$-torsion line bundles.

Proof

The assertion (1) holds by [4, 9.6.17]. The assertion (2) follows from (1). $\square$

As a consequence, we see that the p-rank of the Picard variety is stable under purely inseparable covers.

Proposition 2.3

Let $f: X \rightarrow Y$ be a finite surjective purely inseparable morphism between projective normal varieties. Set $r_X$ and $r_Y$ to be the p-ranks of $\underline{{{\text {Pic}}}}^0(X)_{{{\text {red}}}}$ and $\underline{{{\text {Pic}}}}^0(Y)_{{{\text {red}}}}$, respectively. Then, $r_X=r_Y$.

Proof

We may assume that $[K(X):K(Y)]=p$. Then, the absolute Frobenius morphism $F:Y \rightarrow Y$ factors through $f:X \rightarrow Y$:

$$\begin{aligned} F:Y \overset{g}{\rightarrow }X \overset{f}{\rightarrow }Y. \end{aligned}$$

Thus, it suffices to show $r_Y \le r_X$.

We show that the following inequality

$$\begin{aligned} p^{er_Y} \le |{{\text {Pic}}}(X)[p^{e+1}]| \end{aligned}$$

holds for every $e\in \mathbb Z_{>0}$. Fix $e\in \mathbb Z_{>0}$. Let $L_1, \ldots , L_{p^{er_Y}}$ be mutually distinct $p^e$-torsion line bundles in ${{\text {Pic}}}^0(Y)$. Then, since $\underline{{{\text {Pic}}}}^0(Y)_{{{{\text {red}}}}}$ is an abelian variety, we can find line bundles $M_1, \ldots , M_{p^{er_Y}}$ such that $M_j^p\simeq L_j$ for every $1\le j \le p^{er_Y}$. We see that, for each j,

$$\begin{aligned} L_j \simeq M_j^p =F^*M_j\simeq g^*f^*M_j \end{aligned}$$

and that $f^*M_1, \ldots , f^*M_{p^{er_Y}}$ are mutually distinct $p^{e+1}$-torsion line bundles on X. Thus, we obtain the required inequality $p^{er_Y}\le |{{\text {Pic}}}(X)[p^{e+1}]|.$

By Proposition 2.2(2), we can find $\xi \in \mathbb Z_{>0}$ such that the inequalities

$$\begin{aligned} p^{er_Y} \le |{{\text {Pic}}}(X)[p^{e+1}]| \le p^{(e+1)r_X}\times \xi \end{aligned}$$

hold for every $e\in \mathbb Z_{>0}$. By taking the limit $e\rightarrow \infty$, we obtain $r_Y \le r_X$. $\square$

3 Basic properties

In the main theorem (Theorem 1.1), we treat varieties such that $F_*^e\mathcal {O}_X$ is decomposed into line bundles. In this section, we summarize basic properties of such varieties. Since such varieties are F-split (Lemma 3.2), we also study F-split varieties. First, we give characterizations of F-split varieties.

Lemma 3.1

Let X be a scheme of finite type over k. Then, the following assertions are equivalent.

• (1) X is F-split.

• (2) For every $e \in \mathbb Z_{>0}$, there exists a coherent sheaf E such that $F_*^e\mathcal {O}_X\simeq \mathcal {O}_X\oplus E$.

• (3) $F_*^e\mathcal {O}_X\simeq \mathcal {O}_X\oplus E$ for some $e\in \mathbb Z_{>0}$ and coherent sheaf E.

• (4) $F_*^e\mathcal {O}_X\simeq L\oplus E$ for some $e\in \mathbb Z_{>0}$, $p^e$-torsion line bundle L and coherent sheaf E.

Proof

It is well-known that (1), (2) and (3) are equivalent. It is clear that (3) implies (4). We see that (4) implies (3) by tensoring $L^{-1}$ with $F_*^e\mathcal {O}_X\simeq L\oplus E$. $\square$

We are interested in varieties such that $F_*^e\mathcal {O}_X$ is decomposed into line bundles. By the following lemma, such varieties are F-split.

Lemma 3.2

Let X be a proper normal variety. Assume that $F_*^e\mathcal {O}_X\simeq \bigoplus _j M_j$ for some $e\in \mathbb Z_{>0}$ and divisorial sheaves $M_j$. Then, X is F-split.

Proof

We obtain the following

$$\begin{aligned} 0\ne H^0(X, F_*^e\mathcal {O}_X)\simeq \bigoplus _j H^0(X, M_j). \end{aligned}$$

Therefore, we see $H^0(X, M_{j_0})\ne 0$ for some $j_0$.

We have $M_{j_0}\simeq \mathcal {O}_X(E)$ for some effective divisor E on X. By Lemma 3.1, it is enough to show $E=0$. Tensor $\mathcal {O}_X(-E)$ with

$$\begin{aligned} F_*^e\mathcal {O}_X\simeq \bigoplus _j M_j\simeq \mathcal {O}_X(E)\oplus \left( \bigoplus _{j\ne j_0} M_j\right) \end{aligned}$$

and take the double dual. We obtain the following decomposition:

$$\begin{aligned} F_*^e(\mathcal {O}_X(-p^eE))\simeq \mathcal {O}_X\oplus \left( \bigoplus _{j\ne j_0} \big (M_j\otimes _{\mathcal {O}_X}(-E)\big )^{**}\right) . \end{aligned}$$

Thus, $H^0(X, \mathcal {O}_X(-p^eE))\ne 0$. This implies $E=0$. $\square$

The following result gives an upper bound of the number of $p^e$-torsion line bundles for F-split varieties.

Lemma 3.3

Let X be a proper variety. Assume that X is F-split. Fix $e\in \mathbb Z_{>0}$. Let $F_*^e\mathcal {O}_X\simeq \bigoplus _{j\in J} M_j$ be a decomposition into indecomposable coherent sheaves $M_j$. Then, the following assertions hold.

