An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M _n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁ ² + ... + _n ² . We prove the inequalities {M _nx}c(1-(x/)), {M _n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.     

Syzygies of Veronese Embeddings

We prove that the Veronese embedding _O ⁿ (d): ^{n N} with n2, d3 does not satisfy property N _p (according to Green and Lazarsfeld) if p3d–2. We make the conjecture that also the converse holds. This is true for n=2 and for n=d=3.     

Local Recognition Of Non-Incident Point-Hyperplane Graphs

Let be a projective space. By H() we denote the graph whose vertices are the non-incident point-hyperplane pairs of , two vertices (p,H) and (q,I) being adjacent if and only if p I and q H. In this paper we give a characterization of the graph H() (as well as of some related graphs) by its local structure. We apply this result by two characterizations of groups G with PSL _n ( )GPGL _n ( ), by properties of centralizers of some (generalized) reflections. Here is the (skew) field of coordinates of .     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

Twistor holomorphic Lagrangian surfaces in the complex projective and hyperbolic planes

We completely classify all the twistor holomorphic Lagrangian immersions in the complex projective plane ², i.e. those Lagrangian immersions such that their twistor lifts to the twistor space over ² are holomorphic. This classification provides a one-parameter family of examples of Lagrangian spheres in ².Research partially supported by a DGICYT grant No. PB91-0731.     

A note on generic projections

Let X^N=²ⁿ_K be a subvariety of dimension n and P^N a generic point. If the tangent variety TanX is equal to ^N then for generic points x, y of X the projective tangent spaces t_xX and t_yX meet in one point P=P(x,y). The main result of this paper is that the rational map (x,y)P(x,y) is dominant. In other words, a generic point P is uniquely determined by the ramification locus R(_P) of the linear projection _P:X^N–1.This paper was supported by the DFG Schwerpunkt Global methods in complex analysis, MUR and the Research Group GNSAGA of INDAM. This investigation was also supported by the University of Bologna, funds for selected research topics     

On the smoothness of the moduli space of mathematical instanton bundles

In this paper we prove that the moduli spaces MI _2n+1(k) of mathematical instanton bundles on ²ⁿ⁺¹ with quantum number k are singular for n 2 and k 3 ,giving a positive answer to a conjecture made by Ancona and Ottaviani in 1993.     

3-Mannigfaltigkeiten im ℙ5 und ihre zugehörigen Stabilen Garben

In this paper we begin to study 3-folds in a projective space of dimension 5. Using results from [9] we give a classification of all 3-folds in ⁵ , up to degree 6. There are only 3 different types of 3-folds in ⁵ of degree 6 which are not complete intersections. These manifolds can be represented as zero schemes of sections in certain (extremal) semistable reflexive sheaves of rank 2 on ⁵ . Finally we obtain examples of stable reflexive sheaves on ⁵ with homologieal dimension 1, which do not belong to the extremal sheaves [10].

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Dies ist ein Teil meiner Habilitationsschrift     

Note on Fano 3-folds

LetE be an ample vector bundle of rankr on a compact complex manifoldX of dimension 3 with detE=–K _x, andi(X) the index ofX. Then it is proved in this note thati(X)r unless (X,E)(¹ × ²,p*O(1) q*), wherep,q are the projections and is isomorphic toO(2) O(1) or the tangent bundleT of ². This result gives a counterexample to the conjecture formed by T. Peternell.     

The singularH 2,2-invariant quartic surfaces in ℙ3

AnH _2,2-invariant quartic surface in ₃ is a quartic surface in ₃ invariant under the Heisenberg groupH _2,2 of level (2, 2), the family ofH _2,2-invariant quartic surfaces is parametrized by ₄. For each ₄, the corresponding quartic surfaceX will be a Kummer surface, ifX is singular. The equation for { = 0} ₄ parametrizing all Kummer surfaces is well known. We find another more symmetric form (with respect to a 5-dimensional representation of the symmetric group S₆) for this equation.The aim of this note is to describe all singularH _2,2-invariant quartic surfaces in ₃.     

Differentiable Connections with Poincar{e} Groups of Local Transformations. {III}

In the canonical smooth fiber bundles :ⁿ⁺¹ⁿ, we study generalized differentiable connections constructed by the author in papers [4] and [5]. Special emphasis is laid on the investigation of the behavior of these connections under local transformations of the classical Poincar{e} groups (1,n) and extended Poincar{e} groups canonically acting in the given connections. We found all first-order nonholonomic affine, ₁, ₂, and _1,2-connections with the groups (1,n) and of local transformations and also constructed classes of the corresponding invariant second-order connections.     

On Birational Maps and Jacobian Matrices

One is concerned with Cremona-like transformations, i.e., rational maps from ⁿ to ^m that are birational onto the image Y ^m and, moreover, the inverse map from Y to ⁿ lifts to ^m . We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.     

