Toute variété magnifique est sphérique

LetG be a (connected) reductive group (over C). An algebraicG-varietyX is called “wonderful”, if the following conditions are satisfied:X is (connected) smooth and complete;X containsr irreducible smoothG-invariant divisors having a non void transversal intersection;G has 2 ^r orbits inX. We show that wonderful varieties are necessarily spherical (i.e., they are almost homogeneous under any Borel subgroup ofG).      

A Note on the Infimum of Energy of Unit Vector Fields on a Compact Riemannian Manifold

Our main result in this paper establishes that if G is a compact Lie subgroup of the isometry group of a compact Riemannian manifold M acting with cohomogeneity one in M and either G has no singular orbits or the singular orbits of G have dimension at most n−3, then the unit vector field N orthogonal to the principal orbits of G is weakly smooth and is a critical point of the energy functional acting on the unit normal vector fields of M. A formula for the energy of N in terms of the of integral of the Ricci curvature of M and of the integral of the square of the mean curvature of the principal orbits of G is obtained as well. In the case that M is the sphere and G the orthogonal group it is known that that N is minimizer. It is an open question if N is a minimizer in general.     

Bipolarized threefolds with hyperelliptic curve sections

A classification of smooth complex projective threefoldsX polarized by two very ample line bundlesL andM is given, under the assumption that two general elements of |L| and |M| intersect transversally along a smooth hyperelliptic curve.

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Sunto Si fornisce una classificazione delle terne (X,L,M), doveX è una varietà algebrica proiettiva complessa liscia di dimensione 3 edL,M sono due fibrati lineari molto ampi suX, tali che l’intersezione di due elementi generici di |L| ed |M| sia una curva iperellittica.

     

Parabolic bundles and representations¶of the fundamental group

Let X be a smooth complex projective variety with Neron–Severi group isomorphic to ℤ, and D an irreducible divisor with normal crossing singularities. Assume 1<r≤ 3. We prove that if π₁(X) doesn't have irreducible PU(r) representations, then π₁(X- D) doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth. Received: 20 December 1999 / Revised version: 7 May 2000     

On some smooth projective two-orbit varieties with Picard number 1

We classify all smooth projective horospherical varieties with Picard number 1. We prove that the automorphism group of any such variety X acts with at most two orbits and that this group still acts with only two orbits on X blown up at the closed orbit. We characterize all smooth projective two-orbit varieties with Picard number 1 that satisfy this latter property.     

Versality of linearly normal non-special embeddings

LetX⊂P ⁿ be a smooth non-degenerate non special linearly normal projective curve. Here we classify all such embeddings ofX such that for every hyperplaneM ofP ⁿ the family of all hyperplane sections ofX is a versal deformation of the zerodimensional schemeX∩M.

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Sunto SiaX una curva liscia e proiettiva. Qui si classificano le immersioni non-speciali linearmente normali diX inP ⁿ tali che per ogni iperpianoM diP ⁿ la famiglia delle sezioni iperpiane diX induce una deformazione versale diX∩M.

     

On a motivic interpretation of primitive,variable and fixed cohomology

If M is a smooth projective variety whose motive is Kimura finite‐dimensional and for which the standard Lefschetz Conjecture B holds, then the motive of M splits off a primitive motive whose cohomology is the primitive cohomology. Under the same hypotheses on M, let X be a smooth complete intersection of ample divisors within M. Then the motive of X is the sum of a variable and a fixed motive inducing the corresponding splitting in cohomology. I also give variants with group actions.     

Projective Resolutions for Modules over Infinite Groups

We define a notion of complexity for modules over group rings of infinite groups. This generalizes the notion of complexity for modules over group algebras of finite groups. We show that if M is a module over the group ring kG, where k is any ring and G is any group, and M has f-complexity (where f is some complexity function) over some set of finite index subgroups of G, then M has f-complexity over G (up to a direct summand). This generalizes the Alperin-Evens Theorem, which states that if the group G is finite then the complexity of M over G is the maximal complexity of M over an elementary abelian subgroup of G. We also show how we can use this generalization in order to construct projective resolutions for the integral special linear groups, SL(n, ℤ), where n ≥ 2.     

A generalization of Higgs bundles to higher dimensional varieties

Let X be a smooth n-dimensional projective variety defined over and let L be a line bundle on X. In this paper we shall construct a moduli space parametrizing -cohomology L-twisted Higgs pairs, i.e., pairs where E is a vector bundle on X and . If we take , the canonical line bundle on X, the variety is canonically identified with the cotangent bundle of the smooth locus of the moduli space of stable vector bundles on X and, as such, it has a canonical symplectic structure. We prove that, in the general case, in correspondence to the choice of a non-zero section , one can define, in a natural way, a Poisson structure on . We also analyze the relations between this Poisson structure on and the canonical symplectic structure of the cotangent bundle to the smooth locus of the moduli space of parabolic bundles over X, with parabolic structure over the divisor D defined by the section s. These results generalize to the higher dimensional case similar results proved in [Bo1] in the case of curves. Received November 4, 1997; in final form May 28, 1998     