• (1) Let L be a line bundle with $L^{p^e}\simeq \mathcal {O}_X$. Then, $L\simeq M_{j_1}$ for some $j_1\in J$.

• (2) Let $j_1, j_2\in J$ with $j_1\ne j_2$. If $M_{j_1}$ and $M_{j_2}$ are line bundles and satisfy $M_{j_1}^{p^e}\simeq \mathcal {O}_X$ and $M_{j_2}^{p^e}\simeq \mathcal {O}_X$, then $M_{j_1}\not \simeq M_{j_2}$.

• (3) The number of $p^e$-torsion line bundles on X is at most $p^{e\cdot \dim X}$.

Proof

• (1) Tensor $L^{-1}$ with $F_*^e\mathcal {O}_X\simeq \bigoplus _j M_j$ and we obtain

  $$\begin{aligned} F_*^e\mathcal {O}_X\simeq F_*^e(L^{-p^e})\simeq F_*^e\mathcal {O}_X\otimes _{\mathcal {O}_X} L^{-1}\simeq \bigoplus _j \big (M_j\otimes _{\mathcal {O}_X} L^{-1}\big ). \end{aligned}$$

  Since X is F-split, we have

  $$\begin{aligned} F_*^e\mathcal {O}_X\simeq \mathcal {O}_X\oplus \left( \bigoplus _i N_i\right) \end{aligned}$$

  where each $N_i$ is an indecomposable sheaf. Then, the Krull–Schmidt theorem ([2, Theorem 2]) implies $M_{j_1}\otimes _{\mathcal {O}_X} L^{-1}\simeq \mathcal {O}_X$ for some $j_1$.

• (2) Assume that, for some $j_1\ne j_2$, $M_{j_1}$ and $M_{j_2}$ are line bundles such that $M_{j_1}^{p^e}\simeq \mathcal {O}_X$, $M_{j_2}^{p^e}\simeq \mathcal {O}_X$ and $M_{j_1}\simeq M_{j_2}$. Let us derive a contradiction. Tensor $M_{j_1}^{-1}$ and we obtain

  $$\begin{aligned} F_*^e\mathcal {O}_X\simeq F_*^e(M_{j_1}^{-p^e})\simeq \mathcal {O}_X\oplus \mathcal {O}_X\oplus \left( \bigoplus _{j\ne j_1, j_2} \big (M_j\otimes M_{j_1}^{-1}\big )\right) . \end{aligned}$$

  Taking $H^0$, we obtain a contradiction.

• (3) The assertion immediately follows from (1) and (2).

$\square$

The following lemma is used in the next section and well-known for experts on F-singularities (cf. the proof of [16, Theorem 4.3]).

Lemma 3.4

Let X be a smooth proper variety. Assume that X is F-split. Then, for every $e\in \mathbb Z_{>0}$,

$$\begin{aligned} H^0(X, -(p^e-1)K_X)\ne 0. \end{aligned}$$

In particular, $\kappa (X)\le 0.$

Proof

By the Grothendieck duality, we can check

$$\begin{aligned} \mathcal Hom_{\mathcal {O}_X}(F_*^e\mathcal {O}_X, \omega _X)\simeq F_*^e\omega _X. \end{aligned}$$

This implies that $\omega _X$ is a direct summand of $F_*^e\omega _X$, which is equivalent to the assertion that $\mathcal {O}_X$ is a direct summand of $F_*^e(\omega _X^{1-p^e})$. $\square$

4 A characterization of ordinary abelian varieties

In this section, we show the main theorem of this paper: Theorem 4.7. In the proof, we use [6, Theorem 1.1.1]. For this, it is necessary to show $\kappa _S(X)=0$. We check this in Lemma 4.3. First, we recall the definition of $\kappa _S(X)$.

Definition 4.1

Let X be a smooth proper variety.

• (1) Fix $m\in \mathbb Z_{>0}$. We define

  $$\begin{aligned} S^0(X, mK_X):=\bigcap _{e\ge 0} {{\text {Image}}}\left( \mathrm{Tr}: H^0(X, K_X+(m-1)p^eK_X) \rightarrow H^0(X, mK_X)\right) . \end{aligned}$$

  where $\mathrm{Tr}$ is defined by the trace map $F^e_*\omega _X \rightarrow \omega _X$. For more details, see Remark 4.2 and [6, Lemma 2.2.3].

• (2) We define

  $$\begin{aligned} \kappa _S(X):=\max \{r\,|\,\dim S^0(X, mK_X)=O(m^r)\,\,\mathrm{for\,\,sufficiently\,\,divisible}\,\, m\}. \end{aligned}$$

  This definition is the same as the one of [6, Subsection 4.1].

Remark 4.2

The trace map $F^e_*\omega _X \rightarrow \omega _X$ in Definition 4.1 is obtained by applying the functor $\mathcal Hom_{\mathcal {O}_X}(-, \omega _X)$ to the Frobenius $\mathcal {O}_X \rightarrow F_*^e\mathcal {O}_X$. Indeed, the Grothendieck duality implies $\mathcal Hom_{\mathcal {O}_X}(F_*^e\mathcal {O}_X, \omega _X)\simeq F_*^e\omega _X$. Thus, we obtain the trace map $F^e_*\omega _X \rightarrow \omega _X$.

By the construction, if X is F-split, then the trace map $F^e_*\omega _X \rightarrow \omega _X$ is a split surjection. Therefore, in this case, $H^0(X, mK_X)\ne 0$ (resp. $\kappa (X)\ge 0$) implies $S^0(X, mK_X)\ne 0$ (resp. $\kappa _S(X)\ge 0$).

We check $\kappa _S(X)=0$ to apply [6, Theorem 1.1.1] in the proof of Theorem 4.7.

Lemma 4.3

Let X be a smooth projective variety. If X is F-split and $K_X$ is pseudo-effective, then the following assertions hold.

