We examine the relationship between the notion of Frobenius splitting and ordinarity for varieties. We show the following: a) The de Rham-Witt cohomology groups $H^i(X, W({\mathcal O}_X))$ of a smooth projective Frobenius split variety are finitely generated over $W(k)$. b) we provide counterexamples to a question of V. B. Mehta that Frobenius split varieties are ordinary or even Hodge-Witt. c) a Kummer $K3$ surface associated to an Abelian surface is $F$-split (ordinary) if and only if the associated Abelian surface is $F$-split (ordinary). d) for a $K3$-surface defined over a number field, there is a set of primes of density one in some finite extension of the base field, over which the surface acquires ordinary reduction. This paper should be read along with first author's paper `Exotic torsion, Frobenius splitting and the slope spectral sequence' which is also available in this archive.
In this note we show that Frobenius split smooth, projective threefolds are Hodge-Witt (using a criterion for the degeneration of the slope spectral sequence of smooth projective threefolds which we prove), and that smooth, projective Frobenius split varieties do not have exotic torsion in their slope spectral sequence. We also record a few simple observations on Frobenius split threefolds. This paper complements authors paper with C. S. Rajan ``On Frobenius splitting and ordinarity'' which is also available on this archive.
Prompted by an exercise in the Brion Kumar book on Frobenius splittings, we show compatibility with a few more subvarieties of the Frobenius splitting from our 1992 paper.
We give several mild conditions on a toric bundle on a nonsingular toric variety under which the projectivization of the toric bundle is Frobenius split.
Extending the notion of Frobenius-splitting, we prove that every finite height Calabi-Yau variety defined over an algebraically closed field of positive characteristic can be lifted to the ring of Witt vectors of length two.
The goal of this paper is to construct a Frobenius splitting on $G/U$ via the Poisson geometry of $(G/U,π_{G/U})$, where $G$ is a semi-simple algebraic group of classical type defined over an algebraically closed field of characteristic $p > 3$, $U$ is the uniradical of a Borel subgroup of $G$ and $π_{G/U}$ is the standard Poisson structure on $G/U$. We first study the Poisson geometry of $(G/U,π_{G/U})$. Then, we develop a general theory for Frobenius splittings on $\mathbb{T}$-Poisson varieties, where $\mathbb{T}$ is an algebraic torus. In particular, we prove that compatibly split subvariet…
Let X be a quasiprojective smooth surface defined over an algebraically closed field of positive characteristic. We show that if X is Frobenius split then so is the Hilbert scheme Hilb^n(X) of n points in X. In particular, we get the higher cohomology vanishing for ample line bundles on Hilb^n(X) when X is projective and Frobenius split.
Following the program of algebraic Frobenius splitting begun by Kumar and Littelmann, we use representation-theoretic techniques to construct a Frobenius splitting of the cotangent bundle of the flag variety of a semisimple algebraic group over an algebraically closed field of positive characteristic. We also show that this splitting is the same as one of the splittings constructed by Kumar, Lauritzen, and Thomsen.
The splitting of the Frobenius direct image of line bundles on toric varieties is used to explicitly construct an orthogonal basis of line bundles in the derived category D^b(X) where X is a Fano toric variety with (almost) maximal Picard number.
We show that the fundamental class in K-homology of a Frobenius split scheme can be computed as a certain alternating sum over irreducible varieties, with the coefficients computed using Möbius inversion on a certain poset. If G/P is a generalized flag manifold and X is an irreducible subvariety homologous to a multiplicity-free union of Schubert varieties, then using a result of Brion we show how to compute the K_0-class [X] in K_0(G/P) from the Chow class in A_*(G/P).
Using a local monomialization result of Knaf and Kuhlmann, we prove that the valuation ring of an Abhyankar valuation of a function field over a perfect ground field of prime characteristic is Frobenius split. We show that a Frobenius splitting of a sufficiently well-behaved center lifts to a Frobenius splitting of the valuation ring. We also investigate properties of valuations centered on arbitrary Noetherian domains of prime characteristic. In contrast to [arXiv:1507.06009], this paper emphasizes the role of centers in controlling Frobenius properties of valuations rings in prime characteri…
Two classical rings of invariants are shown to be Frobenius split: for the special linear group acting on the direct sum of several copies of the defining representation and several copies of the dual of the defining representation; and for the special orthogonal group acting on several copies of the defining representation.
