On the second adjunction mapping. The case of a 1-dimensional image

Let be a very ample line bundle on an -dimensional projective manifold , i.e., assume that for some embedding . In this article, a study is made of the meromorphic map, , associated to in the case when the Kodaira dimension of is , and has a -dimensional image. Assume for simplicity that . The first main result of the paper shows that is a morphism if either or . The second main result of this paper shows that if , then the genus, , of a fiber, , of the map induced by on hyperplane sections is . Moreover, if then , a connected component of a general fiber of is either a surface or the blowing up at one point of a surface, and . Finally the structure of the finite to one part of the Remmert-Stein factorization of is worked out.

     

Hecke algebras,

The Donald-Flanigan conjecture asserts that the integral group ring of a finite group can be deformed to an algebra over the power series ring with underlying module such that if is any prime dividing then is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of We prove this for using the natural representation of its Hecke algebra by quantum Yang-Baxter matrices to show that over localized at the multiplicatively closed set generated by and all , the Hecke algebra becomes a direct sum of total matric algebras. The corresponding ``canonical" primitive idempotents are distinct from Wenzl's and in the classical case (), from those of Young.

     

Extensions of modules over Weyl algebras

In this paper we calculate some groups of singular modules over the complex Weyl algebra . In particular we determine conditions under which is an infinite dimensional vector space when or .

     

The second variation of nonorientable minimal submanifolds

Suppose is a complete nonorientable minimal submanifold of a Riemannian manifold . We derive a second variation formula for the area of with respect to certain perturbations, giving a sufficient condition for the instability of . Some simple applications are given: we show that the totally geodesic is the only stable surface in , and we show the non-existence of stable nonorientable cones in . We reproduce and marginally extend some known results in the truly non-compact setting.

     

Strict definiteness of integrals via complete monotonicity of derivatives

Let be a nonnegative integer and let be a function with completely monotone and not constant. If is a signed measure on any euclidean space , with vanishing moments up to order , then the integral is strictly positive whenever it exists. For general no larger class of continuous functions seems to admit the same conclusion. Examples and applications are indicated. A section on 'bilinear integrability' might be of independent interest.

     

Only single twists on unknots can produce composite knots

Let be a knot in the -sphere , and a disc in meeting transversely more than once in the interior. For non-triviality we assume that over all isotopy of . Let () be a knot obtained from by cutting and -twisting along the disc (or equivalently, performing -Dehn surgery on ). Then we prove the following: (1) If is a trivial knot and is a composite knot, then ; (2) if is a composite knot without locally knotted arc in and is also a composite knot, then . We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.

     

Kaehler structures on

Let be a compact connected semi-simple Lie group, let , and let be an Iwasawa decomposition. To a given -invariant Kaehler structure on , there corresponds a pre-quantum line bundle on . Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections as a -representation space. We defined a -invariant -structure on , and let denote the space of square-integrable holomorphic sections. Then is a unitary -representation space, but not all unitary irreducible -representations occur as subrepresentations of . This paper serves as a continuation of that work, by generalizing the space considered. Let be a Borel subgroup containing , with commutator subgroup . Instead of working with , we consider , for all parabolic subgroups containing . We carry out a similar construction, and recover in the unitary irreducible -representations previously missing. As a result, we use these holomorphic sections to construct a model for : a unitary -representation in which every irreducible -representation occurs with multiplicity one.

     

Boundary limits and non-integrability of )

In this paper we consider weighted non-tangential and tangential boundary limits of non-negative functions on the unit ball in that are subharmonic with respect to the Laplace-Beltrami operator on . Since the operator is invariant under the group of holomorphic automorphisms of , functions that are subharmonic with respect to are usually referred to as -subharmonic functions. Our main result is as follows: Let be a non-negative -subharmonic function on satisfying

for some and some , where is the -invariant measure on . Suppose . Then for a.e. ,

uniformly as in each , where for ( when )

We also prove that for the only non-negative -subharmonic function satisfying the above integrability criteria is the zero function.

     

Symmetric powers of complete modules over a two-dimensional regular local ring

Let be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free -module , write for the th symmetric power of , mod torsion. We study the modules , , when is complete (i.e., integrally closed). In particular, we show that , for any minimal reduction and that the ring is Cohen-Macaulay.

     

Differential operators on Stanley-Reisner rings

Let be an algebraically closed field of characteristic zero, and let be a polynomial ring. Suppose that is an ideal in that may be generated by monomials. We investigate the ring of differential operators on the ring , and , the idealiser of in . We show that and are always right Noetherian rings. If is a square-free monomial ideal then we also identify all the two-sided ideals of . To each simplicial complex on there is a corresponding square-free monomial ideal , and the Stanley-Reisner ring associated to is defined to be . We find necessary and sufficient conditions on for to be left Noetherian.

