Irreducible Variety Research Articles

A GOING DOWN THEOREM FOR GROTHENDIECK CHOW MOTIVES

Let X be a geometrically split, geometrically irreducible variety over a field F satisfying Rost nilpotence principle. Consider a field extension E/F and a finite field K. We provide in this note a motivic tool giving sufficient conditions for so-called outer motives of direct summands of the Chow motive of X_E with coefficients in K to be lifted to the base field. This going down result has been used S. Garibaldi, V. Petrov and N. Semenov to give a complete classification of the motivic decompositions of projective homogeneous varieties of inner type E_6 and to answer a conjecture of Rost and Springer.

Chow form for projective differential variety

In this paper, a generic intersection theorem in projective differential algebraic geometry is given. Then, the Chow form for an irreducible projective differential variety is defined and basic properties of the differential Chow form in affine differential case are shown to be valid for its projective differential counterpart. Finally, we apply the differential Chow form to a result of linear dependence over projective varieties given by Kolchin.

The Chow ring of double EPW sextics

A conjecture of Beauville and Voisin states that for an irreducible symplectic variety X, any polynomial relation between classes of divisors and the Chern classes of X which holds in cohomology already holds in the Chow groups. We verify the conjecture for a very general double EPW sextic.

Quiver Grassmannians and degenerate flag varieties

Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the second named author. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and an injective representation of a Dynkin quiver. It is proven that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits, and a cellular decomposition. For type A quivers explicit formulas for the Euler characteristic (the median Genocchi numbers) and the Poincare polynomials are derived.

A simple remark on fields of definition

Let K< L be an extension of fields, in characteristic zero, with L algebraically closed and let ‾K;< L be the algebraic closure of K in L. Let X and Y be irreducible projective algebraic varieties, X defined over ‾K and Y defined over L, and let π : X → Y be a non-constant morphism, defined over L. If we assume that ‾K ≠ L,then one may wonder if Y is definable over ‾K. In the case that K = Q, L = C and that X and Y are smooth curves, a positive answer was obtained by Gonzalez-Diez. In this short note we provide simple conditions to have a positive answer to the above question. We also state a conjecture for a class of varieties of general type.

Special subvarieties of EPW sextics

We study the geometry of EPW sextics in order to produce special subvarieties. In particular we exhibit a (singular) Enriques surface and we compute its class in the Chow ring of the sextic. In order to do this, we produce an explicit degeneration of double EPW sextics to a Hilbert scheme of two points on a quartic surface, both in the smooth and in the singular case (keeping the singularities). The fixed locus of the covering involution on the double EPW sextic degenerates to the surface of bitangents to the quartic, which can be shown to be birational to an Enriques surface, provided the quartic acquires enough nodes. This construction is used in Ferretti (Algebra Number Theory, 2009a) as a starting point to prove a conjecture of Beauville and Voisin on the Chow ring of irreducible symplectic varieties, in the particular case of a very general double EPW sextic.

A proof of the set-theoretic version of the salmon conjecture

We show that the irreducible variety of 4×4×4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 (the Strassen commutative conditions), of degree 6 (the Landsberg–Manivel polynomials), and of degree 9 (the symmetrization conditions).

Asymptotic purity for very general hypersurfaces of ℙn × ℙn of bidegree (k, k)

Abstract For a complex irreducible projective variety, the volume function and the higher asymptotic cohomological functions have proven to be useful in understanding the positivity of divisors as well as other geometric properties of the variety. In this paper, we study the vanishing properties of these functions on hypersurfaces of ℙn × ℙn. In particular, we show that very general hypersurfaces of bidegree (k, k) obey a very strong vanishing property, which we define as asymptotic purity: at most one asymptotic cohomological function is nonzero for each divisor. This provides evidence for the truth of a conjecture of Bogomolov and also suggests some general conditions for asymptotic purity.

Disc functionals and Siciak–Zaharyuta extremal functions on singular varieties

We establish plurisubharmonicity of envelopes of certain classical disc functionals on locally irreducible complex spaces, thereby generalizing the corresponding results for complex manifolds. We also find new formulae expressing the Siciak-Zaharyuta extremal function of an open set in a locally irreducible affine algebraic variety as the envelope of certain disc functionals, similarly to what has been done for open sets in C^n by Lempert and by Larusson and Sigurdsson.

Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity

The prolongation $\mathfrak{g}^{(k)}$ of a linear Lie algebra $\mathfrak{g}\subset \mathfrak{gl}(V)$ plays an important role in the study of symmetries of G-structures. Cartan and Kobayashi-Nagano have given a complete classification of irreducible linear Lie algebras $\mathfrak{g}\subset \mathfrak{gl}(V)$ with non-zero prolongations. If $\mathfrak{g}$ is the Lie algebra $\mathfrak{aut}(\hat{S})$ of infinitesimal linear automorphisms of a projective variety S⊂ℙV, its prolongation $\mathfrak{g}^{(k)}$ is related to the symmetries of cone structures, an important example of which is the variety of minimal rational tangents in the study of uniruled projective manifolds. From this perspective, understanding the prolongation $\mathfrak{aut}(\hat{S})^{(k)}$ is useful in questions related to the automorphism groups of uniruled projective manifolds. Our main result is a complete classification of irreducible non-degenerate nonsingular variety S⊂ℙV with $\mathfrak{aut}(\hat {S})^{(k)}\neq0$ , which can be viewed as a generalization of the result of Cartan and Kobayashi-Nagano. As an application, we show that when S is linearly normal and Sec (S)≠ℙV, the blow-up Bl S (ℙV) has the target rigidity property, i.e., any deformation of a surjective morphism f:Y→Bl S (ℙV) comes from the automorphisms of Bl S (ℙV).

Ordinary varieties and the comparison between multiplier ideals and test ideals

AbstractWe consider the following conjecture: if X is a smooth and irreducible n-dimensional projective variety over a field k of characteristic zero, then there is a dense set of reductions Xs to positive characteristic such that the action of the Frobenius morphism on Hn(Xs, OXs) is bijective. There is another conjecture relating certain invariants of singularities in characteristic zero (the multiplier ideals) with invariants in positive characteristic (the test ideals). We prove that the former conjecture implies the latter one in the case of ambient nonsingular varieties.

Principal Bundles Over Finite Fields

Let M be an irreducible smooth projective variety defined over . Let ϖ(M, x 0) be the fundamental group scheme of M with respect to a base point x 0. Let G be a connected semisimple linear algebraic group over . Fix a parabolic subgroup P ⊊ G, and also fix a strictly antidominant character χ of P. Let E G → M be a principal G-bundle such that the associated line bundle E G (χ) → E G /P is numerically effective. We prove that E G is given by a homomorphism ϖ(M, x 0) → G. As a consequence, there is no principal G-bundle E G → M such that degree(ϕ*E G (χ)) > 0 for every pair (Y, ϕ), where Y is an irreducible smooth projective curve, and ϕ: Y → E G /P is a nonconstant morphism.

ALGEBRO-GEOMETRIC INVARIANTS OF GROUPS (THE DIMENSION SEQUENCE OF A REPRESENTATION VARIETY)

If G is a finitely generated group and A is an algebraic group, then RA(G) = Hom (G, A) is an algebraic variety. Define the "dimension sequence" of G over A as Pd(RA(G)) = (Nd(RA(G)), …, N0(RA(G))), where Ni(RA(G)) is the number of irreducible components of RA(G) of dimension i (0 ≤ i ≤ d) and d = Dim (RA(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofFn = 〈x1, …, xn〉, and letG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety andV-1 = {ρ | ρ ∈ RSL(2, ℂ)(Fn), ρ(w) = -I} ≠ ∅, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). Theorem B.Let w be a nontrivial word in the free group on{x1, …, xn}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). We also show that if G = 〈x1, . ., xn, y; W = yp〉, where p ≥ 1 and W is a word in Fn = 〈x1, …, xn〉, and A = PSL(2, ℂ), then Dim (RA(G)) = Max {3n, Dim (RA(G′)) +2 } ≤ 3n + 1 for G′ = 〈x1, …, xn; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; xp = yt〉 then Pd(RSL(2, ℂ)(G))≠Pd(RPSL(2, ℂ)(G)).