• (1) $(p^e-1)K_X\sim 0$ for every $e\in \mathbb Z_{>0}$.

• (2) $\kappa _S(X)=0$.

Proof

• (1) By Lemma 3.4, we obtain $-(p^e-1)K_X\sim E$ where E is an effective divisor. Then, the pseudo-effectiveness of $K_X$ implies that $E=0$ (cf. [5, Lemma 5.4]).

• (2) By (1), we obtain $\kappa (X)=0$. By [6, Lemma 4.1.3], it suffices to show $\kappa _S(X)\ge 0.$ By Remark 4.2, $\kappa (X)\ge 0$ implies $\kappa _S(X)\ge 0.$

$\square$

The following lemma is a key to show Theorem 4.7.

Lemma 4.4

Let X be a smooth projective variety. Fix $e\in \mathbb Z_{>0}$. Assume the following conditions.

• $F_*^e\mathcal {O}_X\simeq \bigoplus _j M_j$ where each $M_j$ is a line bundle.

• $K_X$ is pseudo-effective.

Then, the following assertions hold.

• (1) $M_j^{p^e}\simeq \mathcal {O}_X$ for every j.

• (2) The number of $p^e$-torsion line bundles on X is equal to $p^{e\cdot \dim X}$.

Proof

• (1) By Lemma 3.2, X is F-split. Thus, Lemma 4.3 implies $(p^{e}-1)K_X\sim 0$. Fix an index $j_0$ and we show $M_{j_0}^{p^e}\simeq \mathcal {O}_X$. We can write

  $$\begin{aligned} F_*^e\mathcal {O}_X=M_{j_0}\oplus \left( \bigoplus _{j\ne j_0} M_j\right) . \end{aligned}$$

  Tensor $M_{j_0}^{-1}$ and we obtain

  $$\begin{aligned} H^0(X, M_{j_0}^{-p^e})\simeq H^0(X, \mathcal {O}_X)\oplus \cdots . \end{aligned}$$

  In particular, we obtain $H^0(X, M_{j_0}^{-p^e})\ne 0$. On the other hand, by applying $\mathcal Hom_{\mathcal {O}_X}(-, \omega _X)$, we have

  $$\begin{aligned} F_*^e\omega _X\simeq & {} \mathcal Hom_{\mathcal {O}_X}\big (F_*^e\mathcal {O}_X, \omega _X\big )\\\simeq & {} \mathcal Hom_{\mathcal {O}_X}\left( \bigoplus _j M_j, \omega _X\right) \\\simeq & {} \bigoplus _j \big (M_j^{-1}\otimes \omega _X\big )\\ \end{aligned}$$

  where the first isomorphism follows from the Grothendieck duality theorem for finite morphisms. Tensor $\omega _X^{-1}$ and we obtain

  $$\begin{aligned} F_*^e\mathcal {O}_X\simeq F_*^e(\omega _X^{1-p^e})\simeq (F_*^e\omega _X)\otimes _{\mathcal {O}_X}\omega _X^{-1}\simeq \bigoplus _j M_j^{-1}. \end{aligned}$$

  Then, tensor $M_{j_0}$, and we obtain $H^0(X, M_{j_0}^{p^e})\ne 0$. Therefore, $M_{j_0}^{p^e}\simeq \mathcal {O}_X$.

• (2) By Lemma 3.2, X is F-split. Then, the assertion follows from (1) and Lemma 3.3.

$\square$

Ordinary abelian varieties satisfy the condition that $F_*^e\mathcal {O}_X$ is decomposed into line bundles.

Lemma 4.5

Let A be a d-dimensional ordinary abelian variety. Fix $e\in \mathbb Z_{>0}$. Let $\{M_j^{(e)}\}_{j\in J}$ be the set of the $p^e$-torsion line bundles on X. Then, the following assertions hold.

• (1) $F_*^e\mathcal {O}_A\simeq \bigoplus _{j\in J} M_j^{(e)}$.

• (2) $M_j^{(e)}\in {{\text {Pic}}}^0(A)$ for every $j\in J$.

Proof

The number of $p^e$-torsion line bundles in ${{\text {Pic}}}^0(X)$ is $p^{ed}$. Apply Lemma 3.3 and we obtain the assertion. $\square$

We also need the following lemma.

Lemma 4.6

Let X be a proper normal variety. Fix $e\in \mathbb Z_{>0}$. Assume that there are mutually distinct $p^e$-torsion line bundles $L_1, \ldots , L_{p^{e\dim X}}$ on X. Let $F_*^e\mathcal {O}_X \simeq E\oplus E'$ where $E\ne 0$ is an indecomposable coherent sheaf and $E'$ is a coherent sheaf. Then, the following assertions hold.

• (1) If ${{\text {rank}}}\,E<p$, then $F_*^e\mathcal {O}_X \simeq \bigoplus _{i=1}^{p^{e\dim X}} L_i.$

• (2) If ${{\text {rank}}}\,E=p$, then $E\otimes _{\mathcal {O}_X} L_i \simeq E\otimes _{\mathcal {O}_X} L_j$ for some $1\le i< j \le p^{e\dim X}$.

Proof

Set $X_{\mathrm{reg}} \subset X$ to be the regular locus of X. Since $(F_*^e\mathcal {O}_X)|_{X_{\mathrm{reg}}}$ is locally free, $E|_{X_{\mathrm{reg}}}$ is also locally free.

We show that E is reflexive. Let

$$\begin{aligned} F_*^e\mathcal {O}_X \simeq E_1 \oplus \cdots \oplus E_s \end{aligned}$$

be a decomposition into indecomposable sheaves with $E_1 \simeq E$. Take the double dual. Since $F_*^e\mathcal {O}_X$ is reflexive, each $E_i$ is reflexive by the Krull–Schmidt theorem ([2, Theorem 2]).

