Frobenius splitting and geometry of $G$-Schubert varieties

TL;DR

This paper demonstrates Frobenius splitting properties of $G$-Schubert varieties in equivariant embeddings, revealing their singularity characteristics and extending results to $ ext{R}$-Schubert varieties, with implications for algebraic geometry in positive characteristic.

Contribution

It proves Frobenius splitting compatibility for all $G$-Schubert varieties in equivariant embeddings and extends these results to $ ext{R}$-Schubert varieties, highlighting their geometric properties.

Findings

$X$ admits a Frobenius splitting compatible with all $G$-Schubert varieties.

Smooth, projective, toroidal $X$ have $G$-Schubert varieties with stable Frobenius splitting.

Existence of a non-normal $G$-Schubert variety in the wonderful compactification of type $G_2$.

Abstract

Let $X$ be an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$ of positive characteristic. Let $B$ denote a Borel subgroup of $G$. A $G$-Schubert variety in $X$ is a subvariety of the form $\diag(G) \cdot V$, where $V$ is a $B \times B$-orbit closure in $X$. In the case where $X$ is the wonderful compactification of a group of adjoint type, the $G$-Schubert varieties are the closures of Lusztig's $G$-stable pieces. We prove that $X$ admits a Frobenius splitting which is compatible with all $G$-Schubert varieties. Moreover, when $X$ is smooth, projective and toroidal, then any $G$-Schubert variety in $X$ admits a stable Frobenius splitting along an ample divisors. Although this indicates that $G$-Schubert varieties have nice singularities we present an example of a non-normal $G$-Schubert variety in the wonderful compactification of a group of type $G_2$. Finally we also extend the Frobenius splitting results to the more general class of $\mathcal R$-Schubert varieties.

Equations

$$<(x,y),(x^{\prime },y^{\prime })>=\ll x,x^{\prime }\gg -\ll y,y^{\prime }\gg .$$

$$p_X:X\to \text{Spec}(k),$$

$$F_k:\text{Spec}(k)\to \text{Spec}(k),$$

$$p_X^{\prime }:X\to \text{Spec}(k),$$

$$F_X:X\to X^{\prime },$$

$$F_X^{♯}:{\mathcal{O}}_{X^{\prime }}\to (F_X{)}_{\ast }{\mathcal{O}}_X,$$

$$(F_X{)}^{\ast }{\mathcal{L}}^{\prime }\simeq {\mathcal{L}}^p$$

$$F_X^{♯}:{\mathcal{O}}_{X^{\prime }}\to (F_X{)}_{\ast }{\mathcal{O}}_X,$$

$$s\in{\rm Hom}_{\mathcal{O}_{X^{\prime}}}\big{(}(F_{X})_{*}\mathcal{O}_{X},\mathcal{O}_{X^{\prime}}\big{)},$$

$$\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}\big{(}(F_{X})_{*}\mathcal{L},\mathcal{O}_{X^{\prime}}\big{)}\rightarrow\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}\big{(}(F_{X})_{*}(\mathcal{L}\otimes\mathcal{I}_{Y}),\mathcal{O}_{Y^{\prime}}\big{)},$$

$$\mathcal{E}n{d}_F^{\mathcal{L}}(X,Y{)}_{|Y}\to \mathcal{E}n{d}_F^{{\mathcal{L}}_{|Y}}(Y),$$

$$s\big{(}(F_{X})_{*}\mathcal{I}_{Y_{i}}\big{)}\subset\mathcal{I}_{Y_{i}^{\prime}}.$$

$$\mathcal{J}=s\big{(}(F_{X})_{*}(\mathcal{I}_{Y_{1}}\otimes\mathcal{L})\big{)}\subset\mathcal{O}_{X^{\prime}}.$$

$${\mathrm{ev}}_X:{\mathrm{End}}_F(X)\to k[X^{\prime }],$$

$$s:(F_X{)}_{\ast }{\mathcal{O}}_X\stackrel{(F_X{)}_{\ast }{\sigma }_D}{\to }(F_X{)}_{\ast }{\mathcal{O}}_X(D)\stackrel{s_D}{\to }{\mathcal{O}}_{X^{\prime }},$$

$${\overline{s}}_D:(F_Y{)}_{\ast }{\mathcal{O}}_X(D{)}_{|Y}\to {\mathcal{O}}_{Y^{\prime }},$$

$$F_X^{(n)}:X\to X^{(n)},$$

$$(F_X^{(n)}{)}^{♯}:{\mathcal{O}}_{X^{(n)}}\to (F_X^{(n)}{)}_{\ast }{\mathcal{O}}_X.$$

$$s\in{\rm Hom}_{\mathcal{O}_{X^{(n)}}}\big{(}(F_{X}^{(n)})_{*}\mathcal{\mathcal{O}}_{X}(D),\mathcal{\mathcal{O}}_{X^{(n)}}\big{)},$$

