We study the projective geometry of homogeneous varieties $X= G/P\subset P(V)$, where $G$ is a complex simple Lie group, $P$ is a maximal parabolic subgroup and $V$ is the minimal $G$-module associated to $P$. Our study began with the observation that Freudenthal's magic chart could be derived from Zak's theorem on Severi varieties and standard geometric constructions. Our attempt to understand this observation led us to discover further connections between projective geometry and representation theory.
Among other things, we calculate the variety of tangent directions to lines on $X$ through a point and determine unirulings of $X$. We show this variety is a Hermitian symmetric space if and only if $P$ does not correspond to a short root. We describe the spaces corresponding to the exceptional short roots and their unirulings using the octonions. Further calculations, in the case $X$ is a Hermitian symmetric space, give rise to a strict prolongation property and the appearance of secant varieties at the infinitesimal level. Our work complements and advances that of Freudenthal and Tits, who studied homogeneous varieties in an abstract/axiomatic setting.
Let $\cP=G/P$ be a homogeneous projective variety with $G$ a reductive group and $P$ a parabolic subgroup. In positive characteristic we exhibit for $G$ of low rank a Karoubian complete strongly exceptional poset of locally free sheaves appearing in the Frobenius direct image of the structure sheaf of $G/P$. These sheaves are all defined over $\bbZ$, so by base change provide a Karoubian complete strongly exceptional poset on $\cP$ over $\bbC$, adding to the list of classical results by Beilinson and Kapranov on the Grassmannians and the quadrics over $\bbC$.
We study global deformations of certain projective bundles over projective spaces. We show that any projective global deformation of a projective bundle over $\bP^1$ carries the structure of a projective bundle over some projective space. Furthermore, we construct examples in arbitrary dimension $\ge 3$ of almost homogeneous Fano projective bundles over $\bP^2$ which can be globally deformed to non-almost homogeneous manifolds.
We give a classification of embedded smooth projective varieties swept out by rational homogeneous varieties whose Picard number and codimension are one.
The purpose of this article is to introduce projective geometry over composition algebras : the equivalent of projective spaces and Grassmannians over them are defined. It will follow from this definition that the projective spaces are in correspondance with Jordan algebras and that the points of a projective space correspond to rank one matrices in the Jordan algebra. A second part thus studies properties of rank one matrices. Finally, subvarieties of projective spaces are discussed.
Answering a problem raised by Lazarsfeld, Hwang and Mok proved that a surjective holomorphic map from a rational homogeneous space of Picard number 1 onto projective manifold different from projective space must be a biholomorphism. THe aim of this paper is to generalized this result to irreducible rational homogeneous space of higher Picard number.
In this article we prove a result comparing rationality of algebraic cycles over the function field of a projective homogeneous variety under a linear algebraic group of type $F_4$ or $E_8$ and over the base field, which can be of any characteristic.
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.
Let H be a closed subgroup of a linear algebraic group G defined over a field F. There is an equivalence of categories between the category of linear finite-dimensional representations of H and the category of finite rank G-homogeneous vectorbundles on G/H. In this short note we study this correspondence for the sheaves of principal parts (=jetbundles) of homogeneous linebundles on projective space in characteristic zero. We describe the representation corresponding to the principal parts of a homogeneous linebundle and as a corollary we obtain known formulas on the splitting type of the princ…
In this addendum we generalize some results of our article "Generically split projective homogeneous varieties", Duke Math. J. 152 (2010), no. 1, 155-173. More precisely, we remove all restrictions on the characteristic of the base field and complete our classification by the last missing case, namely $\PGO_{2n}^+$.
Let $G$ be a semi-simple algebraic group over a perfect field $k$. A lot of progress has been made recently in computing the Chow motives of projective $G$-homogenous varieties. When $k$ has positive characteristic, a broader class of $G$-homogeneous varieties appear. These are varieties over which $G$ acts transitively with possibly non-reduced isotropy subgroup. In this paper we study these varieties which we call ${\it \mbox{projective pseudo-homogeneous varieties}}$ for $G$ inner type over $k$ and prove that their motives satisfy Rost nilpotence. We also find their motivic decompositions a…
The space of smooth rational curves of degree $d$ in a projective variety $X$ has compactifications by taking closures in the Hilbert scheme, the moduli space of stable sheaves or the moduli space of stable maps respectively. In this paper we compare these compactifications by explicit blow-ups and -downs when $X$ is a projective homogeneous variety and $d\leq 3$. Using the comparison result, we calculate the Betti numbers of the compactifications when $X$ is a Grassmannian variety.