Suppose that $π\: Y \to X$ is a finite map of normal varieties over a perfect field of characteristic $p > 0$. Previous work of the authors gave a criterion for when Frobenius splittings on $X$ (or more generally any $p^{-e}$-linear map) extend to $Y$. In this paper we give an alternate and highly explicit proof of this criterion (checking term by term) when $π$ is tamely ramified in codimension 1. Some additional examples are also explored.
An important breakthrough in understanding the geometry of Schubert varieties was the introduction of the notion of Frobenius split varieties and the result that the flag varieties G/P are Frobenius split. The aim of this article is to give in this case a complete and self contained representation theoretic approach to this method. The geometric Frobenius method in (char k=p>0) will here be replaced by Lusztig's Frobenius maps for quantum groups at roots of unity (which exist not only for primes but any odd integer \ell >1).
We discuss a characteristic free version of Frobenius splittings for toric varieties and give a polyhedral criterion for a toric variety to be diagonally split. We apply this criterion to show that section rings of nef line bundles on diagonally split toric varieties are normally presented and Koszul, and that Schubert varieties are not diagonally split in general.
Recently E. Feigin introduced the $\mathbb G_a^N$-degenerations of semisimple algebraic groups and their associated degenerate flag varieties. It has been shown by Feigin, Finkelberg, and Littelmann that the degenerate flag varieties in types $A_n$ and $C_n$ are Frobenius split. In this paper we construct an associated degeneration of homogeneous coordinate rings of classical flag varieties in all types and show that these rings are Frobenius split in most types. It follows that the degenerate flag varieties of types $A_n$, $C_n$ and $G_2$ are Frobenius split. In particular we obtain an altern…
Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal multiplicity vanishing in Frobenius splitting. In this paper we define the algebraic analogue of this concept and construct a Frobenius splitting vanishing with maximal multiplicity on the diagonal of the full flag variety. Our splitting induces a diagonal Frobenius splitting of maximal multiplicity for a special class of smooth Schubert varieties first considered by Kempf. Consequences are Frobenius splitting of tangent bundles, of blow-ups along the diagonal in flag varieties along with the LMP and Wahl conjectures in …
We explain that the Plücker relations provide the defining equations of the thick flag manifold associated to a Kac-Moody algebra. This naturally transplant the result of Kumar-Mathieu-Schwede about the Frobenius splitting of thin flag manifolds to the thick case. As a consequence, we provide a description of the global sections of line bundles of a thick Schubert variety as conjectured in Kashiwara-Shimozono [Duke Math. J. 148 (2009)]. This also yields the existence of a compatible basis of thick Demazure modules, and the projective normality of the thick Schubert varieties.
Let $G_k$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $\neq 2$. Let $K_k \subset G_k$ be a quasi-split symmetric subgroup of $G_k$ with respect to an involution $θ_k$ of $G_k$. The classification of such involutions is independent of the characteristic of $k$ (provided not $2$). We first construct a closed subgroup scheme $\mathbf{G}^\imath$ of the Chevalley group scheme $\mathbf{G}$ over $\mathbb{Z}$. The pair $(\mathbf{G}, \mathbf{G}^\imath)$ parameterizes symmetric pairs of the given type over any algebraically closed field of characteri…
Let $\Fl^a_\la$ be the PBW degeneration of the flag varieties of type $A_{n-1}$. These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$. We prove that $\Fl^a_\la$ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\la$ of $\Fl^a_\la$. The varieties $R_\la$ can be viewed as towers of successive $\bP^1$-fibrations, thus providing an analogue of the classical Bott-Samelson-Demazure-Hansen desingularization. We prove that the varieties $R_\la$ are Frobenius split. This gives us Frobenius splitting for the…
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, proj…
Let G be a simply connected semisimple algebraic group over an algebraically closed field k of positive characteristic. We will untwist the structure of G-modules by a newly found splitting of the Frobenius endomorphism on the algebra of distributions of G, which, for example, explains the subsumed G-equivariant nature of the Frobenius splitting of the flag variety of G.
Let $G$ denote a connected semisimple and simply connected algebraic group over an algebraically closed field $k$ of positive characteristic and let $g$ denote a regular element of $G$. Let $X$ denote any equivariant embedding of $G$. We prove that the closure of the conjugacy class of $g$ within $X$ is normal and Cohen-Macaulay. Moreover, when $X$ is smooth we prove that this closure is a local complete intersection. As a consequence, the closure of the unipotent variety within $X$ share the same geometric properties.