$${\mathcal{O}}_{X^{(n)}}\stackrel{(F_X^{(n)}{)}^{♯}}{\to }(F_X^{(n)}{)}_{\ast }{\mathcal{O}}_X\stackrel{(F_X^{(n)}{)}_{\ast }{\sigma }_D}{\to }(F_X^{(n)}{)}_{\ast }{\mathcal{O}}_X(D)\stackrel{s}{\to }{\mathcal{O}}_{X^{(n)}},$$

$$s\big{(}(F_{X}^{(n)})_{*}(\mathcal{I}_{Y}\otimes\mathcal{O}_{X}(D))\big{)}\subset\mathcal{I}_{Y^{(n)}}.$$

$${\mathrm{H}}^i(X,\mathcal{L})\subset {\mathrm{H}}^i(X,{\mathcal{L}}^{p^n}\otimes {\mathcal{O}}_X(D)).$$

$${\mathrm{H}}^0(X,\mathcal{L})\to {\mathrm{H}}^0(Y,\mathcal{L}),$$

$$(F_{X})_{*}(F_{X})^{!}\mathcal{F}=\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}\big{(}(F_{X})_{*}\mathcal{O}_{X},\mathcal{F}\big{)}.$$

$$\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}\big{(}(F_{X})_{*}\mathcal{O}_{X},\mathcal{F}\big{)},$$

$$(g\cdot \varphi )(f)=\varphi (gf),$$

$$(F_{X})_{*}\mathcal{H}om_{\mathcal{O}_{X}}\big{(}\mathcal{G},(F_{X})^{!}\mathcal{F}\big{)}\simeq\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}\big{(}(F_{X})_{*}\mathcal{G},\mathcal{F}\big{)},$$

$${\rm Hom}_{\mathcal{O}_{X}}\big{(}\mathcal{G},(F_{X})^{!}\mathcal{F}\big{)}\simeq{\rm Hom}_{\mathcal{O}_{X^{\prime}}}\big{(}(F_{X})_{*}\mathcal{G},\mathcal{F}\big{)},$$

$$\eta^{\prime}:(F_{X})_{*}\mathcal{G}\xrightarrow{(F_{X})_{*}\eta}(F_{X})_{*}(F_{X})^{!}\mathcal{F}\simeq\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}\big{(}(F_{X})_{*}\mathcal{O}_{X},\mathcal{F}\big{)}\rightarrow\mathcal{F}.$$

$$(F_X{)}_{\ast }\mathcal{L}\stackrel{(F_X{)}_{\ast }\eta }{\to }(F_X{)}_{\ast }\mathcal{E}n{d}_F^{!}(X).$$

Full text

1. Introduction

Let $G$ denote a connected and reductive group over an algebraically closed field $k$, and let $B$ denote a Borel subgroup of $G$. An equivariant embedding $X$ of $G$ is a $G\times G$-variety which contains $G=(G\times G)/\text{\rm diag}(G)$ as an open $G\times G$-invariant subset, where $\text{\rm diag}(G)$ is the diagonal image of $G$ in $G\times G$. Any equivariant embedding $X$ of $G$ contains finitely many $B\times B$-orbits. In recent years the geometry of closures of $B\times B$-orbits has been studied by several authors. The most general result was obtained in [H-T2] where it was proved that $B\times B$-orbit closures are normal, Cohen-Macaulay and have ($F$-)rational singularities (actually, even stronger results were obtained). In the present paper we will study (closed) subvarieties in $X$ of the form $\rm{diag}(G)\cdot V$, where $V$ denotes the closure of a $B\times B$-orbit. Subvarieties of equivariant embeddings of $G$ of this form will be called $G$-Schubert varieties.

When $G$ is a semisimple group of adjoint type there exists a canonical equivariant embedding ${\bf X}$ of $G$ which is called the wonderful compactification. The wonderful compactifications are of primary interest in this paper. Actually, this work arose from the question of describing the closures of the so-called $G$-stable pieces of ${\bf X}$. The $G$-stable pieces makes up a decomposition of ${\bf X}$ into locally closed subsets. They were introduced by Lusztig in [L] where they were used to construct and study a class of perverse sheaves which generalizes his theory of character sheaves on reductive groups. More precisely, these perverse sheaves are the intermediate extensions of the so-called “character sheaves” on a $G$-stable piece. This motivates the study of closures of $G$-stable pieces which turns out to coincide with the set of $G$-Schubert varieties.

