Decomposability criterion for linear sheaves

We establish a decomposability criterion for linear sheaves on ℙ ⁿ . Applying it to instanton bundles, we show, in particular, that every rank 2n instanton bundle of charge 1 on ℙ ⁿ is decomposable. Moreover, we provide an example of an indecomposable instanton bundle of rank 2n − 1 and charge 1, thus showing that our criterion is sharp.     

Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ<Superscript>3</Superscript>

Symplectic instanton vector bundles on the projective space ℙ³ constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I _n;r of rank-2r symplectic instanton vector bundles on ℙ³ with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I _n;r ^* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).     

Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem

In this article we prove a general result on a nef vector bundle E on a projective manifold X of dimension n depending on the vector space H^n,n(X,E): It is also shown that H^n,n(X,E) = 0 for an indecomposable nef rank 2 vector bundles E on some specific type of n dimensional projective manifold X. The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.     

Weighted integral formulas on manifolds

We present a method of finding weighted Koppelman formulas for (p,q)-forms on n-dimensional complex manifolds X which admit a vector bundle of rank n over X×X, such that the diagonal of X×X has a defining section. We apply the method to ℙ ⁿ and find weighted Koppelman formulas for (p,q)-forms with values in a line bundle over ℙ ⁿ . As an application, we look at the cohomology groups of (p,q)-forms over ℙ ⁿ with values in various line bundles, and find explicit solutions to the -equation in some of the trivial groups. We also look at cohomology groups of (0,q)-forms over ℙ ⁿ ×ℙ ^m with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.     

Factoriality of complete intersections in ℙ<Superscript>5</Superscript>

Let X be a complete intersection of two hypersurfaces F _n and F _k in ℙ⁵ of degree n and k, respectively, with n ≥ k, such that the singularities of X are nodal and F _k is smooth. We prove that if the threefold X has at most (n + k − 2)(n − 1) − 1 singular points, then it is factorial.     

A Criterion for a Vector Bundle Over a Curve to Admit a Connection

Let X be a geometrically connected smooth projective curve defined over a perfect field k. Let E be a vector bundle over X. We prove that E admits a connection if every indecomposable component of E is of degree zero. If the characteristic of k is p, with p > 0, and the rank of each of the indecomposable components of E is not a multiple of p, then E admits a connection if and only if the degree of each indecomposable component of E is a multiple of p. (Received: August 2005)     

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Let X be a Hopf manifolds with an Abelian fundamental group. E is a holomorphic vector bundle of rank r with trivial pull-back to W = ℂ ⁿ –{0}. We prove the existence of a non-vanishing section of L⊗E for some line bundle on X and study the vector bundles filtration structure of E. These generalize the results of D. Mall about structure theorem of such a vector bundle E. The research was supported by 973 Project Foundation of China and the Outstanding Youth Science Grant of NSFC (grant no. 19825105)     

The Dual Quiver of the Auslander–Reiten Quiver of Path Algebras

Let Q be a finite quiver of type A _n , n ≥ 1, D _n , n ≥ 4, E ₆, E ₇ and E ₈, σ ∈ Aut(Q), k be an algebraic closed field whose characteristic does not divide the order of σ. In this article, we prove that the dual quiver [(G_Q)\tilde]\widetilde{\Gamma_{Q}} of the Auslander–Reiten quiver Γ _Q of kQ, the Auslander–Reiten quiver of kQ#kás?kQ\#k\langle\sigma\rangle, and the Auslander–Reiten quiver G_[(Q)\tilde]\Gamma_{\widetilde{Q}} of k[(Q)\tilde]k\widetilde{Q}, where [(Q)\tilde]\widetilde{Q} is the dual quiver of Q, are isomorphic.     

Jumping conics on a smooth quadric in $${\mathbb {P}_3}$$

We investigate the jumping conics of stable vector bundles E of rank 2 on a smooth quadric surface Q with the first Chern class c₁ = O_Q(-1,-1){c_1= \mathcal{O}_Q(-1,-1)} with respect to the ample line bundle O_Q(1,1){\mathcal {O}_Q(1,1)} . We show that the set of jumping conics of E is a hypersurface of degree c ₂(E) − 1 in \mathbb P₃^*{\mathbb {P}_3^{*}} . Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on Q in the cases of lower c ₂(E).     

Rank 4 vector bundles on the quintic threefold

By the results of the author and Chiantini in [3], on a general quintic threefold X⊂P ⁴ the minimum integer p for which there exists a positive dimensional family of irreducible rank p vector bundles on X without intermediate cohomology is at least three. In this paper we show that p≤4, by constructing series of positive dimensional families of rank 4 vector bundles on X without intermediate cohomology. The general member of such family is an indecomposable bundle from the extension class Ext ¹ (E, F), for a suitable choice of the rank 2 ACM bundles E and F on X. The existence of such bundles of rank p=3 remains under question.     

