Showing papers in "Mathematische Annalen in 1982"

Factoring Polynomials with Rational Coefficients

Abstract: In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

Confluent hypergeometric functions on tube domains

Abstract: Our problem specialized to the case of the Siegel upper half space H,. of degree m concerns the Fourier expansion of a series S(z; ~, fl) = ~ det (z + a) -" det (~ + a)~. a~L Here z is a variable on H.~, L is a lattice in the space V of all real symmetric matrices of size m, and (~,fl)eC ~. It can be shown that this is convergent if Re (~+f l )>m, and also that if Re(~)>m/2 and R e ( f l ) > m / 2 , it has a Fourier expansion of the form

Semistable sheaves on projective varieties and their restriction to curves

Abstract: Let X be a nonsingular projective variety of dimension n over an algebraically closed field k. Let H be a very ample line bundle on X. If V is a torsion free coherent sheaf on X we define deg V to be cl(V), c~(H)"~ and/~(V) = deg V/rk V. We call V sernistable (resp. stable) if for all proper subsheaves W of V we have #(W) < #(V) [resp. #(W) < #(V)] (cf. [7, 14]). In this paper we prove that if V is semistable on X then its restriction to a general complete intersection curve of sufficiently high degree is semistable (Theorem 6.1). To give an idea of the proof assume X is a surface and V a vector bundle of rank 2. The restriction of V to a general curve C" of degree m is not semistable if and only if it is not semistable on the generic curve Ym defined over the function field of IPH~ H"). Let/S," be the line bundle on Y,, contradicting the semistability of VI Y,, (cf. Sects. 4.1 and 4.2). First we show that L,, extends uniquely to a line bundle L m on X (Proposition 2.1). If we can get L" as a subbundle of V we are through, for then L" would contradict the semistability of V. So we would like the restriction map H~ Hom(Lm, V))~H~ Hom(L m, V)) to be surjective. Now for fixed L it follows from the lemma of Enriques-Severi (Proposition 3.2; [6, Corollary 7.8]) that H~ Horn(L, V))~H~ Horn(L, V)) is surjective for large m. Therefore it is enough if the L" remain the same line bundle L for infinitely many m. To prove that L,, = L we construct a degenerating family of curves D f ~S, X x S 3 D p ~X, such that the generic fibre is a curve Ct"+ 1) of degree 2 "+ 1 and the special fibre is a reduced curve with two nonsingular components CI") of degree 2" (cf. Sect. 5). Let (m) denote 2". Extending the subbundle Lt,,+~)[Ct"+~) to a subsheaf of p*(V) on D and restricting the extension to CI") gives a lower bound for the maximal degree of a line subbundle of V[CIm ) in terms of that for V[Ctm+I ) (Proposition 4.3). This implies that degL,, is bounded (Lemma 6.5.1) so that for an infinite subsequence of m, degL,, is constant. If degLtm + r)= degLt,,) by refining the above argument with the degenerating family one can prove that Lt"+,)[ CI")

On the geometry of a Siegel modular threefold

Abstract: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 1. Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 2. Humbert Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 3. The Chern Numbers of S~(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 4. Symmetric Quartic Threefolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 5. The Modular Threefold of Level 2 . . . . . . . . . . . . . . . . . . . . . . . . . 335 6. Kummer Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 7. Prym Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 8. Humbert Surfaces Again . . . . . . . . . . . . . . . . : . . . . . . . . . . . . 345

Axial isometries of manifolds of non-positive curvature

Abstract: Let M be a complete C ~~ Riemannian manifold of non-positive sectional curvature. We say that a geodesic 9: IR~ M bounds a fiat strip of width c > 0 (a fiat half plane) if there is a totally geodesic, isometric immersion i: [0, c) x IR~M(i: [0, oo) x IR~M) such that i(0, t) = 9(0. A 9eodesic without fiat strip (without fiat half plane) is a geodesic, which does not bound a flat strip (a flat half plane). We will prove that the existence of a closed geodesic without flat half plane has rather strong consequences for the geometry and topology of M. In fact, many of the properties of a manifold of strictly negative curvature (resp. of a visibility manifold) still remain true if one assumes only the existence of a closed geodesic without flat half plane. We will discuss the existence of free (non-Abelian) subgroups of gl(M), the existence of infinitely many closed geodesics, the density of closed geodesics, and a transitivity property of the geodesic flow. It is, therefore, interesting to give conditions which ensure the existence of a closed geodesic without flat half plane. We will prove that M has a closed geodesic without flat half plane if vol(M)< oo and if M contains a geodesic without flat half plane. Note that a geodesic is not boundary of a flat strip (and a fortiori not boundary of a flat half plane) if it passes through a point p e M such that the sectional curvature of all tangent planes at p is negative. In the proofs of our results we investigate the action of rtl(M ) as group of isometries on the universal covering space H of M. In the proofs of many of our results we do not use the fact that this action is properly discontinuous and free. We, therefore, formulate these results for arbitrary groups D of isometries of H. The paper is organized as follows: In Sect. 1 we fix some definitions and notations and quote some standard results of non-positive curvature. Section 2 is the central section of this paper. We investigate the properties of those isometries of H which correspond to closed geodesics in M. We also prove

Arrangements defined by unitary reflection groups

Abstract: Let V be a complex vector space of dimension I. An arrangement in V is a finite set d of hyperplanes, all containing the origin. Let L = L ( d ) be the set of intersections of elements of ~r Partially order L by reverse inclusion so that L has V as its minimal element and d as its set of atoms. The poset L is a finite geometric lattice with rank function r(X)= dim V/X, XE L. Without loss of generality we assume that N H ~ , H =0 is the maximal element of L and thus L has rank I. The characteristic polynomial z(L, t) of L is defined by

Proper self maps of weakly pseudoconvex domains

Abstract: This theorem was proved by Burns and Shnider I-5] in the case that D is strongly pseudoconvex. In the more general case, Pin6uk 1-9, Corollary 2] has reduced the problem of showing that a proper mapping f : D ~ D is biholomorphic to proving that f is unbranched, i.e., that det [ f ' ] + 0. Our interest in the theorem above stems from the fact that if f : D t ~ D 2 is a proper holomorphic mapping of a weakly pseudoconvex domain D 1 onto a second domain Dz, it is possible for f to branch if D1 +D 2. This is contrary to what may occur if D 1 is assumed to be strongly pseudoconvex (see Alexander [1], Pin6uk [-9, 10], Fornaess 1-8], Bell 1,3], Diederich and Fornaess 1"7]). It has recently been proved in 1-4] and in I-6] that the mapping f in the theorem extends smoothly to/) . This fact is basic to our proof. The theorem may be shown to remain valid if 112" is replaced by a Stein manifold (see [2]), but si~ace-,the arguments of [4] and [6] do not carry over automatically to this case, the methods used in 1-2] are more technical.