     

Coherent functors, with application to torsion in the Picard group

Let be a commutative noetherian ring. We investigate a class of functors from commutative -algebras to sets, which we call coherent. When such a functor in fact takes its values in abelian groups, we show that there are only finitely many prime numbers such that is infinite, and that none of these primes are invertible in . This (and related statements) yield information about torsion in . For example, if is of finite type over , we prove that the torsion in is supported at a finite set of primes, and if is infinite, then the prime is not invertible in . These results use the (already known) fact that if such an is normal, then is finitely generated. We obtain a parallel result for a reduced scheme of finite type over . We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field.

     

Automorphism Groups and Invariant Subspace Lattices

Let be a - dynamical system and let be the analytic subalgebra of . We extend the work of Loebl and the first author that relates the invariant subspace structure of for a -representation on a Hilbert space , to the possibility of implementing on We show that if is irreducible and if lat is trivial, then is ultraweakly dense in We show, too, that if satisfies what we call the strong Dirichlet condition, then the ultraweak closure of is a nest algebra for each irreducible representation Our methods give a new proof of a ``density' theorem of Kaftal, Larson, and Weiss and they sharpen earlier results of ours on the representation theory of certain subalgebras of groupoid -algebras.

     

A bound on the geometric genus of projective varieties verifying certain flag conditions

Fix integers and let be the set of all integral, projective and nondegenerate varieties of degree and dimension in the projective space , such that, for all , does not lie on any variety of dimension and degree . We say that a variety satisfies a flag condition of type if belongs to . In this paper, under the hypotheses , we determine an upper bound , depending only on , for the number , where denotes the geometric genus of . In case and , the study of an upper bound for the geometric genus has a quite long history and, for , and , it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical data . For instance, we extend Halphen's theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension in as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces in . Next we discuss how far is from and show a sort of lifting theorem which states that, at least in certain cases, the varieties of maximal geometric genus must in fact lie on a flag such as , where denotes a subvariety of of degree and dimension . We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo's regularity of varieties verifying flag conditions.

     

An index formula for elliptic systems in the plane

An index formula is proved for elliptic systems of P.D.E.'s with boundary values in a simply connected region in the plane. Let denote the elliptic operator and the boundary operator. In an earlier paper by the author, the algebraic condition for the Fredholm property, i.e. the Lopatinskii condition, was reformulated as follows. On the boundary, a square matrix function defined on the unit cotangent bundle of was constructed from the principal symbols of the coefficients of the boundary operator and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator. The Lopatinskii condition is equivalent to the condition that the function have invertible values. In the present paper, the index of is expressed in terms of the winding number of the determinant of .

     

A Characterization of Finitely Decidable Congruence Modular Varieties

For every finitely generated, congruence modular variety of finite type we find a finite family of finite rings such that the variety is finitely decidable if and only if is congruence permutable and residually small, all solvable congruences in finite algebras from are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebra from is comparable with all congruences of , each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ring from , the variety of -modules is finitely decidable.

     

The Mizohata complex

This paper deals with the local solvability of systems of first order linear partial differential equations defined by a germ at of a -valued, formally integrable (), 1-form with nondegenerate Levi form. More precisely, the size of the obstruction to the solvability, for -forms , of the equation

where is a given -form satisfying is estimated in terms of the De Rham cohomology relative to

     

Wavelet transform and orthogonal decomposition of

Let be the Weyl-Poincaré group and be the Iwasawa decomposition of with . Then the ``affine Weyl-Poincaré group' can be realized as the complex tube domain or the classical Cartan domain . The square-integrable representations of and give the admissible wavelets and wavelet transforms. An orthogonal basis of the set of admissible wavelets associated to is constructed, and it gives an orthogonal decomposition of space on (or the Cartan domain
) with every component being the range of wavelet transforms of functions in with .

     

Liouville type theorems for fourth order elliptic equations in a half plane

Consider an elliptic equation in the half plane with boundary conditions if and if where are second and third order differential operators. It is proved that if and, for some , if if where for some nonnegative integer , then . Results of this type are also established in case under different conditions on and ; furthermore, in one case has a lower order term which depends nonlocally on . Such Liouville type theorems arise in the study of coating flow; in fact, they play a crucial role in the analysis of the linearized version of this problem. The methods developed in this paper are entirely different for the two cases (i) and (ii) ; both methods can be extended to other linear elliptic boundary value problems in a half plane.

     

Partial regularity of solutions to a class of degenerate systems

We consider the system , in coupled with suitable initial-boundary conditions, where is a bounded domain in with smooth boundary and is a continuous and positive function of . Our main result is that under some conditions on there exists a relatively open subset of such that is locally Hölder continuous on , the interior of is empty, and is essentially bounded on .

     

The generalized Berg theorem and BDF-theorem

Let be a separable simple -algebra with finitely many extreme traces. We give a necessary and sufficient condition for an essentially normal element , i.e., is normal ( is the quotient map), having the form for some normal element and We also show that a normal element can be quasi-diagonalized if and only if the Fredholm index for all In the case that is a simple -algebra of real rank zero, with stable rank one and with continuous scale, and has countable rank, we show that a normal element with zero Fredholm index can be written as

where is an (increasing) approximate identity for consisting of projections, is a bounded sequence of numbers and with for any given