• (1) We show that

  $$\begin{aligned} E\otimes _{\mathcal {O}_X} L_i\not \simeq E\otimes _{\mathcal {O}_X} L_j \end{aligned}$$

  for every $1\le i< j\le p^{e\dim X}$. Assume $E\otimes _{\mathcal {O}_X} L_i\simeq E\otimes _{\mathcal {O}_X} L_j$ for some $1\le i< j\le p^{e\dim X}$. Then, we obtain

  $$\begin{aligned} \det \,(E|_{X_{\mathrm{reg}}})\otimes _{\mathcal {O}_{X_{\mathrm{reg}}}} (L_i|_{X_{\mathrm{reg}}})^{{{\text {rank}}}\,E} \simeq \det \,(E|_{X_{\mathrm{reg}}})\otimes _{\mathcal {O}_{X_{\mathrm{reg}}}} (L_j|_{X_{\mathrm{reg}}})^{{{\text {rank}}}\,E}. \end{aligned}$$

  By $1\le {{\text {rank}}}\,E<p$, we obtain $L_i \simeq L_j$, which is a contradiction.

  Thus $E\otimes _{\mathcal {O}_X} L_i$ is also an indecomposable direct summand of $F^e_*\mathcal {O}_X$. Therefore, we see ${{\text {rank}}}\,E=1$ and

  $$\begin{aligned} F^e_*\mathcal {O}_X \simeq \bigoplus _{i=1}^{p^{e\dim X}}E\otimes _{\mathcal {O}_X} L_i. \end{aligned}$$

  Since E is a divisorial sheaf, X is F-split by Lemma 3.2. Then, the assertion follows from Lemma 3.3.

• (2) Assume that $E\otimes _{\mathcal {O}_X} L_i \not \simeq E\otimes _{\mathcal {O}_X} L_j$ for every $1\le i< j \le p^{e\dim X}$. Let us derive a contradiction. Since E is indecomposable, so is $E\otimes _{\mathcal {O}_X} L_i$ for every i. Moreover, $E\otimes _{\mathcal {O}_X} L_i$ is also a direct summand of $F_*^e\mathcal {O}_X$. Thus, by the Krull–Schmidt theorem ([2, Theorem 2]), we obtain

  $$\begin{aligned} F_*^e\mathcal {O}_X \simeq \bigoplus _{i=1}^{p^{e\dim X}}E\otimes _{\mathcal {O}_X} L_i\oplus \cdots . \end{aligned}$$

  Then, we obtain the following contradiction:

  $$\begin{aligned} p^{e \dim X} ={{\text {rank}}}(F_*^e\mathcal {O}_X) \ge p^{e\dim X}\times {{\text {rank}}}\,E=p^{e\dim X}\times p. \end{aligned}$$

$\square$

We show the main theorem of this paper.

Theorem 4.7

Let X be a smooth projective variety. Assume that the following conditions hold.

• For infinitely many $e\in \mathbb Z_{>0}$, $F_*^e\mathcal {O}_X\simeq \bigoplus _j M_j^{(e)}$ where each $M_j^{(e)}$ is a line bundle.

• $K_X$ is pseudo-effective.

Then, X is an ordinary abelian variety.

Proof

Let

$$\begin{aligned} \alpha :X \rightarrow A:={{\text {Alb}}}(X) \end{aligned}$$

be the Albanese morphism.

Step 1. In this step, we show the following assertions.

• (1) The Albanese morphism $\alpha :X\rightarrow A$ is surjective.

• (2) The Albanese variety A is an ordinary abelian variety such that $\dim X=\dim A$.

• (3) For every $e\in \mathbb Z_{>0}$, $F_*^e\mathcal {O}_X\simeq \bigoplus _j M_j^{(e)}$ where each $M_j^{(e)}$ is a $p^e$-torsion line bundle.

Proof of Step 1

• (1) Lemma 3.2 implies that X is F-split. By Lemma 4.3, we see $\kappa _S(X)=0$. Thus we can apply [6, Theorem 1.1.1(1)]. Then, the Albanese morphism $\alpha :X\rightarrow {{\text {Alb}}}(X)$ is surjective.

• (2) By (1), we obtain $\dim {{\text {Pic}}}^0(X)_{{{\text {red}}}}\le \dim X$. Set $r_X$ to be the p-rank of ${{\text {Pic}}}^0(X)_{{{\text {red}}}}$. It suffices to show that $r_X=\dim X$. By Lemma 4.4 and an assumption, the number of $p^e$-torsion line bundles is equal to $p^{e\dim X}$ for infinitely many $e\in \mathbb Z_{>0}$. By Proposition 2.2(2), we can find an integer $\xi >0$ such that

  $$\begin{aligned} p^{er_X} \le p^{e\dim X}=|{{\text {Pic}}}(X)[p^e]| \le p^{er_X}\times \xi , \end{aligned}$$

  for infinitely many $e>0$. Taking the limit $e\rightarrow \infty$, we obtain $r_X=\dim X.$

• (3) The assertion follows from (2) and Lemma 3.3. This completes the proof of Step 1.

$\square$

By Step 1, the Albanese morphism $\alpha :X \rightarrow A$ is a generically finite surjective morphism and A is an ordinary abelian variety. We obtain the following decomposition

$$\begin{aligned} \alpha :X \overset{f}{\rightarrow }Y \overset{g}{\rightarrow }Z \overset{h}{\rightarrow }A \end{aligned}$$

such that

• Y and Z are projective normal varieties.

• f is a birational morphism, and g and h are finite surjective morphisms.

• g is purely inseparable and h is separable.

Note that we can find such a decomposition as follows. First, we take the Stein factorization of $\alpha$ and we obtain Y. Then $f:X \rightarrow Y$ is birational and $Y \rightarrow A$ is finite. Second, take the separable closure L of K(A) in $K(X)=K(Y)$ and consider the normalization Z of A in L.

Step 2. Y is smooth.