We prove that under some assumptions on an algebraic group $G$, indecomposable direct summands of the motive of a projective $G$-homogeneous variety with coefficients in $\mathbb{F}_p$ remain indecomposable if the ring of coefficients is any field of characteristic $p$. In particular for any projective $G$-homogeneous variety $X$, the decomposition of the motive of $X$ in a direct sum of indecomposable motives with coefficients in any finite field of characteristic $p$ corresponds to the decomposition of the motive of $X$ with coefficients in $\mathbb{F}_p$. We also construct a counterexample …
We give a complete classification of anisotropic projective homogeneous varieties of dimension less than 6 up to motivic isomorphism. We give several criteria for anisotropic flag varieties of type A_n to have isomorphic motives.
Let $\mathbb X\subset\mathbb P(V)$ be a projective variety, which is not contained in a hyperplane. Then every vector $v$ in $V$ can be written as a sum of vectors from the affine cone $X$ over $\mathbb X$. The minimal number of summands in such a sum is called the rank of $v$. The set of vectors of rank $r$ is denoted by $X_r$ and its projective image by $\mathbb X_r$. The r-th secant variety of $X$ is defined $σ_r(\mathbb X):=\bar{\sqcup_{s\le r}\mathbb X_s}$; it is called tame if $σ_r(\mathbb X)=\sqcup_{s\le r} \mathbb X_s$ and wild if the closure contains elements of higher rank. In this p…
In this paper we construct a tilting sheaf for Severi-Brauer Varieties and Involution Varieties. This sheaf relates the derived category of each variety to the derived category of modules over a ring whose semisimple component consists of the Tits algebras of the corresponding linear algebraic group.
Assume that R is a local regular ring containing an infinite perfect field, or that R is the local ring of a point on a smooth scheme over an infinite field. Let K be the field of fractions of R and the characteristic of K is not 2. Let X be an exceptional projective homogeneous scheme over R. We prove that in most cases the condition that X has a K-point implies that X has an R-point.
Let A be a central division algebra over a field F with ind A = n. In computing canonical p-dimension of projective PGL_1(A)-homogeneous varieties, for p prime, we can reduce to the case of generalized Severi-Brauer varieties X_e(A) with ind A a power of p divisible by e. We prove that canonical 2-dimension (and hence canonical dimension) equals dimension for all X_e(A) with ind A = 2e a power of 2.
We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential varieties of homogeneous varieties already constructed. Our algorithms make no reference to root systems. Our proofs use properties of root systems, but not their classification.
We consider those projective bundles (or Brauer-Severi varieties) over an abelian variety that are homogeneous, i.e., invariant under translation. We describe the structure of these bundles in terms of projective representations of commutative algebraic groups; the irreducible bundles correspond to Heisenberg groups and their standard representations. Our results extend those of Mukai on semi-homogeneous vector bundles, and yield a geometric view of the Brauer group of abelian varieties.
We study the projective behavior, mainly with respect to osculating spaces and secant varieties, of Lagrangian Grassmannians and Spinor varieties. We prove that these varieties have osculating dimension smaller than expected. Furthermore, we give numerical conditions ensuring the non secant defectivity of Lagrangian Grassmannians in their Plücker embedding and of Spinor varieties in both their Plücker and Spinor embeddings.
Reason for withdrawal: There is a serious mistake in the calculation of the divisor of the rational section used in the proof of Prop. 2.2.1., and with the correct value the argument does not work.
In the present paper, we provide general bounds for the torsion in the codimension 2 Chow groups of the products of projective homogeneous surfaces. In particular, we determine the torsion for the product of four Pfister quadric surfaces and the maximal torsion for the product of three Severi-Brauer surfaces. We also find an upper bound for the torsion of the product of three quadric surfaces with the same discriminant.