Cohomology of the moduli space of Hecke cycles

Let X be a smooth projective curve of genus g?3 and let M₀ be the moduli space of semistable bundles over X of rank 2 with trivial determinant. Three different desingularizations of M₀ have been constructed by Seshadri (Proceedings of the International Symposium on Algebraic Geometry, 1978, 155), Narasimhan-Ramanan (C. P. Ramanujam—A Tribute, 1978, 231), and Kirwan (Proc. London Math. Soc. 65(3) (1992) 474). In this paper, we construct a birational morphism from Kirwan's desingularization to Narasimhan-Ramanan's, and prove that the Narasimhan-Ramanan's desingularization (called the moduli space of Hecke cycles) is the intermediate variety between Kirwan's and Seshadri's as was conjectured recently in (Math. Ann. 330 (2004) 491). As a by-product, we compute the cohomology of the moduli space of Hecke cycles.     

The normality of closures of orbits in a Lie algebra

LexX be the closure of aG-orbit in the Lie algebra of a connected reductive groupG. It seems that the varietyX is always normal. After a reduction to nilpotent orbits, this is proved for some special cases. Results on determinantal schemes are used forGl _n . IfX is small enough we use a resolution and Bott's theorem on the cohomology of homogeneous vector bundles. Our results are conclusive for groups of typeA ₁,A ₂,A ₃ andB ₂.     

Topological and analytical properties of Sobolev bundles,I: The critical case

We study the interrelationship between topological and analytical properties of Sobolev bundles and describe some of their applications to variational problems on principal bundles. We in particular show that the category of Sobolev principal G-bundles of class W ^2,m/2 defined over M ^m is equivalent to the category of smooth principal G-bundles on M and give a characterization of the weak sequential closure of smooth principal G-bundles with prescribed isomorphism class. We also prove a topological compactness result for minimizing sequences of a conformally invariant Yang-Mills functional.      

Holomorphic vector bundles on compact complex non-algebraic surfaces

LetX be a smooth complex compact surface without non-constant meromorphic functions. Here we prove the existence of rank holomorphic vector bundles onX containing exactly one rank one saturated subsheaf.

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Sunto SiaX una superficie complessa compatta non singolare senza funzioni meromorfe non costanti. In questo lavoro si domstra cheX possiede molti fibrati olomorfi di rango 2 contenenti un unico fibrato in rette.

     

Rational surfaces in algebraic models of smooth manifolds

Every compact smooth manifold M is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of M. We study modulo 2 homology classes represented by rational algebraic surfaces in X, as X runs through the class of all algebraic models of M. Received: 16 June 2007     

Degenerations of the moduli spaces of vector bundles on curves II (generalized Gieseker moduli spaces)

LetX ₀ be a projective curve whose singularity is one ordinary double point. We construct a birational modelG(n, d) of the moduli spaceU(n, d) of stable torsion free sheaves in the case (n, d)= 1, such that G(n, d) has normal crossing singularities and behaves well under specialization i.e. if a smooth projective curve specializes toX ₀, then the moduli space of stable vector bundles of rankn and degreed onX specializes toG(n, d). This generalizes an earlier work of Gieseker in the rank two case.     

On non-normalizable orbits of isometric actions on Lorentz manifolds

For an isometric action of a Lie group G on a Lorentz manifold (M, g) we consider non-normalizable orbits, i.e. orbits which do not posses a G-invariant normal bundle. Orbits of this type are lightlike. It is shown, that such orbits contain lightlike homogeneous geodesics. Moreover, conditions are given, under which there exists a set of normalizable orbits having an open dense union.     

Holomorphic and algebraic vector bundles on 0-convex algebraic surfaces

LetX be a smooth complex algebraic surface such that there is a proper birational morphism/:X → Y withY an affine variety. Let X_hol be the 2-dimensional complex manifold associated toX. Here we give conditions onX which imply that every holomorphic vector bundle onX is algebraizable and it is an extension of line bundles. We also give an approximation theorem of holomorphic vector bundles on X_hol (X normal algebraic surface) by algebraic vector bundles.     

Spherical orbit closures in simple projective spaces and their normalizations

Let G be a simply connected semisimple algebraic group over an algebraically closed field k of characteristic 0 and let V be a rational simple G-module. If G/H ⊂ P(V) is a spherical orbit and if X = [`(G/H)] X = \overline {G/H} is its closure, then we describe the orbits of X and those of its normalization [(X)\tilde] \tilde{X} . If, moreover, the wonderful completion of G/H is strict, then we give necessary and sufficient combinatorial conditions so that the normalization morphism [(X)\tilde] ? X \tilde{X} \to X is a homeomorphism. Such conditions are trivially fulfilled if G is simply laced or if H is a symmetric subgroup.