Proof of Step 2

Since $f_*\mathcal {O}_X=\mathcal {O}_Y$, Y is F-split. By Lemma 4.5, there are the mutually distinct p-torsion line bundles $M_1, \ldots , M_{p^{\dim X}}$ on A such that $M_i \in {{\text {Pic}}}^0(A)$. By Sect. 2.2, $\alpha ^*M_1, \ldots , \alpha ^*M_{p^{\dim X}}$ are mutually distinct p-torsion line bundles on X. Thus, the number of p-torsion line bundles on Y is at least $p^{\dim X}=p^{\dim Y}$. Then, by Lemma 3.3, $F_*\mathcal {O}_Y\simeq \bigoplus _{j \in J} L_j$ for some p-torsion line bundles $L_j$ on Y. Therefore Y is smooth by Kunz’s criterion. $\square$

Step 3. f is an isomorphism.

Proof of Step 3

We can write

$$\begin{aligned} K_X=f^*K_Y+E \end{aligned}$$

where E is an f-exceptional divisor. Since Y is smooth and hence terminal (cf. [9, Section 2.3]), E is effective. Since $K_X\equiv 0$, we see that E is f-nef. By the negativity lemma (cf. [9, Lemma 3.39]), we see $E=0$. Therefore, $K_X=f^*K_Y$. Thus, the codimension of ${{\text {Ex}}}(f)$ in X is at least two. Since Y is smooth, f is an isomorphism. $\square$

Now, we have

$$\begin{aligned} \alpha :X \overset{g}{\rightarrow }Z \overset{h}{\rightarrow }A \end{aligned}$$

such that

• Z is projective normal variety.

• g is a finite surjective purely inseparable morphism.

• h is a finite surjective separable morphism.

Step 4. If g is an isomorphism, then $\alpha$ is also an isomorphism.

Proof of Step 4

We see that the albanese morphism

$$\begin{aligned} \alpha =h:X \rightarrow A \end{aligned}$$

is a finite surjective separable morphism. Since $K_X$ is numerically trivial and $K_A \sim 0$, $\alpha :X \rightarrow A$ is etale in codimension one. Then, by the Zariski–Nagata purity, $\alpha$ is etale. By [13, Section 18, Theorem], X is also an ordinary abelian variety. This completes the proof of Step 4. $\square$

Step 5. g is an isomorphism.

Proof of Step 5

Assume that g is not an isomorphism. Then, we can find

$$\begin{aligned} \alpha :X \overset{\varphi }{\rightarrow }W \rightarrow Z \rightarrow A,\,\,\,\, \beta :W \rightarrow A \end{aligned}$$

which satisfies the following properties.

• W is a projective normal variety.

• $\varphi :X \rightarrow W$ and $W \rightarrow Z$ are finite surjective purely inseparable morphisms with $[K(X):K(W)]=p$.

Since A is an ordinary abelian variety, there are mutually distinct p-torsion line bundles $M_1, \ldots , M_{p^{\dim X}}$ on A which form a subgroup of ${{\text {Pic}}}^0 A$ (Lemma 4.5).

Claim We prove the following assertions.

• (a) $F_*\mathcal {O}_{W}\simeq \varphi _*\mathcal {O}_{X}\oplus E$ for some coherent sheaf E.

• (b) $F_*\mathcal {O}_{W}\simeq \beta ^*M_1\oplus \cdots \oplus \beta ^*M_{p^{\dim X}}.$

Proof of Claim (a) Since $[K(X):K(W)]=p$, the Frobenius map $F_{W}$ factors through $\varphi$:

$$\begin{aligned} F_{W}:W \overset{\mu }{\rightarrow }X \overset{\varphi }{\rightarrow }W. \end{aligned}$$

Since $\mu$ is a finite purely inseparable morphism, there is $e\in \mathbb Z_{>0}$ such that $F^{e}_X$ factors through $\mu$:

$$\begin{aligned} F^{e}_X:X \rightarrow W \overset{\mu }{\rightarrow }X. \end{aligned}$$

Since X is F-split, the identity homomorphism $\mathrm{id}_{\mathcal {O}_X}$ factors through $\mu _*\mathcal {O}_{W}$:

$$\begin{aligned} \mathrm{id}_{\mathcal {O}_X}:\mathcal {O}_X \rightarrow \mu _*\mathcal {O}_{W} \rightarrow (F^{e}_X)_*\mathcal {O}_X \rightarrow \mathcal {O}_X. \end{aligned}$$

Thus, we see

$$\begin{aligned} \mu _*\mathcal {O}_{W}\simeq \mathcal {O}_{X}\oplus E_1 \end{aligned}$$

for some coherent sheaf $E_1$ on X. Take the push-forward by $\varphi$ and we obtain

$$\begin{aligned} (F_{W})_*\mathcal {O}_{W}\simeq \varphi _*\mathcal {O}_X \oplus \varphi _*E_1. \end{aligned}$$

(b) Set $L_i:=\beta ^*M_i$. By Sect. 2.2, $L_1, \ldots , L_{p^{\dim X}}$ are mutually distinct p-torsion line bundles on W such that $\{L_1, \ldots , L_{p^{\dim X}}\}$ forms a subgroup of ${{\text {Pic}}}\,W$ and that

$$\begin{aligned} \varphi ^*L_i \not \simeq \varphi ^*L_j \end{aligned}$$

for every $1\le i< j\le p^{\dim X}$. There are the following two cases:

• $\varphi _*\mathcal {O}_{X}$ is not indecomposable.

• $\varphi _*\mathcal {O}_{X}$ is indecomposable.

Assume that $\varphi _*\mathcal {O}_X$ is not indecomposable. Then, $F_*\mathcal {O}_W$ has an indecomposable direct summand of rank $<p$. Therefore, by Lemma 4.6(1), we obtain

$$\begin{aligned} F_*\mathcal {O}_{W}\simeq \beta ^*M_1\oplus \cdots \oplus \beta ^*M_{p^{\dim X}}. \end{aligned}$$

This is what we want to show.