Let G be a semisimple algebraic group over an algebraically closed field of positive characteristic p. Generalizing the construction of the PBW filtration on Weyl modules for G we construct a G-stable filtration on tensor products of Weyl modules which we call the induced PBW filtration. We use this filtration to give some purely representation-theoretic conditions which are equivalent to the existence of a Frobenius splitting of the double flag variety $G/B \times G/B$ that maximally compatibly splits the diagonal. In particular, this gives a sufficient condition for Wahl's conjecture to hold…
We generalize $F$-signature to pairs $(R,D)$ where $D$ is a Cartier subalgebra on $R$ as defined by the first two authors. In particular, we show the existence and positivity of the $F$-signature for any strongly $F$-regular pair. In one application, we answer an open question of I. Aberbach and F. Enescu by showing that the $F$-splitting ratio of an arbitrary $F$-pure local ring is strictly positive. Furthermore, we derive effective methods for computing the $F$-signature and the $F$-splitting ratio in the spirit of the work of R. Fedder.
We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bund…
We study a splitting of the Frobenius map on the whole algebra of distributions of SL_2 (over a finite field) and its relation with the explicit Frobenius descent on arithmetic D-modules over the projective line
Let $R$ be a ring of prime characteristic $p$, and let $F^e_* R$ denote $R$ viewed as an $R$-module via the $e$th iterated Frobenius map. Given a surjective map $ϕ: F^e_* R \to R$ (for example a Frobenius splitting), we exhibit an algorithm which produces all the $ϕ$-compatible ideals. We also explore a variant of this algorithm under the hypothesis that $ϕ$ is not necessarily a Frobenius splitting (or even surjective). This algorithm, and the original, have been implemented in Macaulay2.
This paper is about sheaf cohomology for varieties (schemes) in characteristic $p>0$. We assume the presence of a Frobenius splitting. (See V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Annals of Math. 122 (1985), 27--40). The main result is that a non-zero higher direct image under a proper map of the ideal sheaf of a compatibly Frobenius split subvariety can not have a support whose inverse image is contained in that subvariety. Earlier vanishing theorems for Frobenius split varieties were based on direct limits and Serre's vanishing theor…
We study the structure of Frobenius splittings (and generalizations thereof) induced on compatible subvarieties $W \subseteq X$. In particular, if the compatible splitting comes from a compatible splitting of a divisor on some birational model $E \subseteq X' \to X$ (ie, this is a log canonical center), then we show that the divisor corresponding to the splitting on $W$ is bounded below by the divisorial part of the different as studied by Kawamata, Shokurov, Ambro and others. We also show that difference between the divisor associated to the splitting and the divisorial part of the different …
Let f be a polynomial of degree n in ZZ[x_1,..,x_n], typically reducible but squarefree. From the hypersurface {f=0} one may construct a number of other subschemes {Y} by extracting prime components, taking intersections, taking unions, and iterating this procedure. We prove that if the number of solutions to f=0 in \FF_p^n is not a multiple of p, then all these intersections in Å^n_{\FF_p} just described are reduced. (If this holds for infinitely many p, then it holds over \QQ as well.) More specifically, there is a_Frobenius splitting_ on Å^n_{\FF_p} compatibly splitting all these subschemes…
We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field $\mathbb K$ of characteristic $\neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind)scheme structure, its projective coordinate ring has a $\mathbb Z$-model, and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of th…
In this article, a sequel to "Global Frobenius Liftability I" (math:1708:03777v2), we continue the development of a comprehensive theory of Frobenius liftings modulo $p^2$. We study compatibility of divisors and closed subschemes with Frobenius liftings, Frobenius liftings of blow-ups, descent under quotients by some group actions, stability under base change, and the properties of associated F-splittings. Consequently, we characterise Frobenius liftable surfaces and Fano threefolds, confirming the conjecture stated in our previous paper.
The purpose of this paper is to describe several applications of finiteness properties of $F$-finite $F$-modules recently discovered by M. Hochster to the study of Frobenius maps on injective hulls, Frobenius near-splittings and to the nature of morphisms of $F$-finite $F$-modules. Among the results in the paper we show that morphisms of $F$-finite $F$-modules have a particularly simple form, and we show that certain Frobenius near-splittings have finitely many compatible submodules, thus generalizing a result of M. Blickle and G. Böckle to the non-$F$-finite case.
