Displaying similar documents to “The finiteness of compact varieties in C n

Asymptotics of eigensections on toric varieties

A. Huckleberry, H. Sebert (2013)

Annales de l’institut Fourier

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Using exhaustion properties of invariant plurisubharmonic functions along with basic combinatorial information on toric varieties, we prove convergence results for sequences of densities | ϕ n | 2 = | s N | 2 / | | s N | | L 2 2 for eigensections s N Γ ( X , L N ) approaching a semiclassical ray. Here X is a normal compact toric variety and L is an ample line bundle equipped with an arbitrary positive bundle metric which is invariant with respect to the compact form of the torus. Our work was motivated by and extends that of Shiffman, Tate...

Finiteness of cominuscule quantum K -theory

Anders S. Buch, Pierre-Emmanuel Chaput, Leonardo C. Mihalcea, Nicolas Perrin (2013)

Annales scientifiques de l'École Normale Supérieure

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The product of two Schubert classes in the quantum K -theory ring of a homogeneous space X = G / P is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on  X . We show that if X is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to  X that take the marked points to general Schubert varieties and...

On the dimension of secant varieties

Luca Chiantini, Ciro Ciliberto (2010)

Journal of the European Mathematical Society

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In this paper we generalize Zak’s theorems on tangencies and on linear normality as well as Zak’s definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumptions on X , and we classify varieties for which the bound is attained.

Lagrangian fibrations on generalized Kummer varieties

Martin G. Gulbrandsen (2007)

Bulletin de la Société Mathématique de France

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We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface A of Picard number one we find the following: The Kummer variety K n A is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if n is a perfect square. And this is the case if and only if K n A carries a divisor with vanishing Beauville-Bogomolov square.

Equivariant K-theory of flag varieties revisited and related results

V. Uma (2013)

Colloquium Mathematicae

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We obtain several several results on the multiplicative structure constants of the T-equivariant Grothendieck ring K T ( G / B ) of the flag variety G/B. We do this by lifting the classes of the structure sheaves of Schubert varieties in K T ( G / B ) to R(T) ⊗ R(T), where R(T) denotes the representation ring of the torus T. We further apply our results to describe the multiplicative structure constants of K ( X ) where X denotes the wonderful compactification of the adjoint group of G, in terms of the structure...

On varieties of Hilbert type

Lior Bary-Soroker, Arno Fehm, Sebastian Petersen (2014)

Annales de l’institut Fourier

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A variety X over a field K is of Hilbert type if X ( K ) is not thin. We prove that if f : X S is a dominant morphism of K -varieties and both S and all fibers f - 1 ( s ) , s S ( K ) , are of Hilbert type, then so is X . We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.

Varieties of Algebras of Polynomial Growth

Daniela La Mattina (2008)

Bollettino dell'Unione Matematica Italiana

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Let 𝒱 be a proper variety of associative algebras over a field F of characteristic zero. It is well-known that 𝒱 can have polynomial or exponential growth and here we present some classification results of varieties of polynomial growth. In particular we classify all subvarieties of the varieties of almost polynomial growth, i.e., the subvarieties of 𝐯𝐚𝐫 ( G ) and 𝐯𝐚𝐫 ( U T 2 ) , where G is the Grassmann algebra and U T 2 is the algebra of 2 × 2 upper triangular matrices.

On Zariski's theorem in positive characteristic

Ilya Tyomkin (2013)

Journal of the European Mathematical Society

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In the current paper we show that the dimension of a family V of irreducible reduced curves in a given ample linear system on a toric surface S over an algebraically closed field is bounded from above by - K S . C + p g ( C ) - 1 , where C denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality 𝚍𝚒𝚖 ( V ) = - K S . C + p g ( C ) - 1 does not imply the nodality of C even if C belongs...

Quiver varieties and the character ring of general linear groups over finite fields

Emmanuel Letellier (2013)

Journal of the European Mathematical Society

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Given a tuple ( 𝒳 1 , ... , 𝒳 k ) of irreducible characters of G L n ( F q ) we define a star-shaped quiver Γ together with a dimension vector v . Assume that ( 𝒳 1 , ... , 𝒳 k ) is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product 𝒳 1 𝒳 k as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to ( Γ , v ) . The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied,...

Birational Finite Extensions of Mappings from a Smooth Variety

Marek Karaś (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We present an example of finite mappings of algebraic varieties f:V → W, where V ⊂ kⁿ, W k n + 1 , and F : k k n + 1 such that F | V = f and gdeg F = 1 < gdeg f (gdeg h means the number of points in the generic fiber of h). Thus, in some sense, the result of this note improves our result in J. Pure Appl. Algebra 148 (2000) where it was shown that this phenomenon can occur when V ⊂ kⁿ, W k m with m ≥ n+2. In the case V,W ⊂ kⁿ a similar example does not exist.

