Motivated by Demailly's strategy towards the Kobayashi hyperbolicity conjecture, we study the action on the k-jets of germs of holomorphic discs in a complex manifold X of the reparametrization group of k-jets of germs of biholomorphisms of the source. This reparametrization group is a subgroup of the general linear group GL(k) which is not reductive, but nonetheless we show that its invariants for any linear action which extends to GL(k) form a finitely generated algebra, and give a new geometric description of the Demailly-Semple algebra of invariant jet differentials.
In this article we briefly discuss the finite generation of fiber rings of invariant k-jets of holomorphic curves in a complex projective manifold, using differential Galois theory.
We give explicit estimates for the volume of the Green-Griffiths jet differentials of any order on a toroidal compactification of a ball quotient. To this end, we first determine the growth of the logarithmic Green-Griffiths jet differentials on these objects, using a natural deformation of the logarithmic jet space of a given order, to a suitable weighted projective bundle. Then, we estimate the growth of the vanishing conditions that a logarithmic jet differential must satisfy over the boundary to be a standard one.
In this paper, we collect the fundamental basic properties of jet modules in algebraic geometry and related properties of differential operators. We claim no originality but we want to provide a reference work for own research and the research of other people.
This paper shows how properties of jet schemes relate to those of the singularity on the base scheme. We will see that the jet scheme's properties of being Q-factorial, Q-Gorenstein, canonical, terminal and so on are inherited by the base scheme.
We formulate and prove the existence of an asymptotic duality along the fibers of the Green-Griffiths jet bundles over projective manifolds. The existence of global sections for these bundles and also for their dual sheaves has been established by the author and J. P. Demailly in a previous article, [3].
We provide detailed holomorphic Morse estimates for the cohomology of sheaves of jet differentials and their dual sheaves. These estimates apply on arbitrary directed varieties, and a special attention has been given to the analysis of the singular situation. As a consequence, we obtain existence results for global jet differentials and global differential operators under positivity conditions for the canonical or anticanonical sheaf of the directedstructure.
The main goal of this paper is to prove the Schmidt--Kolchin conjecture. This conjecture says the following: the vector space of degree \(d\) differentially homogeneous polynomials in \((N+1)\) variables is of dimension \((N+1)^{d}\). Next, we establish a one-to-one correspondance between differentially homogeneous polynomials in \((N+1)\) variables, and twisted jet differentials on projective spaces. As a by-product of our study of differentially homogeneous polynomials, we are thus able to understand explicitly twisted jet differentials on projective spaces.
We give a fully algebraic proof of an important theorem of Demailly, stating the existence of many Green-Griffiths jet differentials on a complex projective manifold of general type. To this end, we introduce a new algebraic version of the Morse inequalities, which we then use in our proof as an algebraic counterpart to Demailly's and Bonavero's holomorphic Morse inequalities.
We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle TX. In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by D and multiplicities are bounded by the ramification indices of the components of D. We estimate precisely the curvature tensor of the co…
Let $X\subset\mathbb P^{n+1}$ be a smooth complex projective hypersurface. In this paper we show that, if the degree of $X$ is large enough, then there exist global sections of the bundle of invariant jet differentials of order $n$ on $X$, vanishing on an ample divisor. We also prove a logarithmic version, effective in low dimension, for the log-pair $(\mathbb P^n,D)$, where $D$ is a smooth irreducible divisor of high degree. Moreover, these result are sharp, \emph{i.e.} one cannot have such jet differentials of order less than $n$.
By implementing jet differential techniques in non-archimedean geometry, we obtain a big Picard type extension theorem, which generalizes a previous result of Cherry and Ru. As applications, we establish two hyperbolicity-related results. Firstly, we prove a non-archimedean Ax-Lindemann theorem for totally degenerate abelian varieties. Secondly, we show the pseudo-Borel hyperbolicity for subvarieties of general type in abelian varieties.
This note is intended to provide a general reference for jet spaces and jet differentials, valid in maximal generality (at the level of EGA). The approach is rather concrete, using Hasse-Schmidt (divided) higher differentials. Discussion of projectivized jet spaces (as in Green and Griffiths (1980)) is included.
On a generic complete intersection surface X^2 in P^4(C) having polynomial equations z^d = R(x,y) and t^e = S(x,y) with 752 <= d <= e <= d^2/648, there exist extrinsic meromorphic jet differentials of the form J(x,y,x',y') / [y^d z^{m(d-1)} t^{m(e-1)}] where J(x,y,x',y') = sum_{j+k+p+q=m} A_{j,k,p,q}(x,y) (x')^j (y')^k (R')^p (S')^q (R)^{m-p} (S)^{m-q} with the complex coefficients of the polynomials A_{j,k,p,q}(x,y) satisfying a certain system of linear equations depending explicitly on R, S, the restriction to X^2 of which provides nonzero intrinsic global holomorphic sections of the bundle …
This work provides a unified formalism for studying difference and (Hasse-) differential algebraic geometry, by introducing a theory of "iterative Hasse rings and schemes". As an application, Hasse jet spaces are constructed generally, allowing the development of the theory for arbitrary systems of algebraic partial difference/differential equations, where constructions by earlier authors applied only to the finite dimensional case. In particular, it is shown that under appropriate separability assumptions a Hasse variety is determined by its jet spaces at a point.
We develop the theory of higher prolongations of algebraic varieties over fields in arbitrary characteristic with commuting Hasse-Schmidt derivations. Prolongations were introduced by Buium in the context of fields of characteristic 0 with a single derivation. Inspired by work of Vojta, we give a new construction of higher prolongations in a more general context. Generalizing a result of Buium in characteristic 0, we prove that these prolongations are represented by a certain functor, which shows that they can be viewed as `twisted jet spaces.' We give a new proof of a theorem of Moosa, Pillay…
Demailly-Semple jets are studied using the invariant theory of non reductive groups. The geometric characterization of the 3-jets bundle in dimension 3 is given and provides a Riemann-Roch computation.
Using Buium's theory of arithmetic differential characters, we construct a filtered $F$-isocrystal ${\bf H}(A)_K$ associated to an abelian scheme $A$ over a $p$-adically complete discrete valuation ring with perfect residue field. As a filtered vector space, ${\bf H}(A)_K$ admits a natural map to the usual de Rham cohomology of $A$, but the Frobenius operator comes from arithmetic differential theory and is not the same as the usual crystalline one. When $A$ is an elliptic curve, we show that ${\bf H}(A)_K$ has a natural integral model ${\bf H}(A)$, which implies an integral refinement of a re…