Assume that $\varphi _*\mathcal {O}_{X}$ is indecomposable. Since ${{\text {rank}}}(\varphi _*\mathcal {O}_{X})=p$, we can apply Lemma 4.6(2) and can find

$$\begin{aligned} \varphi _*\mathcal {O}_{X}\otimes L_i \simeq \varphi _*\mathcal {O}_{X}\otimes L_j \end{aligned}$$

for some $1\le i< j \le p^{\dim X}$. Since $\{L_1, \ldots , L_{p^{\dim X}}\}$ is a group, we obtain $L_i^{-1}\otimes _{\mathcal {O}_X} L_j \simeq L_r$ for some $1\le r \le p^{\dim X}$ with $\varphi ^*L_r\not \simeq \mathcal {O}_X$. Tensor $L_i^{-1}$ and we see

$$\begin{aligned} \varphi _*\mathcal {O}_X \simeq \varphi _*\mathcal {O}_X\otimes L_r \simeq \varphi _*(\varphi ^*L_r). \end{aligned}$$

Then, taking $H^0$, we obtain the following contradiction

$$\begin{aligned} 0\ne H^0(X, \mathcal {O}_X) \simeq H^0(X, \varphi ^*L_r)=0, \end{aligned}$$

where the last equality holds because $\varphi ^*L_r$ is a non-trivial p-torsion line bundle. This completes the proof of Claim.

By the Krull–Schmidt theorem ([2, Theorem 2]), the assertions (a) and (b) in Claim imply

$$\begin{aligned} \varphi _*\mathcal {O}_X=\bigoplus _{j\in J} \beta ^*M_j \end{aligned}$$

for some $J \subset \{1, \ldots , p^{\dim X}\}$. Since $\#J=p$, we obtain $M_{j_0} \not \simeq \mathcal {O}_A$ for some $j_0\in J$.

By Sect. 2.2, we see that $\alpha ^*M_{j_0}\not \simeq \mathcal {O}_X$. Since $\alpha ^*M_{j_0}$ is a non-trivial p-torsion line bundle, we obtain

$$\begin{aligned} H^0\big (X, \alpha ^*M_{j_0}^{-1}\big )=0. \end{aligned}$$

On the other hand, we obtain

$$\begin{aligned} \varphi _*\alpha ^*M_{j_0}^{-1} \simeq \varphi _*\varphi ^*\beta ^*M_{j_0}^{-1} \simeq \varphi _*\mathcal {O}_X \otimes \beta ^*M_{j_0}^{-1} \end{aligned}$$

$$\begin{aligned} \simeq \left( \bigoplus _{j\in J} \beta ^*M_j\right) \otimes \beta ^*M_{j_0}^{-1} \simeq \mathcal {O}_W\oplus \left( \bigoplus _{j\ne j_0} \beta ^*M_j\otimes \beta ^*M_{j_0}^{-1}\right) , \end{aligned}$$

which implies

$$\begin{aligned} H^0(X, \alpha ^*M_{j_0}^{-1})\ne 0. \end{aligned}$$

This is a contradiction. Thus, $g:X \rightarrow Z$ is an isomorphism. This completes the proof of Step 5. $\square$

Step 4 and Step 5 imply the assertion in the theorem. $\square$

5 On the behavior of $F^e_*\mathcal {O}_X$ for some special varieties

In the former sections, we investigate varieties X such that $F_*\mathcal {O}_X$ is decomposed into line bundles. In this section, we study the behavior of $F_*\mathcal {O}_X$ for some special varieties.

5.1 Abelian varieties

In this subsection, we show Theorem 5.3. We recall some results essentially obtained by [15].

Theorem 5.1

(Oda) Let $f:X \rightarrow Y$ be an isogeny of abelian varieties over k. Set $\hat{f}:\hat{Y} \rightarrow \hat{X}$ to be the dual of f. Let $L\in {{\text {Pic}}}^0(X)$. Then,

$$\begin{aligned} f_*L \simeq \mathrm{pr}_{1*}(\mathcal P_Y|_{Y\times {\hat{f}^{-1}([L])}}) \end{aligned}$$

where $\mathcal P_Y$ is the normalized Poincare line bundle of (Y, 0) and $\mathrm{pr}_1$ is the first projection.

Proof

We can apply the same argument as [15, Corollary 1.7]. $\square$

Theorem 5.2

(Oda) Let X be an abelian variety. Let $S\subset \hat{X}$ be a closed subscheme of the dual abelian variety $\hat{X}$. If S is zero-dimensional and Gorenstein, then the following assertions hold.

• (1) There exists an isomorphism between non-commutative k-algebras:

  $$\begin{aligned}\mathrm{End}_{\mathcal {O}_X}(\mathrm{pr}_{1*}(\mathcal P_X|_{X\times S})) \simeq \Gamma (S, \mathcal {O}_S).\end{aligned}$$

  In particular, $\mathrm{End}_{\mathcal {O}_X}(\mathrm{pr}_{1*}(\mathcal P_X|_{X\times S}))$ is a commutative ring.

• (2) If S is one point, that is, $\Gamma (S, \mathcal {O}_S)$ is a local ring, then $\mathrm{pr}_{1*}(\mathcal P_X|_{X\times S})$ is an indecomposable sheaf.

