Chen-Ruan cohomology of some moduli spaces of parabolic vector bundles

Let (X,D) be an ?-pointed compact Riemann surface of genus at least two. For each point x∈D, fix parabolic weights such that . Fix a holomorphic line bundle ξ over X of degree one. Let PMξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights . The group of order two line bundles over X acts on PMξ by the rule E_∗⊗L?E_∗⊗L. We compute the Chen-Ruan cohomology ring of the corresponding orbifold.     

Higher syzygies of hyperelliptic curves

Let X be a hyperelliptic curve of arithmetic genus g and let f:X→P¹ be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d≤2g. Note that the minimal free resolution of X of degree ≥2g+1 is already completely known. Let A=f^∗O_P¹(1), and let L be a very ample line bundle on X of degree d≤2g. For , we call the pair (m,d−2m)the factorization type ofL. Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by |L| are precisely determined by the factorization type of L.     

Quantization of Donaldson’s heat flow over projective manifolds

Consider E a holomorphic vector bundle over a projective manifold X polarized by an ample line bundle L. Fix k large enough, the holomorphic sections \(H^0(E\otimes L^k)\) provide embeddings of X in a Grassmanian space. We define the balancing flow for bundles as a flow on the space of projectively equivalent embeddings of X. This flow can be seen as a flow of algebraic type hermitian metrics on E. At the quantum limit \(k\rightarrow \infty \), we prove the convergence of the balancing flow towards the Donaldson heat flow, up to a conformal change. As a by-product, we obtain a numerical scheme to approximate the Yang–Mills flow in that context.     

Bergman kernels and local holomorphic Morse inequalities

Let (X,) be a hermitian manifold and let L^k be a high power of a hermitian holomorphic line bundle over X. Local versions of Demaillys holomorphic Morse inequalities (that give bounds on the dimension of the Dolbeault cohomology groups associated to L^k), are presented - after integration they give the usual holomorphic Morse inequalities. The local weak inequalities hold on any hermitian manifold (X,), regardless of compactness and completeness. The proofs, which are elementary, are based on a new approach to pointwise Bergman kernel estimates, where the kernels are estimated by a model kernel in Mathematics Subject Classification (2000): 32A25, 32L10, 32L20in final form: 19 June 2003     

Compact almost automorphic solutions to integral equations with infinite delay

Given a∈L¹(R) and the generator A of an L¹-integrable resolvent family of linear bounded operators defined on a Banach space X, we prove the existence of compact almost automorphic solutions of the semilinear integral equation for each f:R×X→X compact almost automorphic in t, for each x∈X, and satisfying Lipschitz and Hölder type conditions. In the scalar linear case, we prove that a∈L¹(R) positive, nonincreasing and log-convex is sufficient to obtain the existence of compact almost automorphic solutions.     

The inversion formulae for automorphisms of polynomial algebras and rings of differential operators in prime characteristic

Let K be an arbitrary field of characteristic p>0. We find an explicit formula for the inverse of any algebra automorphism of any of the following algebras: the polynomial algebra P_n?K[x₁,…,x_n], the ring of differential operators D(P_n) on P_n, D(P_n)⊗P_m, the n’th Weyl algebra A_n or A_n⊗P_m, the power series algebra K[[x₁,…,x_n]], the subalgebra T_k1,…,kn of D(P_n) generated by P_n and the higher derivations , 0≤j<p^k_i, i=1,…,n (where k₁,…,k_n∈N), T_k1,…,kn⊗P_m or an arbitrary central simple (countably generated) algebra over an arbitrary field.     

On holomorphic domination, I

Let X be a separable Banach space and u:X→R locally upper bounded. We show that there are a Banach space Z and a holomorphic function h:X→Z with u(x)<‖h(x)‖ for x∈X. As a consequence we find that the sheaf cohomology group Hq(X,O) vanishes if X has the bounded approximation property (i.e., X is a direct summand of a Banach space with a Schauder basis), O is the sheaf of germs of holomorphic functions on X, and q?1. As another consequence we prove that if f is a C¹-smooth -closed (0,1)-form on the space X=L₁[0,1] of summable functions, then there is a C¹-smooth function u on X with on X.     

