15 research outputs found
On the canonical degrees of curves in varieties of general type
A widely believed conjecture predicts that curves of bounded geometric genus
lying on a variety of general type form a bounded family. One may even ask
whether the canonical degree of a curve in a variety of general type is
bounded from above by some expression , where and are
positive constants, with the possible exceptions corresponding to curves lying
in a strict closed subset (depending on and ). A theorem of Miyaoka
proves this for smooth curves in minimal surfaces, with . A conjecture
of Vojta claims in essence that any constant is possible provided one
restricts oneself to curves of bounded gonality.
We show by explicit examples coming from the theory of Shimura varieties that
in general, the constant has to be at least equal to the dimension of the
ambient variety.
We also prove the desired inequality in the case of compact Shimura
varieties.Comment: 10 pages, to appear in Geometric and Functional Analysi
Logarithmic Moduli Spaces for Surfaces of Class VII
In this paper we describe logarithmic moduli spaces of pairs (S,D) consisting
of minimal surfaces S of class VII with positive second Betti number b_2
together with reduced divisors D of b_2 rational curves. The special case of
Enoki surfaces has already been considered by Dloussky and Kohler. We use
normal forms for the action of the fundamental group of the complement of D and
for the associated holomorphic contraction germ from (C^2,0) to (C^2,0).Comment: Minor correction of the dimension of the moduli spac
On semistable principal bundles over a complex projective manifold, II
Let (X, \omega) be a compact connected Kaehler manifold of complex dimension
d and E_G a holomorphic principal G-bundle on X, where G is a connected
reductive linear algebraic group defined over C. Let Z (G) denote the center of
G. We prove that the following three statements are equivalent: (1) There is a
parabolic subgroup P of G and a holomorphic reduction of the structure group of
E_G to P (say, E_P) such that the bundle obtained by extending the structure
group of E_P to L(P)/Z(G) (where L(P) is the Levi quotient of P) admits a flat
connection; (2) The adjoint vector bundle ad(E_G) is numerically flat; (3) The
principal G-bundle E_G is pseudostable, and the degree of the charateristic
class c_2(ad(E_G) is zero.Comment: 15 page
Stability of the Minimal Heterotic Standard Model Bundle
The observable sector of the "minimal heterotic standard model" has precisely
the matter spectrum of the MSSM: three families of quarks and leptons, each
with a right-handed neutrino, and one Higgs-Higgs conjugate pair. In this
paper, it is explicitly proven that the SU(4) holomorphic vector bundle leading
to the MSSM spectrum in the observable sector is slope-stable.Comment: LaTeX, 19 page
The Tate conjecture for K3 surfaces over finite fields
Artin's conjecture states that supersingular K3 surfaces over finite fields
have Picard number 22. In this paper, we prove Artin's conjecture over fields
of characteristic p>3. This implies Tate's conjecture for K3 surfaces over
finite fields of characteristic p>3. Our results also yield the Tate conjecture
for divisors on certain holomorphic symplectic varieties over finite fields,
with some restrictions on the characteristic. As a consequence, we prove the
Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite
fields of characteristic p>3.Comment: 20 pages, minor changes. Theorem 4 is stated in greater generality,
but proofs don't change. Comments still welcom
Integral points on punctured abelian surfaces
We study the density of integral points on punctured abelian surfaces. Linear growth rates are observed experimentally