364 research outputs found
Effective algebraic degeneracy
We prove that any nonconstant entire holomorphic curve from the complex line
C into a projective algebraic hypersurface X = X^n in P^{n+1}(C) of arbitrary
dimension n (at least 2) must be algebraically degenerate provided X is generic
if its degree d = deg(X) satisfies the effective lower bound: d larger than or
equal to n^{{(n+1)}^{n+5}}
The Moduli of Reducible Vector Bundles
A procedure for computing the dimensions of the moduli spaces of reducible,
holomorphic vector bundles on elliptically fibered Calabi-Yau threefolds X is
presented. This procedure is applied to poly-stable rank n+m bundles of the
form V + pi* M, where V is a stable vector bundle with structure group SU(n) on
X and M is a stable vector bundle with structure group SU(m) on the base
surface B of X. Such bundles arise from small instanton transitions involving
five-branes wrapped on fibers of the elliptic fibration. The structure and
physical meaning of these transitions are discussed.Comment: 33+1 page
Small bound for birational automorphism groups of algebraic varieties (with an Appendix by Yujiro Kawamata)
We give an effective upper bound of |Bir(X)| for the birational automorphism
group of an irregular n-fold (with n = 3) of general type in terms of the
volume V = V(X) under an ''albanese smoothness and simplicity'' condition. To
be precise, |Bir(X)| < d_3 V^{10}. An optimum linear bound |Bir(X)|-1 <
(1/3)(42)^3 V is obtained for those 3-folds with non-maximal albanese
dimension. For all n > 2, a bound |Bir(X)| < d_n V^{10} is obtained when alb_X
is generically finite, alb(X) is smooth and Alb(X) is simple.Comment: Mathematische Annalen, to appea
Anti-Pluricanonical Systems On Q-Fano Threefolds
We investigate birationality of the anti-pluricanonical map , the
rational map defined by the anti-pluricanonical system , on
-Fano threefolds.Comment: 18 page
Analytic curves in algebraic varieties over number fields
We establish algebraicity criteria for formal germs of curves in algebraic
varieties over number fields and apply them to derive a rationality criterion
for formal germs of functions, which extends the classical rationality theorems
of Borel-Dwork and P\'olya-Bertrandias valid over the projective line to
arbitrary algebraic curves over a number field.
The formulation and the proof of these criteria involve some basic notions in
Arakelov geometry, combined with complex and rigid analytic geometry (notably,
potential theory over complex and -adic curves). We also discuss geometric
analogues, pertaining to the algebraic geometry of projective surfaces, of
these arithmetic criteria.Comment: 55 pages. To appear in "Algebra, Arithmetic, and Geometry: In Honor
of Y.i. Manin", Y. Tschinkel & Yu. Manin editors, Birkh\"auser, 200
Simple geometrical interpretation of the linear character for the Zeno-line and the rectilinear diameter
The unified geometrical interpretation of the linear character of the
Zeno-line (unit compressibility line Z=1) and the rectilinear diameter is
proposed. We show that recent findings about the properties of the Zeno-line
and striking correlation with the rectilinear diameter line as well as other
empirical relations can be naturally considered as the consequences of the
projective isomorphism between the real molecular fluids and the lattice gas
(Ising) model.Comment: 7 pages, 2 figure
Big Line Bundles over Arithmetic Varieties
We prove a Hilbert-Samuel type result of arithmetic big line bundles in
Arakelov geometry, which is an analogue of a classical theorem of Siu. An
application of this result gives equidistribution of small points over
algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also
generalize Chambert-Loir's non-archimedean equidistribution
Borcherds symmetries in M-theory
It is well known but rather mysterious that root spaces of the Lie
groups appear in the second integral cohomology of regular, complex, compact,
del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms)
of toroidal compactifications of M theory. Their Borel subgroups are actually
subgroups of supergroups of finite dimension over the Grassmann algebra of
differential forms on spacetime that have been shown to preserve the
self-duality equation obeyed by all bosonic form-fields of the theory. We show
here that the corresponding duality superalgebras are nothing but Borcherds
superalgebras truncated by the above choice of Grassmann coefficients. The full
Borcherds' root lattices are the second integral cohomology of the del Pezzo
surfaces. Our choice of simple roots uses the anti-canonical form and its known
orthogonal complement. Another result is the determination of del Pezzo
surfaces associated to other string and field theory models. Dimensional
reduction on corresponds to blow-up of points in general position
with respect to each other. All theories of the Magic triangle that reduce to
the sigma model in three dimensions correspond to singular del Pezzo
surfaces with (normal) singularity at a point. The case of type I and
heterotic theories if one drops their gauge sector corresponds to non-normal
(singular along a curve) del Pezzo's. We comment on previous encounters with
Borcherds algebras at the end of the paper.Comment: 30 pages. Besides expository improvements, we exclude by hand real
fermionic simple roots when they would naively aris
Stability conditions and positivity of invariants of fibrations
We study three methods that prove the positivity of a natural numerical
invariant associated to parameter families of polarized varieties. All
these methods involve different stability conditions. In dimension 2 we prove
that there is a natural connection between them, related to a yet another
stability condition, the linear stability. Finally we make some speculations
and prove new results in higher dimension.Comment: Final version, to appear in the Springer volume dedicated to Klaus
Hulek on the occasion of his 60-th birthda
Complex-Temperature Properties of the Ising Model on 2D Heteropolygonal Lattices
Using exact results, we determine the complex-temperature phase diagrams of
the 2D Ising model on three regular heteropolygonal lattices, (kagom\'{e}), , and (bathroom
tile), where the notation denotes the regular -sided polygons adjacent to
each vertex. We also work out the exact complex-temperature singularities of
the spontaneous magnetisation. A comparison with the properties on the square,
triangular, and hexagonal lattices is given. In particular, we find the first
case where, even for isotropic spin-spin exchange couplings, the nontrivial
non-analyticities of the free energy of the Ising model lie in a
two-dimensional, rather than one-dimensional, algebraic variety in the
plane.Comment: 31 pages, latex, postscript figure
- …