Refinements of Milnor's fibration theorem for complex singularities

Let X be an analytic subset of an open neighbourhood U of the origin in Cn. Let be holomorphic and set V=f−1(0). Let Bε be a ball in U of sufficiently small radius ε>0, centred at . We show that f has an associated canonical pencil of real analytic hypersurfaces Xθ, with axis V, which leads to a fibration Φ of the whole space (X∩Bε)?V over S¹. Its restriction to (X∩Sε)?V is the usual Milnor fibration , while its restriction to the Milnor tube f−1(∂Dη)∩Bε is the Milnor-Lê fibration of f. Each element of the pencil Xθ meets transversally the boundary sphere Sε=∂Bε, and the intersection is the union of the link of f and two homeomorphic fibres of ? over antipodal points in the circle. Furthermore, the space obtained by the real blow up of the ideal (Re(f),Im(f)) is a fibre bundle over RP¹ with the Xθ as fibres. These constructions work also, to some extent, for real analytic map-germs, and give us a clear picture of the differences, concerning Milnor fibrations, between real and complex analytic singularities.     

Grauert’s line bundle convexity,reduction and Riemann domains

We consider a convexity notion for complex spaces X with respect to a holomorphic line bundle L over X. This definition has been introduced by Grauert and, when L is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if H⁰(X,L) separates each point of X, then X can be realized as a Riemann domain over the complex projective space Pⁿ, where n is the complex dimension of X and L is the pull-back of O(1).     

Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces

Let K(n) be the nth Morava K-theory at a prime p, and let T(n) be the telescope of a v_n-self map of a finite complex of type n. In this paper we study the K(n)_*-homology of Ω^∞X, the 0th space of a spectrum X, and many related matters.We give a sampling of our results.Let PX be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map

s_n(X):L_T(n)P(X)→L_T(n)Σ^∞(Ω^∞X)₊     

Secant varieties of toric varieties

Let X_P be a smooth projective toric variety of dimension n embedded in P^r using all of the lattice points of the polytope P. We compute the dimension and degree of the secant variety . We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties X_A embedded using a set of lattice points A⊂P∩Zⁿ containing the vertices of P and their nearest neighbors.     

Szegö kernel asymptotics and Morse inequalities on CR manifolds

Let X be an abstract compact orientable CR manifold of dimension ${2n-1, n\,\geqslant\,2}$ , and let L ^k be the k-th tensor power of a CR complex line bundle L over X. We assume that condition Y(q) holds at each point of X. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in L ^k , for large k. After integration, this gives weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities. We apply the strong Morse inequalities to the embedding of some convex–concave manifolds.     

On a question of Sean Keel

Let X be a smooth projective surface defined over , and let L be a line bundle over X such that for every complete curve Y contained in X. A question of Keel asks whether L is ample. If X is a P¹-bundle over a curve, we prove that this question has an affirmative answer.     

Balanced Metrics and Chow Stability of Projective Bundles over Kähler Manifolds II

In the previous article (Seyyedali, Duke Math. J. 153(3):573–605, 2010), we proved that slope stability of a holomorphic vector bundle E over a polarized manifold (X,L) implies Chow stability of $(\mathbb{P}E^{*},\mathcal{O}_{\mathbb{P}E^{*}}(1)\otimes\pi^{*} L^{k})$ for k?0 if the base manifold has no nontrivial holomorphic vector field and admits a constant scalar curvature metric in the class of 2πc ₁(L). In this article, using asymptotic expansions of the Bergman kernel on Sym ^d E, we generalize the main theorem of Seyyedali (Duke Math. J. 153(3):573–605, 2010) to polarizations $(\mathbb{P}E^{*},\mathcal {O}_{\mathbb{P}E^{*}}(d)\otimes\pi^{*} L^{k})$ for k?0, where d is a positive integer.     

Some properties of the injective tensor product of L[0,1] and a Banach space

Let X be a (real or complex) Banach space and 1<p,p′<∞ such that 1/p+1/p′=1. Then , the injective tensor product of L^p[0,1] and X, has the Radon-Nikodym property (resp. the analytic Radon-Nikodym property, the near Radon-Nikodym property, contains no copy of c₀, is weakly sequentially complete) if and only if X has the same property and each continuous linear operator from L^p′[0,1] to X is compact.     

Erd?s-Ko-Rado for three sets

Fix integers k?3 and n?3k/2. Let F be a family of k-sets of an n-element set so that whenever A,B,C∈F satisfy |A∪B∪C|?2k, we have A∩B∩C≠∅. We prove that with equality only when ?_F∈FF≠∅. This settles a conjecture of Frankl and Füredi [2], who proved the result for n?k²+3k.     

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Let X be a compact connected Kähler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly et al. (1994) [11] says that there is a finite unramified Galois covering M→X, a complex torus T, and a holomorphic surjective submersion f:M→T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We also prove that the holomorphic principal G-bundle over T given by f, where G is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.     

Postulation of disjoint unions of lines and a few planes

Let X⊂Pⁿ, n≥5, be a general union of x planes and y lines. Here we prove that X has the expected postulation if x is small with respect to y.