Before discussing the closures of $G$-stable pieces in details, let us make a short digression and discuss some other motivations for studying $G$-stable pieces and $G$-Schubert varieties (in wonderful compactifications):

1. (1)

When $G$ is a simple group, the boundary of the closure of the unipotent subvariety of $G$ in the wonderful compactification ${\bf X}$, is a union of certain $G$-Schubert varieties (see [He] and [H-T]). Thus knowing the geometry of these $G$-Schubert varieties will help us to understand the geometry of the closure of the unipotent variety within ${\bf X}$. 2. (2)

Let $\text{\rm Lie}(G)$ denote the Lie algebra of a simple group $G$ over a field of characteristic zero. Let $\ll,\gg$ denote a fixed symmetric non-degenerate ad-invariant bilinear form. Let $<,>$ be the bilinear form on $\text{\rm Lie}(G)\oplus\text{\rm Lie}(G)$ defined by

[TABLE]

In [E-L], Evens and Lu showed that each splitting $\text{\rm Lie}(G)\oplus\text{\rm Lie}(G)=l\oplus l^{\prime}$, where $l$ and $l^{\prime}$ are Lagrangian subalgebras of $\text{\rm Lie}(G)\oplus\text{\rm Lie}(G)$, gives rise to a Poisson structure $\Pi_{l,l^{\prime}}$ on ${\bf X}$. If moreover, one starts with the Belavin-Drinfeld splitting, then all the $G$-stable pieces/$G$-Schubert varieties and $B\times B^{-}$-orbits of ${\bf X}$ are Poisson subvarieties, where $B^{-}$ is a Borel subgroup opposite to $B$. Thus to understand the Poisson structure on ${\bf X}$ corresponding to the Belavin-Drinfeld splitting, one needs to understand the geometry of the $G$-stable pieces/$G$-Schubert varieties. If we start with another splitting, then we obtain a different Poisson structure on ${\bf X}$ and in order to understand these Poisson structures, one needs to study the $\mathcal{R}$-stable pieces [L-Y] instead (see Section 12), which generalize both the $G$-stable pieces and the $B\times B^{-}$-orbits.

The main technical ingredient in this paper is the positive characteristic notion of Frobenius splitting. Frobenius splitting is a powerful tool which has been proved to be very useful in obtaining strong geometric conclusions for e.g. Schubert varieties and closures of $B\times B$-orbits in equivariant embeddings. In the present paper we obtain two types of results related to $G$-Schubert varieties over fields of positive characteristic. First of all, if we fix an equivariant embedding $X$ of a reductive group $G$ then we prove that all $G$-Schubert varieties in $X$ are simultaneously compatibly Frobenius split by a Frobenius splitting of $X$. Secondly, concentrating on a single $G$-Schubert variety $\mathcal{X}$, in a smooth projective and toroidal embedding $X$, we prove that this admits a stable Frobenius splitting along an ample divisor. Statements of this form put strong conditions on the intertwined behavior of cohomology groups of line bundles on $X$ and its $G$-Schubert varieties. As this is related to geometric properties it therefore seems natural to expect that $G$-Schubert varieties should have nice singularities. It therefore comes as a complete surprise that $G$-Schubert varieties, in general, are not even normal. We only provide a single example of this phenomenon (in the wonderful compactification of a group of type $G_{2}$), but expect that this absence of normality is the general picture.

In obtaining the Frobenius splitting result mentioned above, we have developed some general theory of how to construct Frobenius splitting of varieties of the form $G\times_{P}X$ (see Section 4.2 for the definition). This part of the paper is influenced by the theory of $B$-canonical Frobenius splitting as discussed in [B-K, Chap.4]; in particular the proof of [B-K, Prop.4.1.17]. The presentation we provide is more general and makes it possible to extract even better result from the ideas of $B$-canonical Frobenius splittings. This theory is presented in Chapter 5 in a generality which is more than necessary for obtaining the described Frobenius splitting results for $G$-Schubert varieties. However, we hope that this theory could be useful elsewhere and we certainly consider it to be of independent interest.

This paper is organized in the following way. In Section 2 we introduce notation, and in Section 3 we briefly define Frobenius splitting and explain its fundamental ideas. Section 4 is devoted to some results on linearized sheaves which should all be well known. In Section 5 we study the Frobenius splitting of varieties of the form $G\times_{P}X$ for a variety $X$ with an action by a parabolic subgroup $P$. The main idea is to decompose the Frobenius morphism on $G\times_{P}X$ into maps associated to the Frobenius morphism on the base $\nicefrac{{G}}{{P}}$ and the fiber $X$ of the natural morphism $G\times_{P}X\rightarrow\nicefrac{{G}}{{P}}$. In Section 6 we relate $B$-canonical Frobenius splittings to the material in Section 5. Section 7 contains applications of Section 5 to general $G\times G$-varieties. In section 8 we define the $G$-stable pieces and $G$-Schubert varieties. In Section 9 we apply the material of the previous sections to the class of equivariant embeddings and obtain Frobenius splitting results for $G$-Schubert varieties. Section 10 contains results related to cohomology of line bundles on $G$-Schubert varieties. Section 11 contains an example of a non-normal $G$-Schubert variety. Finally Section 12 contains generalizations and variations of the previous sections.