In the present paper we provide a general algorithm to compute multiplicative cohomological operations on algebraic oriented cohomology of projective homogeneous G-varieties, where G is a split reductive algebraic group over a field of characteristic 0. More precisely, we extend such operations to the respective T-equivariant (T is a maximal split torus of G) oriented theories, and then compute them using equivariant Schubert calculus techniques. This generalizes an approach suggested by Garibaldi-Petrov-Semenov for Steenrod operations. We also show that operations on the theories of additive …
We give bounds on the order of torsion in the Chow group of zero dimensional cycles for isotropic Grassmannians and Brauer-Severi flag varieties. To do this, we introduce tools to understand the behavior of torsion in Chow groups with coefficients under morphisms of proper varieties
I. Panin proved in the nineties that the algebraic K-theory of twisted projective homogeneous varieties can be expressed in terms of central simple algebras. Later, Merkurjev and Panin described the algebraic K-theory of toric varieties as a direct summand of the algebraic K-theory of separable algebras. In this article, making use of the recent theory of noncommutative motives, we extend Panin and Merkurjev-Panin computations from algebraic K-theory to every additive invariant. As a first application, we fully compute the cyclic homology (and all its variants) of twisted projective homogeneou…
We will consider an explicit birational map between a quadric and the projective variety X(J) of traceless rank one elements in a simple reduced Jordan algebra J. X(J) is a homogeneous G-variety for the automorphism group G=Aut(J). We will show that the birational map is a blow up followed by a blow down. This will allow us to use the blow up formula for motives together with Vishik's work on the motives of quadrics to give a motivic decomposition of X(J).
By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface, a projective homogeneous space admits a rational point if the elementary obstruction vanishes.
We give an elementary introduction to our papers relating the geometry of rational homogeneous varieties to representation theory. We also describe related work and recent progress.
We study open equivariant projective embeddings of homogeneous spaces such that the complement of the open orbit does not contain divisors. Criterions of existence of such an embedding are considered and finiteness of isomorphism classes of embeddings for a given homogeneous space is proved. Any embedding with small boundary is realized as a GIT-quotient associated with a linearization of the trivial line bundle on the space of the canonical embedding. The generalized Cox's construction and the theory of bunched rings allow us to describe basic geometric properties of embeddings with small bou…
We explore connections between birational anabelian geometry and abstract projective geometry. One of the applications is a proof of a version of the birational section conjecture.
We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G of rank 2 over a field. We exhibit exceptional collections of the expected length for types A_2 and B_2=C_2 and prove that no such collection exists for type G_2. This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G-varieties for split linear algebraic groups G of rank at most 2.
Let $f(\bf z,\bar{\bf z})$ be a strongly mixed homogeneous polynomial of 3 variables $\bf z=(z_1,z_2,z_3)$ of polar degree $q$ with an isolated singularity at the origin. It defines a smooth Riemann surface $C$ in the complex projective space $\mathbb P^2$. The fundamental group of the complement $\mathbb P^2\setminus C$ is cyclic group of order $q$ if $f$ is homogeneous polynomial without $\bar{\bf z}$. We propose a conjecture that this may be even true for mixed homogeneous polynomials by giving several supporting examples.
We consider possibly singular rational projective k*-surfaces and provide an explicit description of the unit component of the automorphism group in terms of isotropy group orders and intersection numbers of suitable invariant curves. As an application, we characterize the almost homogeneous rational projective k*-surfaces and we specify the two-dimensional groups acting almost transitively as well as the corresponding homogeneous spaces.
Let $M_1$ and $M_2$ be two Kähler manifolds. One says that $M_1$ and $M_2$ are "relatives" if they share a non-trivial Kähler submanifold $S$, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) $h_1: S -> M_1$ and $h_2: S -> M_2$. In this paper we show that a bounded homogeneous domain with a homogeneous Kähler metric and a projective Kähler manifold (i.e. a projective manifold endowed with the restriction of the Fubini-Study metric) are not relatives. Our result is a generalization of the result obtained by A. J. Di Scala and A. Loi (in "Kähler manifolds and t…
Grothendieck-Chow motives of quadric hypersurfaces have provided many insights into the theory of quadratic forms. Subsequently, the landscape of motives of more general projective homogeneous varieties has begun to emerge. In particular, there have been many results which relate the motive of a one homogeneous variety to motives of other simpler or smaller ones. In this paper, we exhibit a relationship between motives of two homogeneous varieties by producing a natural rational map between them which becomes a projective bundle morphism after being resolved. This allows one to use formulas fo…
We consider the construction of Frobenius manifolds associated to projective special geometry and analyse the dependence on choices involved. In particular, we prove that the underlying F-manifold is canonical. We then apply this construction to integrable systems of Hitchin type.