Proof

(1) holds from [15, Corollary 1.12]. We show (2). Assuming $\mathrm{pr}_{1*}(\mathcal P_X|_{X\times S})\simeq E_1 \oplus E_2$ with $E_i\ne 0$, we derive a contradiction. By (1), the ring $\mathrm{End}_{\mathcal {O}_X}(\mathrm{pr}_{1*}(\mathcal P_X|_{X\times S}))$ is a commutative ring. We obtain idempotents $\mathrm{id}_{E_1}\times 0_{E_2}$ and $0_{E_1}\times \mathrm{id}_{E_2}$ such that $\mathrm{id}_{E_1}\times 0_{E_2}+0_{E_1}\times \mathrm{id}_{E_2}$ is the unity of the ring $\mathrm{End}_{\mathcal {O}_X}(\mathrm{pr}_{1*}(\mathcal P_X|_{X\times S}))$. Therefore, we obtain

$$\begin{aligned} \Gamma (S, \mathcal {O}_S) \simeq \mathrm{End}_{\mathcal {O}_X}(\mathrm{pr}_{1*}(\mathcal P_X|_{X\times S})) \simeq A\times B \end{aligned}$$

for some non-zero rings A and B. But, $\Gamma (S, \mathcal {O}_S)$ is a local ring. This is a contradiction. $\square$

We show the main theorem of this subsection.

Theorem 5.3

Let X be an abelian variety. Set $r_X$ to be the p-rank of X. Let $L\in {{\text {Pic}}}^0(X)$. Then, for every $e\in \mathbb Z_{>0}$, we obtain

$$\begin{aligned} F_*^eL \simeq E_1 \oplus \cdots \oplus E_{p^{er_X}} \end{aligned}$$

where each $E_i$ is an indecomposable locally free sheaf of rank $p^{e(\dim X-r_X)}$.

Proof

Fix $e\in \mathbb Z_{>0}$. Consider the absolute Frobenius morphism $F^e_X:X \rightarrow X.$ Set $X^{(p^e)}:=X \times _{k, F_k^e} k$ and we obtain

$$\begin{aligned} F^e_X:X \xrightarrow {F^{e, \mathrm{rel}}_X} X^{(p^e)} \overset{\beta }{\rightarrow }X. \end{aligned}$$

where $\beta$ is a non-k-linear isomorphism of schemes and

$$\begin{aligned} F^{e, \mathrm{rel}}_X:X \rightarrow X^{(p^e)} \end{aligned}$$

is k-linear. Thus, it suffices to show that

$$\begin{aligned} (F^{e, \mathrm{rel}}_X)_*L \simeq E'_1 \oplus \cdots \oplus E'_{p^{er_X}} \end{aligned}$$

for some indecomposable locally free sheaves $E'_i$ of rank $p^{e(\dim X-r_X)}$. Take the dual of $F^{e, \mathrm{rel}}_X$:

$$\begin{aligned} \widehat{(F^{e, \mathrm{rel}}_X)}:\widehat{X^{(p^e)}} \rightarrow \hat{X}. \end{aligned}$$

We show that the number of the fiber of every closed point of $\widehat{\big (F^{e, \mathrm{rel}}_X\big )}$ is $p^{er_X}$. Since $\widehat{\big (F^{e, \mathrm{rel}}_X\big )}(k)$ is a group homomorphism, the numbers of all the fibers are the same. Thus, it suffices to prove that the number of $\widehat{\big (F^{e, \mathrm{rel}}_X\big )}^{-1}(0_{\hat{X}})={{\text {Ker}}}\big (\widehat{\big (F^{e, \mathrm{rel}}_X\big )}(k)\big )$ is $p^{er_X}$. This is equivalent to show that the number of line bundles $M\in {{\text {Pic}}}^0(X^{(p^e)})=\widehat{X^{(p^e)}}(k)$ such that $(F^{e, \mathrm{rel}}_X)^*M\simeq \mathcal {O}_X$ is $p^{er_X}$. Since $\beta :X^{(p^e)}\rightarrow X$ is an isomorphism, we prove that the number of line bundles $N\in {{\text {Pic}}}^0(X)$ such that $N^{p^e}=(F^{e}_X)^*N\simeq \mathcal {O}_X$ is $p^{er_X}$. This follows from the definition of the p-rank.

Taking the separable closure, we obtain

$$\begin{aligned} \widehat{\big (F^{e, \mathrm{rel}}_X\big )}:\widehat{X^{(p^e)}} \overset{g}{\rightarrow }Y \overset{h}{\rightarrow }\hat{X}, \end{aligned}$$

where Y is a normal projective variety, g is a finite surjective purely inseparable morphism and h is a finite surjective separable morphism. Since the numbers of every fiber of $\widehat{\big (F^{e, \mathrm{rel}}_X\big )}$ are the same, h is an etale morphism. In particular, Y is an abelian variety ([13, Section 18, Theorem]) and we may assume that g and h are isogenies. Take the duals again and we obtain

$$\begin{aligned} F^{e, \mathrm{rel}}_X:X \xrightarrow {\hat{h}} \hat{Y} \xrightarrow {\hat{g}} X^{(p^e)}. \end{aligned}$$

Let

$$\begin{aligned} {{\text {Ker}}}(\widehat{F^{e, \mathrm{rel}}_X})=g^{-1}([M_1]) \amalg \cdots \amalg g^{-1}([M_{p^{er_X}}]) \end{aligned}$$

be the decomposition into one point schemes. By Theorem 5.1, we obtain

$$\begin{aligned} (F^{e, \mathrm{rel}}_X)_*\mathcal {O}_X\simeq & {} \mathrm{pr}_{1*}(\mathcal P_{X^{(p^e)}}|_{X^{(p^e)}\times {{\text {Ker}}}(\widehat{F^{e, \mathrm{rel}}_X})})\\\simeq & {} \mathrm{pr}_{1*}(\mathcal P_{X^{(p^e)}}|_{X^{(p^e)}\times g^{-1}([M_1])})\oplus \cdots \oplus \mathrm{pr}_{1*}(\mathcal P_{X^{(p^e)}}|_{X^{(p^e)}\times g^{-1}([M_{p^{er_X}}])})\\\simeq & {} \hat{g}_*M_1\oplus \cdots \oplus \hat{g}_*M_{p^{er_X}}. \end{aligned}$$

Thus, it suffices to show that each locally free sheaf

$$\begin{aligned} \mathrm{pr}_{1*}(\mathcal P_{X^{(p^e)}}|_{X^{(p^e)}\times g^{-1}([M_j])}) \simeq \hat{g}_*M_j \end{aligned}$$

is indecomposable. We see that $g^{-1}([M_j])$ is one point. Thus, if $g^{-1}([M_j])$ is Gorenstein, then $\mathrm{pr}_{1*}(\mathcal P_{X^{(p^e)}}|_{X^{(p^e)}\times g^{-1}([M_j])})$ is indecomposable by Theorem 5.2(2). Since g is finite and Y is smooth, $g^{-1}([M_j])$ is a local complete intersection scheme. In particular, $g^{-1}([M_j])$ is Gorenstein. $\square$

5.2 Curves

In this subsection, we show Theorem 5.5. We need the following result from the theory of stable vector bundles.