We would like to thank the referee for a careful reading of this paper and for numerous suggestions concerning the presentation.

2. Notation

We will work over a fixed algebraically closed field $k$. The characteristic of $k$ will depend on the application. By a variety we mean a reduced and separated scheme of finite type over $k$. In particular, we allow a variety to have several irreducible components.

2.1. Group setup

We let $G$ denote a connected linear algebraic group over $k$. We fix a Borel subgroup $B$ and a maximal torus $T\subset B$. The notation $P$ is used for a parabolic subgroup of $G$ containing $B$. The set of $T$-characters is denoted by $X^{*}(T)$ and we identify this set with the set $X^{*}(B)$ of $B$-characters.

2.2. Reductive case

In many cases we will specialize to the case where $G$ is reductive. In this case we will also use the following notation : the set of roots determined by $T$ is denoted by $R\subseteq X^{*}(T)$ while the set of positive roots determined by $(B,T)$ is denoted by $R^{+}$. The simple roots are denoted by $\alpha_{1},\dots,\alpha_{l}$, and we let $\Delta=\{1,\dots,l\}$ denote the associated index set. The simple reflection associated to the simple root $\alpha_{i}$ is then denoted by $s_{i}$. The Weyl group $W=\nicefrac{{N_{G}(T)}}{{T}}$ is generated by the simple reflections $s_{i}$, for $i\in\Delta$. The length of $w\in W$ will be denoted by $l(w)$. For $J\subset\Delta$, let $W_{J}$ denote the subgroup of $W$ generated by the simple reflection associated with the elements in $J$, and let $W^{J}$ (resp. ${}^{J}W$) denote the set of minimal length coset representatives for ${W}/{W_{J}}$ (resp. $W_{J}\backslash W$). The element in $W$ of maximal length will be denoted by $w_{0}$, while $w_{0}^{J}$ is used for the same kind of element in $W_{J}$. For any $w\in W$, we let $\dot{w}$ denote a representative of $w$ in $N_{G}(T)$. For $J\subset\Delta$, let $P_{J}\supset B$ denote the corresponding standard parabolic subgroup and $P^{-}_{J}\supset B^{-}$ denote its opposite parabolic. Let $L_{J}=P_{J}\cap P^{-}_{J}$ be the common Levi subgroup of $P_{J}$ and $P^{-}_{J}$ containing $T$. Let $U_{J}$ (resp. $U_{J}^{-}$) denote the unipotent radical of $P_{J}$ (resp. $P_{j}^{-}$). When $J=\emptyset$ we also use the notation $U$ and $U^{-}$ for $U_{J}$ and $U_{J}^{-}$ respectively. When $G$ is semisimple and simply connected we may associate a fundamental character $\omega_{i}$ to each simple root $\alpha_{i}$. The sum of the fundamental characters is then denoted by $\rho$. Then $\rho$ also equals half the sum of the positive roots.

3. The relative Frobenius morphism

In this section we collect some results related to the Frobenius morphism and to the concept of Frobenius splitting. Compared to other presentations on the same subject, this presentation differs only in its emphasis on the set {\rm Hom}_{\mathcal{O}_{X^{\prime}}}\big{(}(F_{X})_{*}\mathcal{O}_{X},\mathcal{O}_{X^{\prime}}\big{)} (to be defined below) and not just the set of Frobenius splittings. Thus, the obtained results are only small variations of already known results as can be found in e.g. [B-K].

3.1. The Frobenius morphism

By definition a variety $X$ comes with an associated morphism

[TABLE]

of schemes. Assume that the field $k$ has positive characteristic $p>0$. Then the Frobenius morphism on $\textrm{Spec}(k)$ is the morphism of schemes

[TABLE]

which on the level of coordinate rings is defined by $a\mapsto a^{p}$. As $k$ is assumed to be algebraically closed the morphism $F_{k}$ is actually an isomorphism and we let $F_{k}^{-1}$ denote the inverse morphism. Composing $p_{X}$ with $F_{k}^{-1}$ we obtain a new variety

[TABLE]

with underlying scheme $X$. In the following we suppress the morphism $p_{X}$ from the notation and simply use $X$ as the notation for the variety defined by $p_{X}$. The variety defined by $p^{\prime}_{X}$ is then denoted by $X^{\prime}$.

The relative Frobenius morphism on $X$ is then the morphism of varieties :

[TABLE]

which as a morphism of schemes is the identity map on the level of points and where the associated map of sheaves

[TABLE]

is the $p$-th power map. A key property of the Frobenius morphism is the relation

[TABLE]

which is satisfied for every line bundle $\mathcal{L}$ on $X$ (here $\mathcal{L}^{\prime}$ denotes the corresponding line bundle on $X^{\prime}$).