ON THE ALGEBRAIC HOLONOMY OF STABLE PRINCIPAL BUNDLES

Let EG be a stable principal G–bundle over a compact connected Kähler manifold, where G is a connected reductive linear algebraic group defined over ℂ. Let H ⊂ G be a complex reductive subgroup which is not necessarily connected, and let EH ⊂ EG be a holomorphic reduction of structure group to H. We prove that EH is preserved by the Einstein–Hermitian connection on EG. Using this we show that if EH is a minimal reductive reduction (which means that there is no complex reductive proper subgroup of H to which EH admits a holomorphic reduction of structure group), then EH is unique in the following sense: For any other minimal reduction of structure group (H′, EH′) of EG to some reductive subgroup H′, there is some element g ∈ G such that H′ = g-1Hg and EH′ = EHg. As an application, we show the following: Let M be a simply connected, irreducible smooth complex projective variety of dimension n such that the Picard number of M is one. If the canonical line bundle KM is ample, then the algebraic holonomy of the holomorphic tangent bundle T1, 0M is GL (n, ℂ). If [Formula: see text] is ample, the rank of the Picard group of M is one, the biholomorphic automorphism group of M is finite, and M admits a Kähler–Einstein metric, then the algebraic holonomy of T1, 0M is GL (n, ℂ). These answer some questions posed in V. Balaji and J. Kollár, Publ. Res. Inst. Math. Sci.44 (2008) 183–211.

Singularities of duals of Grassmannians

Let X ⊂ P N be a smooth irreducible nondegenerate projective variety and let X ⁎ ⊂ P N denote its dual variety. The locus of bitangent hyperplanes, i.e. hyperplanes tangent to at least two points of X, is a component of the singular locus of X ⁎ . In this paper we provide a sufficient condition for this component to be of maximal dimension and show how it can be used to determine which dual varieties of Grassmannians are normal. That last part may be compared to what has been done for hyperdeterminants by J. Weyman and A. Zelevinsky (1996) in [23].

On the André motive of certain irreducible symplectic varieties

We show that if Y is an algebraic deformation of the Hilbert scheme of points on a K3 surface, then the André motive of Y is an object of the category generated be the motive of Y truncated in degree 2.

Wronskian of derivations

An associative multilinear polynomial depending on 16 variables and being skew-symmetric with respect to 12 of them is presented. This polynomial provides us with a mapping recovering the algebra of regular functions of an irreducible affine variety from any smooth involutive distribution of dimension 2.

Wild automorphisms of varieties with Kodaira dimension 0

An automorphism σ of a projective variety X is said to be wild if σ(Y) ≠ Y for every non-empty subvariety \({Y \subsetneq X}\) . In [1] Z. Reichstein, D. Rogalski, and J.J. Zhang conjectured that if X is an irreducible projective variety admitting a wild automorphism then X is an abelian variety, and proved this conjecture for dim(X) ≤ 2. As a step toward answering this conjecture in higher dimensions we prove a structure theorem for projective varieties of Kodaira dimension 0 admitting wild automorphisms. This essentially reduces the Kodaira dimension 0 case to a study of Calabi-Yau varieties, which we also investigate. In support of this conjecture, we show that there are no wild automorphisms of certain Calabi-Yau varieties.

The Beilinson equivalence for differential operators

Let D be the ring of differential operators on a smooth irreducible affine variety X over C , or, more generally, the enveloping algebra of any locally free Lie algebroid on X . The category of finitely generated graded modules of the Rees algebra D ˜ has a natural quotient category P D which imitates the category of modules on P r o j of a graded commutative ring. We show that the derived category D b ( P D ) is equivalent to the derived category of finitely generated modules of a sheaf of algebras E on X which is coherent over X . This generalizes the usual Beilinson equivalence for projective space, and also the Beilinson equivalence for differential operators on a smooth curve used by Ben-Zvi and Nevins in [6] to describe the moduli space of left ideals in D .

On varieties of almost minimal degree I: Secant loci of rational normal scrolls

To provide a geometrical description of the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let X ̃ ⊂ P K r + 1 be a variety of minimal degree and of codimension at least 2, and consider X p = π p ( X ̃ ) ⊂ P K r where p ∈ P K r + 1 ∖ X ̃ . By Brodmann and Schenzel (2007) [1], it turns out that the cohomological and local properties of X p are governed by the secant locus Σ p ( X ̃ ) of X ̃ with respect to p . Along these lines, the present paper is devoted to giving a geometric description of the secant stratification of X ̃ , that is of the decomposition of P K r + 1 via the types of secant loci. We show that there are at most six possibilities for the secant locus Σ p ( X ̃ ) , and we precisely describe each stratum of the secant stratification of X ̃ , each of which turns out to be a quasi-projective variety. As an application, we obtain a different geometrical description of non-normal del Pezzo varieties X ⊂ P K r , first classified by Fujita (1985) [3, Theorem 2.1(a)] by providing a complete list of pairs ( X ̃ , p ) , where X ̃ ⊂ P K r + 1 is a variety of minimal degree, p ∈ P K r + 1 ∖ X ̃ and X p = X ⊂ P K r .