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     

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.     

Maximal arcs in the complex projective plane and completely 2-valent functions

In this note we construct maximal arcs in the projective plane over the complex numbers ₂() that are essentially sliced open 2-spheres. In doing so we rely heavily on results from the theory of multivalent analytic functions.     

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.     

Legitimate colorings of projective planes

For a projective plane _n of ordern, let( _n ) denote the minimum numberk, so that there is a coloring of the points of _n ink colors such that no two distinct lines contain precisely the same number of points of each color. Answering a question of A. Rosa, we show that for all sufficiently largen, 5 ( _n ) 8 for every projective plane _n of ordern. Research supported in part by Allon Fellowship and by a grant from the United States Israel Binational Science Foundation     

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.     

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.     

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.     

A sharp bound for the number of points where a rank 3 stable reflexive sheaf is not free

Let ³ be the projective space over an algebraically closed field k and let E be a rank 3 stable reflexive sheaf on ³ such that its restriction to a general plane is stable. The aim of this paper is to give a sharp bound of s= lengthExt ¹ (E,G)in terms of the Chern classes of E.     

The trace to subsets ofR n of Besov spaces in the general case

R ⁿ. , , , F R ⁿ, F , R ⁿ R ⁿ . ^p,q (Rⁿ), >0, 1, q, — ( ) Rⁿ. , ^p,q (Rⁿ) ^F Rⁿ. , q B ^p,q (F), = – (n–)/, >0, — « », ad — F, . , . : , F=R ^d,F— « » F— R ⁿ, « », F. .

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This work has been supported in part by the Swedish Natural Science Research Council.     

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.     

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 .     

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     

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.     

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.     

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     

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.     

On the asymptotic behaviour of first passage times for transient random walk

Summary Let _x denote the time at which a random walk with finite positive mean first passes into (x, ), wherex0. This paper establishes the asymptotic behaviour of Pr { _x >n} asn for fixedx in two cases. In the first case the left hand tail of the step-distribution is regularly varying, and in the second the step-distribution satisfies a one-sided Cramér type condition. As a corollary, it follows that in the first case Pr { _x >n}/Pr{ ₀ >n} coincides with the limit of the same quantity for recurrent random walk satisfying Spitzer's condition, but in the second case the limit is more complicated.     

Automorphismen und Modulraum galoisscher dreiblättriger Überlagerungen

The points of the modulus space M _g ² of compact hyperelliptic Riemann surfaces of genus^g g can be identified, with the PGL (2,)-orbits in the set of all subsets of ¹, which contain 2g+2 elements. The automorphism group Aut(X) of such a surface contains a canonical involution J, and Aut(X)/J is isomorphic to the finite subgroup of Aut(¹), whose elements permute the critical values of some twofold cover. In this paper we derive similar results for surfaces X, which admit a threefold Galois cover over ¹.

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Herrn Prof. Dr. Dr. h.c. Karl Stein in Dankbarkeit gewidmet     

Rate of approximation by rectangular partial sums of double orthogonal series

( ) . .

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Dedicated to Professor K. Tandori on his seventieth birthday

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This research was supported in part by Grant # K41 100 of the Joint Fund of the Government of Ukraine and the International Science Foundation.