We study locally free sheaves of rank two on the projective line over the integers, especially indecomposable ones. Subsequently we apply various concepts of Arakelov geometry to these sheaves. We compute for example the arithmetic Chern classes and use the arithmetic Riemann-Roch theorem.
We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial.
Over a field of positive characteristic, a semisimple algebraic group $G$ may have some nonreduced parabolic subgroup $P$. In this paper, we study the Schubert and Bott-Samelson-Demazure-Hansen (BSDH) varieties of $G/P$, with $P$ nonreduced, when the base field is perfect. It is shown that in general the Schubert and BSDH varieties of such a $G/P$ are not normal, and the projection of the BSDH variety onto the Schubert variety has nonreduced fibers at closed points. When the base field is finite, the generalized convolution morphisms between BSDH varieties are also studied. It is shown that th…
We show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.
We classify minimal-degree curves in the Hilbert schemes of points on algebraic surfaces. When the algebraic surface is the projective plane, the nef cone and a flip structure of these Hilbert schemes are determined.
Automorphism groups and Homogeneous fibrations on projective varieties with nef anticanonical divisors
Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）；Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University（西交利物浦大学数学科学系）
The Jacobian ideal provides the set of infinitesimally trivial deformations for a homogeneous polynomial, or for the corresponding complex projective hypersurface. In this article, we investigate whether the associated linear deformation is indeed trivial, and show that the answer is no in a general situation. We also give a characterization of tangentially smoothable hypersurfaces with isolated singularities. Our results have applications in the local study of variations of projective hypersurfaces, complementing the global versions given by J. Carlson and P. Griffiths, R. Donagi and the auth…
We describe a close relation between wall crossings in the birational geometry of moduli space of Gieseker stable sheaves $M_H(v)$ on $\bb{P}^2$ and mini-wall crossings in the stability manifold $Stab(D^b(\bb{P}^2))$.
This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.
Consider a finite dimensional (generally reducible) polynomial representation ρof GL_n. A projective compactification of GL_n is the closure of ρ(GL_n) in the space of all operators defined up to a factor (this class of spaces can be characterized as equivariant projective normal compactifications of GL_n). We give an expicit description for all projective compactifications. We also construct explicitly (in elementary geometrical terms) a universal object for all projective compactifications of GL_n.
In this article, we will introduce methods of non-standard analysis into projective geometry. Especially, we will analyze the properties of a projective space over a non-Archimedean field. Non-Archimedean fields contain numbers that are smaller than every real number: the so called "infinitesimal" numbers. The theory is well known from non-standard analysis. This enables us to define projective objects that deviate only infinitesimally (in an appropriate metric) from each other. And we show that in most cases operations involving such objects that deviate infinitesimally also experience only i…
These are lecture notes on the rigidity of submanifolds of projective space "resembling" compact Hermitian symmetric spaces in their homogeneous embeddings. Recent results are surveyed, along with their classical predecessors. The notes include an introduction to moving frames in projective geometry, an exposition of the Hwang-Yamaguchi ridgidity theorem and a new variant of the Hwang-Yamaguchi theorem.
Any set of $σ$-Hermitian matrices of size $n \times n$ over a field with involution $σ$ gives rise to a projective line in the sense of ring geometry and a projective space in the sense of matrix geometry. It is shown that the two concepts are based upon the same set of points, up to some notational differences.
To understand the structure of an algebraic variety we often embed it in various projective spaces. This develops the notion of projective geometry which has been an invaluable tool in algebraic geometry. We develop a perfectoid analog of projective geometry, and explore how equipping a perfectoid space with a map to a certain analog of projective space can be a powerful tool to understand its geometric and arithmetic structure. In particular, we show that maps from a perfectoid space $X$ to the perfectoid analog of projective space correspond to line bundles on $X$ together with some extra da…
In the first part of the thesis, we study a classical invariant of projective varieties, the secant defectivity. The second part is devoted to modern algebraic geometry, we study the birational geometry of blow-ups of Grassmannians at points.