Theorem 5.4

Let X be a smooth projective curve of genus $g\ge 2$. Let L be a line bundle on X. Then, $F_*^eL$ is indecomposable for every $e\in \mathbb Z_{>0}$.

Proof

Since L is a line bundle, L is a stable vector bundle. Then, by [17, Theorem 2.2], $F_*^eL$ is also a stable vector bundle. Since stable vector bundles are indecomposable, $F_*^eL$ is indecomposable. $\square$

We show the main theorem of this subsection.

Theorem 5.5

Let X be a smooth projective curve of genus g. Fix an arbitrary positive integer e. Then the following assertions hold.

(0):

If $g=0$, then $F^e_*\mathcal {O}_X \simeq \bigoplus L_j$ where every $L_j$ is a line bundle.

$(1\mathrm{or})$ :

If $g=1$ and X is an ordinary elliptic curve, then $F^e_*\mathcal {O}_X \simeq \bigoplus L_j$ where every $L_j$ is a line bundle.

$(1{ \mathrm ss})$ :

If $g=1$ and X is a supersingular elliptic curve, then $F^e_*\mathcal {O}_X$ is indecomposable.

(2):

If $g\ge 2$, then $F^e_*\mathcal {O}_X$ is indecomposable.

Proof

The assertion (0) immediately follows from the fact that every locally free sheaf of finite rank on $\mathbb P^1$ is decomposed into the direct sum of line bundles.

The assertions $(1\mathrm{or})$ and $(1{ \mathrm ss})$ hold by Theorem 5.3. The assertion (2) follows from Theorem 5.4. $\square$

By Theorem 5.5, it is natural to ask the following question.

Question 5.6

If X is a smooth projective surface X of general type, then is $F_*\mathcal {O}_X$ indecomposable?

As far as the authors know, this question is open. On the other hand, if we drop the assumption that X is smooth, then there exists a counter-example as follows. For a related result, see also [7, Example 3.5].

Theorem 5.7

There exists a projective normal surface X which satisfies the following properties.

• (1) The singularities of X are at worst canonical.

• (2) $K_X$ is ample.

• (3) $F_*\mathcal {O}_X$ is not indecomposable.

Proof

Let S be an ordinary abelian surface. Fix a very ample line bundle H on S. Let $s\in H^0(X, H^p)$ be a general element and set

$$\begin{aligned} \pi :X:={{\text {Spec}}}_S\big (\mathcal {O}_S\oplus H^{-1} \oplus \cdots \oplus H^{-(p-1)}\big ) \rightarrow S \end{aligned}$$

to be the finite purely inseparable morphism where the $\mathcal {O}_S$-algebra $\mathcal {O}_S\oplus H^{-1} \oplus \cdots \oplus H^{-(p-1)}$ is defined by $s\in H^0(X, H^p)$. By [10, Remark 3.5(1)], we can apply [10, Theorem 3.4] for $\mathcal L:=H$. Since the scheme X constructed above is the same as the $\alpha _{\mathcal L}$-torsor $\delta (s)$ appearing in [10, Theorem 3.4]. Therefore, X is normal and has at worst $A_{p-1}$-singularities. Thus (1) holds. We see

$$\begin{aligned} K_X=\pi ^*K_S+(p-1)\pi ^*H \sim (p-1)\pi ^*H, \end{aligned}$$

which implies (2).

We show (3). Since $\pi :X \rightarrow S$ is a finite purely inseparable morphism of degree p, the absolute Frobenius morphisms of X and S factors through $\pi$:

$$\begin{aligned} F_S:S \rightarrow X \overset{\pi }{\rightarrow }S, \,\,\, F_X:X \overset{\pi }{\rightarrow }S \overset{\varphi }{\rightarrow }X. \end{aligned}$$

Since S is F-split, the identity homomorphism $\mathrm{id}_{\mathcal {O}_S}$ factors through $\pi _*\mathcal {O}_X$:

$$\begin{aligned} \mathrm{id}_{\mathcal {O}_S}:\mathcal {O}_S \rightarrow \pi _*\mathcal {O}_X \rightarrow (F_S)_*\mathcal {O}_S \rightarrow \mathcal {O}_S. \end{aligned}$$

This implies

$$\begin{aligned} \pi _*\mathcal {O}_X\simeq \mathcal {O}_S \oplus E \end{aligned}$$

for some coherent sheaf E. Taking the push-forward by $\varphi$, we see

$$\begin{aligned} (F_X)_*\mathcal {O}_X=\varphi _*\pi _*\mathcal {O}_X\simeq \varphi _*\mathcal {O}_S \oplus \varphi _*E. \end{aligned}$$

This implies (3). $\square$

Remark 5.8

If X is a smooth projective curve of general type, then $F_*\mathcal {O}_X$ is indecomposable by Theorem 5.4. Theorem 5.4 depends on the theory of the stable vector bundles. For the 2-dimensional case, a similar result is obtained by Kitadai–Sumihiro [8], Liu–Zhou [11], and Sun [18]. For example, [18, Theorem 4.9 and Remark 4.10] imply that $F_*\mathcal {O}_X$ is indecomposable under the assumptions that $\mu (\Omega _X^1)>0$ and $\Omega _X^1$ is semi-stable.