3.2. Frobenius splitting

A variety $X$ is said to be Frobenius split if the $\mathcal{O}_{X^{\prime}}$-linear map of sheaves :

[TABLE]

has a section; i.e. if there exists an element

[TABLE]

such that the composition $s\circ F_{X}^{\sharp}$ is the identity endomorphism of $\mathcal{O}_{X^{\prime}}$. The section $s$ will be called a Frobenius splitting of $X$.

3.3. Compatibility with line bundles and closed subvarieties

Fix a line bundle $\mathcal{L}$ on $X$ and a closed subvariety $Y$ in $X$ with sheaf of ideals $\mathcal{I}_{Y}$. Let $Y^{\prime}$ denote the closed subvariety of $X^{\prime}$ associated to $Y$ with sheaf of ideals denoted by $\mathcal{I}_{Y^{\prime}}$. The kernel of the natural morphism

[TABLE]

induced by the inclusion $\mathcal{L}\otimes\mathcal{I}_{Y}\subset\mathcal{L}$ and the projection $\mathcal{O}_{X^{\prime}}\rightarrow\mathcal{O}_{Y^{\prime}}$, will be denoted by $\mathcal{E}nd_{F}^{\mathcal{L}}(X,Y)$. The associated space of global sections will be denoted by ${\rm End}_{F}^{\mathcal{L}}(X,Y)$. When $Y=X$ we simply denote $\mathcal{E}nd_{F}^{\mathcal{L}}(X,Y)$ (resp. ${\rm End}_{F}^{\mathcal{L}}(X,Y)$) by $\mathcal{E}nd_{F}^{\mathcal{L}}(X)$ (resp. ${\rm End}_{F}^{\mathcal{L}}(X)$). The sheaf $\mathcal{E}nd_{F}^{\mathcal{L}}(X,Y)$ is a subsheaf of $\mathcal{E}nd_{F}^{\mathcal{L}}(X)$ consisting of the elements compatible with $Y$. Moreover, there is a natural morphism

[TABLE]

where the notation |Y means restriction to $Y$.

If $Y_{1},Y_{2},\dots,Y_{m}$ is a collection of closed subvarieties of $X$ then the notation $\mathcal{E}nd_{F}^{\mathcal{L}}(X,Y_{1},\dots,Y_{m})$ (or sometimes $\mathcal{E}nd_{F}^{\mathcal{L}}(X,\{Y_{i}\}_{i=1}^{m})$) will denote the intersection of the subsheaves $\mathcal{E}nd_{F}^{\mathcal{L}}(X,Y_{i})$ for $i=1,\dots,m$. The set of global sections of the sheaf $\mathcal{E}nd_{F}^{\mathcal{L}}(X,Y_{1},\dots,Y_{m})$ will be denoted by ${\rm End}_{F}^{\mathcal{L}}(X,Y_{1},\dots,Y_{m})$.

When $\mathcal{L}=\mathcal{O}_{X}$ we remove $\mathcal{L}$ from all of the above notation. In particular, the vectorspace ${\rm End}_{F}(X)$ denotes the set of morphisms from $(F_{X})_{*}\mathcal{O}_{X}$ to $\mathcal{O}_{X^{\prime}}$ and thus contains the set of Frobenius splittings of $X$. A Frobenius splitting $s$ of $X$ contained in ${\rm End}_{F}(X,\{Y_{i}\}_{i})$ is said to be compatible with the subvarieties $Y_{1},\dots,Y_{m}$. When $s$ is compatible in this sense it induces a Frobenius splitting of each $Y_{i}$ for $i=1\dots,m$. In this case we also say that $s$* compatibly Frobenius splits $Y_{1},\dots,Y_{m}$*. In concrete terms, this is equivalent to

[TABLE]

for all $i$.

Lemma 3.1**.**

Let $Y$ and $Z$ denote closed subvarieties in $X$ and let $s$ denote a global section of $\mathcal{E}nd_{F}^{\mathcal{L}}(X,Z,Y)$.

1. (1)

$s\in{\rm End}_{F}^{\mathcal{L}}(X,Y_{1})$* for every irreducible component $Y_{1}$ of $Y$.* 2. (2)

If the scheme theoretic intersection $Z\cap Y$ is reduced then $s$ is contained in ${\rm End}_{F}^{\mathcal{L}}(X,Y\cap Z)$.

Proof.