A Remark on Bernstein,Prokhorov, Bennett,Hoeffding, and Talagrand Inequalities

Let X ₁,..., X_n be independent random variables such that {X_j 1}=1 and E X _j=0 for all j. We prove an upper bound for the tail probabilities of the sum M _n=X₁+...+ X_n. Namely, we prove the inequality {M _nx} 3.7 {S_n x}, where S _n=₁+...+ _n is a sum of centered independent identically distributed Bernoulli random variables such that E S _n ² =ME M _n ² and {_k=1}=E S _n ² /(n+E S _n ² ) for all k (we call a random variable Bernoulli if it assumes at most two values). The inequality holds for x at which the survival function x{S _nx} has a jump down. For remaining x, the inequality still holds provided that we interpolate the function between the adjacent jump points linearly or log-linearly. If necessary, in order to estimate {S _nx} one can use special bounds for binomial probabilities. Up to the factor at most 2.375, the inequality is final. The inequality improves the classical Bernstein, Prokhorov, Bennett, Hoeffding, Talagrand, and other bounds.     

The truncated Witten genus

In this paper we define and examine the truncated Witten genus. It is defined as the equivariant index of the Dirac operator on the manifold Map(C_p,M) with its natural C_p-action. Here, Map(C_p,M) is the space of maps from the cyclic group of order p into a closed, connected, spin manifold, M. By applying the Atiyah-Singer index theorem we give a topological formula for the truncated Witten genus which is related to the formula for the Witten genus by truncation of the infinite products. We also show that the equivariant index of the Dirac operator on the projective space Map(C_p,ⁿ⁺¹) is closely related to the truncated Witten genus of ⁿ. The spaces Map(C_p,ⁿ⁺¹) define a filtration of the space Map(S¹,ⁿ⁺¹) which has been used to study equivariant objects on the smooth loop space of ⁿ.Acknowledgement This work was mainly done while the author was studying for a Ph.D. at Warwick University under the supervision of J. D. S. Jones. The author would like to thank John Jones for originally suggesting the problem and for his guidance whilst the author was at Warwick, to thank the EPSRC for funding his Ph.D., and to thank the referee for his detailed comments which helped greatly with the writing of this paper.     

A rigidity theorem for ℙ3(ℂ)

In this paper the following theorem is proved. Let X be a compact complex manifold of dimension 3 which carries 3 algebraically independent meromorphic functions. Assume that X is topologically isomorphic to the 3-dimensional projective space ₃. Then X is biholomorphically equivalent to ₃. In particular any deformation of ₃ is ₃.Dedicated to Karl Stein     

Northcott's theorem on heights I. A general estimate

Given a pointP=(a _o:...:a _n) of projective space ⁿ = ⁿ (A) whereA is the field of algebraic numbers, letd(P) be its degree andH(P) its absolute multiplicative height. Northcott's Theorem says that givend, n andX, there are only finitely many pointsP ⁿ withd(P)d andH(P)X. We will show that there are at mostc(d, n)X ^d(d+n) such points.Supported in part by NSF grant DMS-9108581.     

Normal surfaces of degree 7 in ℙ n

Let Y be a surface of degree 7 of the complex projective space ⁿ , linearly normal and not contained in any hyperplane of ⁿ . The purpose of this paper is to describe the normal singular surfaces of degree 7 embedded in ⁿ , as above. The main contribution of this paper is to describe all normal singular projective surfaces with sectional genus 4.     

Monads and moduli of vector bundles

Horrocks has shown that every vector bundle on ₂ and ₃ admits a certain double-ended resolution by line bundles, which he called a monad. We reprove Horrocks' results taking much care of uniqueness of the monads so obtained. This technique should be useful for constructing moduli spaces of stable vector bundles.     

Colouring finite incidence structures

One of our results: Let denote a finite projective plane of ordern. Colour its vertices by colours. If no colour appears more than twice on any of the lines, then n + 1. Explicit constructions show that this bound is sharp when is desarguesian.     

Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps

Summary. Let u and u_V V be the solution and, respectively, the discrete solution of the non-homogeneous Dirichlet problem u=f on , u|=0. For any m and any bounded polygonal domain , we provide a construction of a new sequence of finite dimensional subspaces V_n such that where f H^m–1() is arbitrary and C is a constant that depends only on and not on n (we do not assume u H^m+1()). The existence of such a sequence of subspaces was first proved in a ground–breaking paper by Babuka [8]. Our method is different; it is based on the homogeneity properties of Sobolev spaces with weights and the well–posedness of non-homogeneous Dirichlet problem in suitable Sobolev spaces with weights, for which we provide a new proof, and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary. Our results extended right away to domains whose boundaries have conical points. We also indicate some of the changes necessary to deal with domains with cusps. Our numerical computation are in agreement with our theoretical results.The authors were supported in part by the NSF grant DMS 02-09497. Victor Nistor was also partially supported by NSF grant DMS 02-00808.