Schwede proved very recently in arXiv:0901.1154 that in a quasiprojective scheme X with a fixed Frobenius splitting, there are only finitely many subschemes {Y} that are compatibly split. (A simpler proof has already since been given in arXiv:0901.2098, by Kumar and Mehta.) It follows that their deformations (as compatibly split subschemes) are obstructed. We give a short proof that if X is projective, its compatibly split subschemes {Y} have no deformations at all (again, as compatibly split subschemes). This reproves Schwede's result in some simple cases.
Let G be a semisimple algebraic group over an algebraically closed field of positive characteristic. In this note, we show that an irreducible closed subvariety of the flag variety of G is compatibly split by the unique canonical Frobenius splitting if and only if it is a Richardson variety, i.e. an intersection of a Schubert and an opposite Schubert variety.
Classical Serre-Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips its local moduli space with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo $p^2$ of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Fr…
Let $X$ be a projective Frobenius split variety over an algebraically closed field with splitting $θ: F_* Ø_X \to Ø_X$. In this paper we give a sharp bound on the number of subvarieties of $X$ compatibly split by $θ$. In particular, suppose $\sL$ is a sufficiently ample line bundle on $X$ (for example, if $\sL$ induces a projectively normal embedding) with $n = \dim H^0(X, \sL)$. We show that the number of $d$-dimensional irreducible subvarieties of $X$ that are compatibly split by $θ$ is less than or equal to ${n \choose d+1}$. This generalizes a well known result on the number of closed poin…
Let k be an algebraically closed field of characteristic p>2. By a result of Kumar and Thomsen, the standard Frobenius splitting of the affine plane induces a Frobenius splitting of the Hilbert scheme of n points in the plane. In this thesis, we investigate the question, "what is the stratification of the Hilbert scheme of points in the plane by all compatibly Frobenius split subvarieties?" We provide the answer to this question when n is at most 4 and we give a conjectural answer when n=5. We prove that this conjectural answer is correct up to the possible inclusion of one particular one-dime…
Let $X$ be a smooth projective variety over a perfect field $k$ of characteristic $p>0$, and $V$ be a vector bundle over $X$. It is well known that if $X$ is a curve and $V$ is not strongly semistable, then some Frobenius pullback $(F^t)^*V$ is a direct sum of strongly semistable bundles. A natural question to ask is whether this still holds in higher dimension. Indranil Biswas, Yogish I. Holla, A.J. Parameswaran, and S. Subramanian showed that there is always a counterexample to this over any algebraically closed field of positive characteristic which is uncountable. However, we will produce …
Using the theory of spherical varieties and especially Frobenius splitting results for symmetric varieties, we give a type independent very short proof of Wahl's conjecture for cominuscule homogeneous spaces for all primes different from 2.
Mehta and van der Kallen put a Frobenius splitting on the type A cotangent bundle $T^* GL_n/B$, thereby defining a stratification by compatibly split subvarieties, and they determined a few of the elements of this stratification. We embed $T^* GL_n/B$ as a stratum in a larger stratified (and Frobenius split) space $GL_n/B \times Mat_n$ whose stratification we determine, thereby giving a full description of the one of Mehta-van der Kallen. The main technique is to endow $GL_n/B \times Mat_n$ with a Bruhat atlas, covering it with open sets that are stratified-isomorphic to Bruhat cells (in $GL_{…
Using reduction to positive characteristic and the method of Frobenius splitting of diagonals, due to Mehta and Ramanathan, it is shown that homogeneous coordinate rings for either proper and smooth toric varieties or Schubert varieties are Koszul algebras.
Let G be a connected reductive group and X an equivariant compactifiction of G. In X, we study generalised and opposite generalised Schubert varieties, their intersections called generalised Richardson varieties and projected generalised Richardson varieties. Any complete G-embedding has a canonical Frobenius splitting and we prove that the compatibly split subvarieties are the generalised projected Richardson varieties extending a result of Knutson, Lam and Speyer to the situation.
Building on the results of Deligne and Illusie on liftings to truncated Witt vectors, we give a criterion for non-liftability that involves only the dimension of certain cohomology groups of vector bundles arising from the Frobenius pushforward of the de Rham complex. Using vector bundle methods, we apply this to show that exceptional Enriques surfaces, a class introduced by Ekedahl and Shepherd-Barron, do not lift to truncated Witt vectors, yet the base of the miniversal formal deformation over the Witt vectors is regular. Using the classification of Bombieri and Mumford, we also show that bi…