Let $Y_{1}$ denote an irreducible component of $Y$ and let

[TABLE]

Let $U$ denote the open complement (in $X^{\prime}$) of the irreducible components of $Y^{\prime}$ which are different from $Y_{1}^{\prime}$. Then $\mathcal{I}_{Y_{1}^{\prime}}$ coincides with $\mathcal{I}_{Y^{\prime}}$ on $U$ and consequently $\mathcal{J}_{|U}\subset(\mathcal{I}_{Y^{\prime}})_{|U}$ as $s$ is compatible with $Y$. In particular, $\mathcal{J}_{|U}\subset(\mathcal{I}_{Y_{1}^{\prime}})_{|U}$. We claim that this implies that $\mathcal{J}\subset\mathcal{I}_{Y_{1}^{\prime}}$ : let $V$ denote an open subset of $X^{\prime}$ and let $f$ be a section of $\mathcal{J}$ over $V$. As $\mathcal{J}$ is a subsheaf of $\mathcal{O}_{X^{\prime}}$, we may consider $f$ as a function on $V$, and it suffices to prove that $f$ vanishes on $Y_{1}^{\prime}\cap V$. If $Y_{1}^{\prime}\cap V$ is empty then this is clear. Otherwise, $U\cap V\cap Y_{1}^{\prime}$ is a dense subset of $Y_{1}^{\prime}$ and it suffices to prove that $f$ vanishes on this set. But this follows from the inclusion $\mathcal{J}_{|U}\subset(\mathcal{I}_{Y_{1}^{\prime}})_{|U}$. As a consequence $s$ is compatible with $Y_{1}$. The second claim follows as the sheaf of ideals of the intersection $Z\cap Y$ is $\mathcal{I}_{Y}+\mathcal{I}_{Z}$. ∎

The condition that $Z\cap Y$ is reduced, in Lemma 3.1, only ensures that $Z\cap Y$ is a variety. When $\mathcal{L}=\mathcal{O}_{X}$ and $s$ is a Frobenius splitting this is always satisfied [B-K, Prop.1.2.1].

3.4. The evaluation map

Let $k[X^{\prime}]$ denote the space of global regular functions on $X^{\prime}$. Evaluating an element $s:(F_{X})_{*}\mathcal{O}_{X}\rightarrow\mathcal{O}_{X^{\prime}}$ of ${\rm End}_{F}(X)$ at the constant global function $1$ on $X$ defines an element in $k[X^{\prime}]$ which we denote by ${\rm ev}_{X}(s)$. This defines a morphism

[TABLE]

with the property that ${\rm ev}_{X}(s)=1$ if and only if $s$ is a Frobenius splitting of $X$.

3.5. Frobenius $D$-splittings

Consider an effective Cartier divisor $D$ on $X$, and let $\sigma_{D}$ denote the associated global section of the associated line bundle $\mathcal{O}_{X}(D)$. A Frobenius splitting $s$ of $X$ is said to be a Frobenius $D$-splitting if $s$ factorizes as

[TABLE]

for some element $s_{D}$ in {\rm End}_{F}^{\mathcal{O}_{X}(D)}\big{(}X\big{)}. We furthermore say that the Frobenius $D$-splitting $s$ is compatible with a subvariety $Y$ if $s_{D}$ is compatible with $Y$. The following result assures that, in this case, the compatibility with closed subvarieties agrees with the usual definition [R, Defn.1.2].

Lemma 3.2**.**

Assume that $s$ defines a Frobenius $D$-splitting of $X$. Then $s_{D}$ is compatible with $Y$ if and only if (i) $s$ compatibly Frobenius splits $Y$ and (ii) the support of $D$ does not contain any irreducible components of $Y$.

Proof.

The if part of the statement follows from [R, Prop.1.4]. So assume that $s_{D}$ is compatible with $Y$. Then $s_{D}$ induces a morphism

[TABLE]

satisfying $\overline{s}_{D}((\sigma_{D})_{|Y})$ is the constant function $1$ on $Y^{\prime}$. As a consequence $(\sigma_{D})_{|Y}$ does not vanish on any of the irreducible components of $Y$. This proves part (ii) of the statement. Part (i) is clearly satisfied. ∎

It follows that if $s$ is compatible with $Y$ and, moreover, defines a Frobenius $D$-splitting of $X$ then $D\cap Y$ makes sense as an effective Cartier divisor on $Y$ and, in this case, $s$ induces a Frobenius $D\cap Y$-splitting of $Y$.

3.6. Stable Frobenius splittings along divisors

Let $X^{(0)}=X$ and define recursively $X^{(n)}=(X^{(n-1)})^{\prime}$ for $n\geq 1$. Composing the Frobenius morphisms on $X^{(i)}$ for $i=0,\dots,n$, we obtain a morphism

[TABLE]

with an associated map of sheaves

[TABLE]

Let, as in Section 3.5, $D$ denote an effective Cartier divisor on $X$ with associated canonical section $\sigma_{D}$ of $\mathcal{O}_{X}(D)$. We say that $X$* admits a stable Frobenius splitting along $D$* if there exists a positive integer $n$ and an element

[TABLE]

such that the composed map

[TABLE]

is the identity map on $\mathcal{O}_{X^{(n)}}$. The element $s$ is called a stable Frobenius splitting of $X$ along $D$. When $Y$ is a closed subvariety of $X$ we say that the stable Frobenius splitting $s$* is compatible with $Y$* if