Generalizations of the odd degree theorem and applications

LetV ⊂ ℙℝ ⁿ be an algebraic variety, such that its complexificationV _ℂ ⊂ ℙ ⁿ is irreducible of codimensionm ≥ 1. We use a sufficient condition on a linear spaceL ⊂ ℙℝ ⁿ of dimensionm + 2r to have a nonempty intersection withV, to show that any six dimensional subspace of 5 × 5 real symmetric matrices contains a nonzero matrix of rank at most 3.     

Uniform vector bundles on quadrics

Summary In this paper we prove that a rankr uniform vector bundle on a nonsingular quadricQ with dimQ≥2r+2 is a direct sum of line bundles. We study also rank 2 uniform vector bundles onP ₁×P₁. We prove that they are not all homogeneous and that any rank 2 homogeneous vector bundles onP ₁×P₁ is decomposable.

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Riassunto In questo lavoro si dimostra che un fibrato uniforme di rangor su una quadrica non singolareQ con dimQ≥2r+2 è somma diretta di fibrati in rette. Si studiano poi i fibrati uniformi di rango 2 suP ₁×P₁. Si dimostra che non sono tutti omogenei e che ogni fibrato omogeneo di rango 2 suP ₁×P₁ è decomponibile.

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The author is member of G.N.S.A.G.A. of C.N.R.     

Postulation of subschemes of irreducible curves on a quadric surface

We study all the possible Hilbert functions of 0-dimensional subschemes of irreducible curves of a smooth quadric of ℙ³. We obtain characterizations in case of complete intersection, arithmetically Cohen-Macaulay and arithmetically Buchsbaum curves and other necessary conditions in the general cases.     

Notes on varieties of codimension 3 in ℙ N

In this article I describe construction methods for smooth subvarieties of codimension 3 in projective spaces or other ambient spaces. The methods include liaison of 3-folds in ℙ⁶, sections in smooth reflexive sheaves, and Pfaffians of twisted skew-symmetric vector bundle morphisms. I use these methods to construct new families of 3-folds in ℙ⁶, and new codimension 3 submanifolds in ℙ⁸ and ℙ⁹. This article was processed using the LAT_EX macro package with LMAMULT style     

Rational curves on moduli spaces of vector bundles

We identify the spaces Hom_i(ℙ¹,M) fori = 1, 2, whereM is the moduli space of vector bundles of rank 2 and determinant isomorphic to ,x ₀ ∈X, on a compact Riemann surface of genusg ≥ 2.     

On globally generated vector bundles on projective spaces

A classification is given for globally generated vector bundles E of rank k on Pⁿ having first Chern class c₁(E)=2. In particular, we get that they split if k<n unless E is a twisted null-correlation bundle on P³. In view of the well-known correspondence between globally generated vector bundles and maps to Grassmannians, we obtain, as a corollary, a classification of double Veronese embeddings of Pⁿ into a Grassmannian G(k−1,N) of (k−1)-planes in P^N.     

On the Hilbert scheme of curves in higher-dimensional projective space

In this paper we prove that, for anyn≥3, there exist infinitely manyr∈N and for each of them a smooth, connected curveC _r in ℙ ^r such thatC _r lies on exactlyn irreducible components of the Hilbert scheme Hilb(ℙ ^r ). This is proven by reducing the problem to an analogous statement for the moduli of surfaces of general type.     

Gorenstein liaison of 0-dimensional schemes

 The theory of Gorenstein liaison has been developed during the last 3 years to generalize liaison theory of codimension 2 schemes to schemes of codimension ≥ 3 in a projective space. One of the main open questions in Gorenstein liaison theory is whether any arithmetically Cohen-Macaulay subscheme of ℙ ⁿ is in the Gorenstein liaison class of a complete intersection. In this paper we prove that any set of general points lying on a rational normal scroll surface is in the Gorenstein liaison class of a complete intersection. Received: 21 November 2001 Published online: 24 April 2003 Mathematics Subject Classification (2000): 14M06, 14C20, 14M05     

A generalization of the existence theorem of special line bundles

LetX be a generic smooth irreducible complex projective curve of genusg withg4. In this paper, we generalize the existence theorem of special divisors to high dimensional indecomposable vector bundles. We give a necessary and sufficient condition on the existence ofn-dimensional indecomposable vector bundlesE onX with det(E)=d, dimH ⁰(X,E)h. We also determine under what condition the set of all such vector bundles will be finite and how many elements it contains.Project partly supported by the National Natural Science Foundation of China.     

Morse theory for the space of Higgs <Emphasis Type="Italic">G</Emphasis>–bundles

Fix a C ^∞ principal G–bundle E⁰_G{E^0_G} on a compact connected Riemann surface X, where G is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang–Mills–Higgs functional on the cotangent bundle of the space of all smooth connections on E⁰_G{E^0_G}. We prove that this flow preserves the subset of Higgs G–bundles, and, furthermore, the flow emanating from any point of this subset has a limit. Given a Higgs G–bundle, we identify the limit point of the integral curve passing through it. These generalize the results of the second named author on Higgs vector bundles.