Polarizations of Prym varieties for Weyl groups via abelianization

Herbert Lange, Christian Pauly (2009)

Journal of the European Mathematical Society

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Let π : Z X be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group G . For any dominant weight λ consider the curve Y = Z / Stab ( λ ) . The Kanev correspondence defines an abelian subvariety P λ of the Jacobian of Y . We compute the type of the polarization of the restriction of the canonical principal polarization of Jac ( Y ) to P λ in some cases. In particular, in the case of the group E 8 we obtain families of Prym-Tyurin varieties. The main idea is...

The Brauer group and the Brauer–Manin set of products of varieties

Alexei N. Skorobogatov, Yuri G. Zahrin (2014)

Journal of the European Mathematical Society

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Let X and Y be smooth and projective varieties over a field k finitely generated over Q , and let X ¯ and Y ¯ be the varieties over an algebraic closure of k obtained from X and Y , respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br ( X ¯ ) Br( Y ¯ ) has finite index in the Galois invariant subgroup of Br ( X ¯ × Y ¯ ) . This implies that the cokernel of the natural map Br ( X ) Br ( Y ) Br ( X × Y ) is finite when k is a number field. In this case we prove that the Brauer–Manin set of the...

Singularities of theta divisors and the geometry of 𝒜 5

Gavril Farkas, Samuele Grushevsky, Salvati R. Manni, Alessandro Verra (2014)

Journal of the European Mathematical Society

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We study the codimension two locus H in 𝒜 g consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class [ H ] C H 2 ( 𝒜 g ) for every g . For g = 4 , this turns out to be the locus of Jacobians with a vanishing theta-null. For g = 5 , via the Prym map we show that H 𝒜 5 has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of 𝒜 5 ¯ and show that the component N 0 ' ¯ of the Andreotti-Mayer...

Generating varieties for the triple loop space of classical Lie groups

Yasuhiko Kamiyama (2003)

Fundamenta Mathematicae

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For G = SU(n), Sp(n) or Spin(n), let C G ( S U ( 2 ) ) be the centralizer of a certain SU(2) in G. We have a natural map J : G / C G ( S U ( 2 ) ) Ω ³ G . For a generator α of H ( G / C G ( S U ( 2 ) ) ; / 2 ) , we describe J⁎(α). In particular, it is proved that J : H ( G / C G ( S U ( 2 ) ) ; / 2 ) H ( Ω ³ G ; / 2 ) is injective.

J -invariant of linear algebraic groups

Viktor Petrov, Nikita Semenov, Kirill Zainoulline (2008)

Annales scientifiques de l'École Normale Supérieure

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Let G be a semisimple linear algebraic group of inner type over a field F , and let X be a projective homogeneous G -variety such that G splits over the function field of X . We introduce the J -invariant of G which characterizes the motivic behavior of X , and generalizes the J -invariant defined by A. Vishik in the context of quadratic forms. We use this J -invariant to provide motivic decompositions of all generically split projective homogeneous G -varieties, e.g. Severi-Brauer varieties,...

Special m-hyperidentities in biregular leftmost graph varieties of type (2,0)

Apinant Anantpinitwatna, Tiang Poomsa-ard (2009)

Discussiones Mathematicae - General Algebra and Applications

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Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra A ( G ) ̲ satisfies s ≈ t. A class of graph algebras V is called a graph variety if V = M o d g Σ where Σ is a subset of T(X) × T(X). A graph variety V ' = M o d g Σ ' is called a biregular leftmost graph variety if Σ’ is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity...

Unique decomposition for a polynomial of low rank

Edoardo Ballico, Alessandra Bernardi (2013)

Annales Polonici Mathematici

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Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the d-uple Veronese embedding of m into m + d d - 1 but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s ≤ d then F can be uniquely written as F = M d + + M t d + Q , where M , . . . , M t are linear forms with t ≤ (d-1)/2, and Q is a binary form such that Q = i = 1 q l i d - d i m i with l i ’s linear forms...

Geometry of Syzygies via Poncelet Varieties

Giovanna Ilardi, Paola Supino, Jean Vallès (2009)

Bollettino dell'Unione Matematica Italiana

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We consider the Grassmannian 𝔾 r ( k , n ) of ( k + 1 ) -dimensional linear subspaces of V n = H 0 ( 1 , 𝒪 1 ( n ) ) . We define 𝔛 k , r , d as the classifying space of the k -dimensional linear systems of degree n on 1 , whose bases realize a fixed number r of polynomial relations of fixed degree d , say r syzygies of degree d . Firstly, we compute the dimension of 𝔛 k , r , d . In the second part we make a link between 𝔛 k , r , d and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the...

Birational positivity in dimension 4

Behrouz Taji (2014)

Annales de l’institut Fourier

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In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of Ω p is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an X provided that X has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.

Construction of Mendelsohn designs by using quasigroups of ( 2 , q ) -varieties

Lidija Goračinova-Ilieva, Smile Markovski (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let q be a positive integer. An algebra is said to have the property ( 2 , q ) if all of its subalgebras generated by two distinct elements have exactly q elements. A variety 𝒱 of algebras is a variety with the property ( 2 , q ) if every member of 𝒱 has the property ( 2 , q ) . Such varieties exist only in the case of q prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property ( 2 , q ) , 2 < q , blocks of Steiner system of type ( 2 , q ) are obtained. The stated correspondence...