[TABLE]

Notice that this condition necessarily implies that the support of $D$ does not contain any of the irreducible components of $Y$ (cf. proof of Lemma 3.2). Notice also that if $X$ admits a Frobenius $D$-splitting which is compatible with $Y$ then $X$ admits a stable Frobenius splitting along $D$ which is compatible with $Y$. The following is well known (see e.g. [T, Lem.4.4])

Lemma 3.3**.**

Let $D_{1}$ and $D_{2}$ denote effective Cartier divisors on $X$ and let $Y$ denote a closed subvariety of $X$. Then $X$ admits stable Frobenius splittings along $D_{1}$ and $D_{2}$ which are compatible with $Y$ if and only if $X$ admits a stable Frobenius splitting along $D_{1}+D_{2}$ which is compatible with $Y$.

The following result explains one of the main applications of (stable) Frobenius splitting. Remember that a line bundle $\mathcal{L}$ is nef if $\mathcal{L}\otimes\mathcal{M}$ is ample whenever $\mathcal{M}$ is ample.

Proposition 3.4**.**

Assume that $X$ admits a stable Frobenius splitting along an effective Cartier divisor $D$. Then there exists a positive integer $n$ such that for each line bundle $\mathcal{L}$ on $X$ we have an inclusion of abelian groups

[TABLE]

In particular, if $D$ is ample and $\mathcal{L}$ is nef, then ${\rm H}^{i}(X,\mathcal{L})=0$ for $i>0$. Moreover, if the stable Frobenius splitting of $X$ is compatible with a subvariety $Y$, $D$ is ample and $\mathcal{L}$ is nef then the restriction morphism

[TABLE]

is surjective.

Proof.

Argue as in the proof [R, Prop.1.13(i)]. ∎

3.7. Duality for $F_{X}$

By duality (see [Har2, Ex.III.6.10]) for the finite morphism $F_{X}$ we may to each quasi-coherent $\mathcal{O}_{X^{\prime}}$-module $\mathcal{F}$ associate an $\mathcal{O}_{X}$-module denoted by $(F_{X})^{!}\mathcal{F}$ and satisfying

[TABLE]

Actually, as $F_{X}$ is the identity on the level of points we may define $(F_{X})^{!}\mathcal{F}$ as the sheaf of abelian groups

[TABLE]

with $\mathcal{O}_{X}$-module structure defined by

[TABLE]

for $g,f\in\mathcal{O}_{X}$ and \phi\in\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}\big{(}(F_{X})_{*}\mathcal{O}_{X},\mathcal{F}\big{)}. When $\mathcal{F}=\mathcal{O}_{X}$ we will also use the notation $\mathcal{E}nd^{!}_{F}(X)$ for $(F_{X})^{!}\mathcal{O}_{X}$. This sheaf is particularly nice when $X$ is smooth as $(F_{X})^{!}\mathcal{O}_{X}$ then coincides with the line bundle $\omega_{X}^{1-p}$, where $\omega_{X}$ denotes the dualizing sheaf of $X$ (see e.g. [B-K, Sect.1.3]). If $Y_{1},Y_{2},\dots,Y_{m}$ is a collection of closed subvarieties of $X$ then $\mathcal{E}nd^{!}_{F}(X,Y_{1},\dots,Y_{m})$ (or $\mathcal{E}nd^{!}_{F}(X,\{Y_{i}\}_{i=1}^{m})$) will denote the subsheaf of $\mathcal{E}nd^{!}_{F}(X)$ consisting of the elements mapping the sheaf of ideals $\mathcal{I}_{Y_{i}}$ to $\mathcal{I}_{Y_{i}^{\prime}}$ for all $i=1,\dots,m$. We say that $\mathcal{E}nd^{!}_{F}(X,\{Y_{i}\}_{i=1}^{m})$ is the subsheaf of elements compatible with $Y_{1},\dots,Y_{m}$.

More generally, duality for $F_{X}$ implies that we have a natural identification

[TABLE]

whenever $\mathcal{G}$ (resp. $\mathcal{F}$) is a quasicoherent sheaf on $X$ (resp. $X^{\prime}$). This leads to the identification

[TABLE]

where a morphism $\eta:\mathcal{G}\rightarrow(F_{X})^{!}\mathcal{F}$ is identified with the composed morphism

[TABLE]

Here the latter map is the natural evaluation map at the element $1$ in $\mathcal{O}_{X}$. From now on we will specialize to the case where $\mathcal{F}=\mathcal{O}_{X^{\prime}}$ and $\mathcal{G}$ equals a line bundle $\mathcal{L}$ on $X$. In this case, an element in {\rm Hom}_{\mathcal{O}_{X}}\big{(}\mathcal{L},\mathcal{E}nd^{!}_{F}(X)\big{)} may also be considered as a global section of the sheaf $\mathcal{E}nd^{!}_{F}(X)\otimes\mathcal{L}^{-1}$. For later use we emphasize

