Serdica Mathematical Journal
Volume 23, Number 2, 1997
C O N T E N T S

Conference on Geometry and Mathematical Physics, Bulgaria, Zlatograd, 28.0801.09.1995
(pp. iiv)

Bakalov, B., E. Horozov, M. Yakimov.
Highest weight modules of W_{1+¥},
Darboux transformations and the bispectral problem
(pp. 95112)

Christov, O.
On the KAM  theory conditions for the Kirchhoff top
(pp. 113124)

Dragnev, Dr.
On the isoenergetical nondegeneracy of the problem of two centers
of gravitation
(pp. 125130)

Johnsen, T., S. L. Kleiman.
Toward Clemens' conjecture in degrees between 10 and 24
(pp. 131142)

Holzapfel, R.P.
Symplectic representation of a braid group on 3sheeted
covers of the Riemann sphere
(pp. 143164)

Kasparian, A.
When does a bounded domain cover a projective manifold? (Survey)
(pp. 165176)

Kuratsuji, H., T. Tochishita, M. Mizui.
Quantum mechanics of guiding center motion
in strong magnetic field (Coherent state path integral approach)
(pp. 177186)

Yotov, M.
Nadel's subschemes of Fano manifolds X with a
Picard group Pic(X) isomorphic to Z
(pp. 187202)
A B S T R A C T S
HIGHEST WEIGHT MODULES OF W_{1+¥},
DARBOUX TRANSFORMATIONS AND THE BISPECTRAL PROBLEM
B. Bakalov, E. Horozov, M. Yakimov
1991 Mathematics Subject Classification:
Primary  81R10, 58F07, 58F37; secondary  17B68, 17B69, 33C60.
Key words:
Bispectral operators, Darboux transformations, $W$algebras,
highest weight representations, KPhierarchy.
This paper is a survey of our recent results on the bispectral
problem. We describe a new method for constructing bispectral
algebras of any rank and illustrate the method by a series of
new examples as well as by all previously known ones. Next we
exhibit a close connection of the bispectral problem to the
representation theory of W_{1+¥}algerba. This connection
allows us to explain and generalise to any rank the result of
Magri and Zubelli on the symmetries of the manifold of the bispectral
operators of rank and order two.
ON THE KAM  THEORY CONDITIONS FOR THE KIRCHHOFF TOP
Ognyan Christov
1991 Mathematics Subject Classification:
58F07, 70Exx.
Key words:
KAMtheory, Kirchhoff top, Abelian integrals.
In this paper the classical Kirchhoff case of motion of a
rigid body in an infinite ideal fluid is considered. Then for
the corresponding Hamiltonian system on the zero integral level,
the KAM theory conditions are checked. In contrast to the known
similar results, there
exists a curve in the bifurcation diagram along which the
Kolmogorov's condition vanishes for certain values of the parameters.
ON THE ISOENERGETICAL NONDEGENERACY OF THE PROBLEM OF TWO CENTERS
OF GRAVITATION
Dragomir Dragnev
1991 Mathematics Subject Classification:
58F07, 70Exx.
Key words:
two centers of gravitation,
isoenergetical nondegeneracy.
For the system describing the motion of a moss point under the action of
two static gravity centers (with equal masses), we find a subset of the set
of the regular values of the energy and momentum, where the condition of
isoenergetical nondegeneracy is fulfilled.
TOWARD CLEMENS' CONJECTURE
IN DEGREES BETWEEN 10 AND 24
Trygve Johnsen, Steven L. Kleiman
1991 Mathematics Subject Classification:
Primary 14J30, secondary: 14H45, 14N10.
Key words:
rational curves, quintic threefold.
We introduce and study a likely condition that implies the following
form of Clemens' conjecture in degrees d between 10 and 24: given a general quintic threefold F in complex P^{4}, the Hilbert scheme of rational, smooth and irreducible curves C of degree d on F is
finite, nonempty, and reduced; moreover, each C is embedded in F
with balanced normal sheaf O(1)ÅO(1), and in P^{4} with
maximal rank.
SYMPLECTIC REPRESENTATION OF A BRAID GROUP ON 3SHEETED
COVERS OF THE RIEMANN SPHERE
RolfPeter Holzapfel
1991 Mathematics Subject Classification:
11G15, 11G18, 11H56, 11R11, 14D05, 14D22, 14E20, 14G35, 14H10,
14H30, 14H40, 14H5, 14J10, 14J15, 14J25, 14K10, 14K20, 14K22, 14M17,
20C12, 20H05, 20H10, 32G20, 32M15, 33C40
Key words:
algebraic curves, abelian threefolds, period matrices, moduli spaces,
Shimura surface, Siegel domain, complex unit ball, uniformization,
braid group, monodromy group, modular group, gundamental groups,
PicardFuchsian groups, symplectic group, aritmetic group,
representation, quadratic number field.
We define Picard cycles on each smooth threesheeted Galois cover C
of the Riemann sphere. The moduli space of all these algebraic curves
is a nice Shimura surface, namely a symmetric quotient of the projective
plane uniformized by the complex twodimensional unit ball. We show that
all Picard cycles on C form a simple orbit of the Picard modular group
of Eisenstein numbers. The proof uses a special surface classification in
connection with the uniformization of a classical PicardFuchs system. It
yields an explicit symplectic representation of the braid groups (coloured
or not) of four strings.
WHEN DOES A BOUNDED DOMAIN COVER A PROJECTIVE MANIFOLD?
(SURVEY)
Azniv Kasparian
1991 Mathematics Subject Classification:
32J25, 14E20, 32M05.
Key words:
Uniformization of compact complex manifolds
with ample canonical bundle, lattices in semisimple
algebraic groups, bounded
domains with and without compact discrete quotients.
The present survey introduces in some classical properties of the
universal coverings of the projective algebraic manifolds. All the
results are nonoriginal. A forthcoming note is intended to discuss
the corresponding fundamental groups.
QUANTUM MECHANICS OF GUIDING CENTER MOTION
IN STRONG MAGNETIC FIELD (COHERENT STATE PATH INTEGRAL APPROACH)
Hiroshi Kuratsuji, Teruyuki Tochishita, Masayuki Mizui
1991 Mathematics Subject Classification:
58D30.
Key words:
Landau level, guiding ceter, coherent state.
A new approach is proposed for the quantum mechanics of
guiding center motion in strong magnetic field.
This is achieved by use of the coherent state path integral
for the coupled systems of the cyclotron and the guiding center
motion. We are specifically concerned with the effective action
for the guiding center degree, which can be used to
get the BohrSommerfeld quantization scheme.
The quantization rule is similar to
the one for the vortex motion as a dynamics of
point particles.
NADEL'S SUBSCHEMES OF FANO MANIFOLDS X WITH A
PICARD GROUP Pic(X) ISOMORPHIC TO Z
M. Yotov
1991 Mathematics Subject Classification:
32L10, 32L20, 32F05.
Key words:
Fano manifolds,
holomorphic vector bundles, vanishing theorems, Nadel's subschemes.
In this paper we find a global sufficient condition for suitable
subschemes of
Fano manifolds to be Nadel's subschemes. We apply this condition to
onedimensional subschemes of a projective space.
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