90 research outputs found
On the complete classification of extremal log Enriques surfaces
We show that there are exactly, up to isomorphisms, seven extremal log
Enriques surfaces Z and construct all of them; among them types D_{19} and
A_{19} have been shown of certain uniqueness by M. Reid. We also prove that the
(degree 3 or 2) canonical covering of each of these seven Z has either X_3 or
X_4 as its minimal resolution. Here X_3 (resp. X_4) is the unique K3 surface
with Picard number 20 and discriminant 3 (resp. 4), which are called the most
algebraic K3 surfaces by Vinberg and have infinite automorphism groups (by
Shioda-Inose and Vinberg).Comment: 22 pages. Math. Z. to appea
Automorphism groups of smooth quintic threefolds
We study automorphism groups of smooth quintic threefolds. Especially, we describe all the maximal ones with explicit examples of target quintic threefolds. There are exactly such groups
A simple remark on a flat projective morphism with a Calabi-Yau fiber
If a K3 surface is a fiber of a flat projective morphisms over a connected
noetherian scheme over the complex number field, then any smooth connected
fiber is also a K3 surface. Observing this, Professor Nam-Hoon Lee asked if the
same is true for higher dimensional Calabi-Yau fibers. We shall give an
explicit negative answer to his question as well as a proof of his initial
observation.Comment: 8 pages, main theorem is generalized, one more remark is added,
mis-calculation and typos are corrected etc
Birational automorphism groups of projective varieties of Picard number two
We slightly extend a result of Oguiso on birational or automorphism groups
(resp. of Lazi\'c - Peternell on Morrison-Kawamata cone conjecture) from
Calabi-Yau manifolds of Picard number two to arbitrary singular varieties X
(resp. to klt Calabi-Yau pairs in broad sense) of Picard number two. When X has
only klt singularities and is not a complex torus, we show that either Aut(X)
is almost cyclic, or it has only finitely many connected components.Comment: title slightly changed to this; some proof simplified; submitted to
the Proceedings of Groups of Automorphisms in Birational and Affine Geometry,
28 October - 3 November 2012, C.I.R.M., Trento, Ital
Cohomologically hyperbolic endomorphisms of complex manifolds
We show that if a compact Kahler manifold X admits a cohomologically
hyperbolic surjective endomorphism then its Kodaira dimension is non-positive.
This gives an affirmative answer to a conjecture of Guedj in the holomorphic
case. The main part of the paper is to determine the geometric structure and
the fundamental groups (up to finite index) for those X of dimension 3.Comment: International Journal of Mathematics (to appear
Points of Low Height on Elliptic Curves and Surfaces, I: Elliptic surfaces over P^1 with small d
For each of n=1,2,3 we find the minimal height h^(P) of a nontorsion point P
of an elliptic curve E over C(T) of discriminant degree d=12n (equivalently, of
arithmetic genus n), and exhibit all (E,P) attaining this minimum. The minimal
h^(P) was known to equal 1/30 for n=1 (Oguiso-Shioda) and 11/420 for n=2
(Nishiyama), but the formulas for the general (E,P) were not known, nor was the
fact that these are also the minima for an elliptic curve of discriminant
degree 12n over a function field of any genus. For n=3 both the minimal height
(23/840) and the explicit curves are new. These (E,P) also have the property
that that mP is an integral point (a point of naive height zero) for each
m=1,2,...,M, where M=6,8,9 for n=1,2,3; this, too, is maximal in each of the
three cases.Comment: 15 pages; some lines in the TeX source are commented out with "%" to
meet the 15-page limit for ANTS proceeding
Black Holes and Large Order Quantum Geometry
We study five-dimensional black holes obtained by compactifying M theory on
Calabi-Yau threefolds. Recent progress in solving topological string theory on
compact, one-parameter models allows us to test numerically various conjectures
about these black holes. We give convincing evidence that a microscopic
description based on Gopakumar-Vafa invariants accounts correctly for their
macroscopic entropy, and we check that highly nontrivial cancellations -which
seem necessary to resolve the so-called entropy enigma in the OSV conjecture-
do in fact occur. We also study analytically small 5d black holes obtained by
wrapping M2 branes in the fiber of K3 fibrations. By using heterotic/type II
duality we obtain exact formulae for the microscopic degeneracies in various
geometries, and we compute their asymptotic expansion for large charges.Comment: 42 pages, 20 eps figures, small correction
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