Lemma 3.5**.**

Let $\eta$ be an element in {\rm Hom}_{\mathcal{O}_{X}}\big{(}\mathcal{L},\mathcal{E}nd^{!}_{F}(X)\big{)} and let $\eta^{\prime}$ denote the corresponding element in {\rm Hom}_{\mathcal{O}_{X^{\prime}}}\big{(}(F_{X})_{*}\mathcal{L},\mathcal{O}_{X^{\prime}}\big{)} by the above identification. Then $\eta^{\prime}$ factors through the morphism

[TABLE]

Moreover, the element $\eta^{\prime}$ is compatible with a collection of closed subvarieties $Y_{1},\dots,Y_{m}$ of $X$ if and only if the image of $\eta$ is contained in $\mathcal{E}nd^{!}_{F}(X,Y_{1},\dots,Y_{m})$.

Proof.

The first part of the statement follows directly from the discussion above. To prove the second statement we may assume that $m=1$. We use the notation $Y=Y_{1}$. Let $\sigma$ denote a section of $\mathcal{L}$ over an open subset $U$ of $X$, and consider $s=\eta(\sigma)$ as a map

[TABLE]

That $s$ is compatible with $Y$ means that $s(f)$ vanishes on $Y^{\prime}$ whenever $f$ vanishes on $Y$ for a function $f$ on $U$. Alternatively, the evaluation of $f\cdot s$ at $1$, which coincides with $\eta^{\prime}(f\cdot\sigma)$, should vanish on $Y^{\prime}$. In particular, the image of $\eta$ is contained in $\mathcal{E}nd^{!}_{F}(X,Y)$ if and only if the restriction of $\eta^{\prime}$ to (F_{X})_{*}\big{(}\mathcal{I}_{Y}\otimes\mathcal{L}\big{)} maps into $\mathcal{I}_{Y^{\prime}}$. This ends the proof. ∎

We will also need the following remark

Lemma 3.6**.**

Let $D$ denote a reduced effective Cartier divisor on $X$ and $\mathcal{L}$ denote a line bundle on $X$. Let $\mathcal{M}=\mathcal{O}_{X}((p-1)D)\otimes\mathcal{L}$ and assume that we have an $\mathcal{O}_{X}$-linear morphism $\eta:\mathcal{M}\rightarrow\mathcal{E}nd^{!}_{F}(X)$. Let $\sigma_{D}$ denote the canonical section of $\mathcal{O}_{X}(D)$ and consider the map

[TABLE]

induced by $\sigma_{D}^{p-1}$. Then the element

[TABLE]

induced by $\eta_{D}$, is compatible with the support of $D$. In particular, the image of $\eta_{D}$ is contained in $\mathcal{E}nd^{!}_{F}(X,D)$.

Proof.

Notice that $\eta_{D}^{\prime}$ is the composition

[TABLE]

where $\eta^{\prime}$ is the element corresponding to $\eta$. Hence, the restriction of $\eta_{D}^{\prime}$ to $\mathcal{L}\otimes\mathcal{O}_{X}(-D)$ coincides with the map

[TABLE]

But the restriction of $\eta^{\prime}$ to (cf. (1))

[TABLE]

maps by linearity into $\mathcal{O}_{X^{\prime}}(-D^{\prime})$. The in particular part follows by Lemma 3.5. ∎

3.8. Push-forward operation

Assume that $f:X\rightarrow Z$ is a morphism of varieties satisfying that the associated map $f^{\sharp}:\mathcal{O}_{Z}\rightarrow f_{*}\mathcal{O}_{X}$ is an isomorphism. Let $f^{\prime}:X^{\prime}\rightarrow Z^{\prime}$ denote the associated morphism. Then $f^{\prime}_{*}$ induces a morphism

[TABLE]

If $Y\subset X$ is a closed subset then the subsheaf $f^{\prime}_{*}\mathcal{E}nd_{F}(X,Y)$ is mapped to $\mathcal{E}nd_{F}(Z,\overline{f(Y)})$, where $\overline{f(Y)}$ denotes the variety associated to the closure of the image of $Y$. On the level of global sections this means that every Frobenius splitting $s$ of $X$ induces a Frobenius splitting $f^{\prime}_{*}s$ of $Z$ such that when $s$ is compatible with $Y$ then $f^{\prime}_{*}s$ is compatible with $\overline{f(Y)}$. Likewise

Lemma 3.7**.**

With notation as above, let $\mathcal{L}$ denote a line bundle on $Z$ and let $s$ be an element of {\rm End}_{F}^{f^{*}(\mathcal{L})}\big{(}X\big{)}. Then $f^{\prime}_{*}s$ is an element of {\rm End}_{F}^{\mathcal{L}}\big{(}Z\big{)}. Moreover, if $s$