## Recursos de colección

1. #### Canonical filtrations and stability of direct images by Frobenius morphisms

We study the stability of direct images by Frobenius morphisms. First, we compute the first Chern classes of direct images of vector bundles by Frobenius morphisms modulo rational equivalence up to torsions. Next, introducing the canonical filtrations, we prove that if $X$ is a nonsingular projective minimal surface of general type with semistable $\Omega_X^1$ with respect to the canonical line bundle $K_X$, then the direct images of line bundles on $X$ by Frobenius morphisms are semistable with respect to $K_X$.

2. #### Associate and conjugate minimal immersions in $\boldsymbol{M} \times \boldsymbol{R}$

Hauswirth, Laurent; Sa Earp, Ricardo; Toubiana, Eric
We establish the definition of associate and conjugate conformal minimal isometric immersions into the product spaces, where the first factor is a Riemannian surface and the other is the set of real numbers. When the Gaussian curvature of the first factor is nonpositive, we prove that an associate surface of a minimal vertical graph over a convex domain is still a vertical graph. This generalizes a well-known result due to R. Krust. Focusing the case when the first factor is the hyperbolic plane, it is known that in certain class of surfaces, two minimal isometric immersions are associate. We show...

3. #### Maximal slices in anti-de Sitter spaces

Li, Zhenyang; Shi, Yuguang
We prove the existence of maximal slices in anti-de Sitter spaces (ADS spaces) with small boundary data at spatial infinity. The main argument is carried out by implicit function theorem. We also get a necessary and sufficient condition for the boundary behavior of totally geodesic slices in ADS spaces. Moreover, we show that any isometric and maximal embedding of hyperbolic spaces into ADS spaces must be totally geodesic. Combined with this, we see that most of maximal slices obtained in this paper are not isometric to hyperbolic spaces, which implies that the Bernstein Theorem in ADS space fails.

4. #### Commutation relations of Hecke operators for Arakawa lifting

Murase, Atsushi; Narita, Hiro-aki
T. Arakawa, in his unpublished note, constructed and studied a theta lifting from elliptic cusp forms to automorphic forms on the quaternion unitary group of signature $(1, q)$. The second named author proved that such a lifting provides bounded (or cuspidal) automorphic forms generating quaternionic discrete series. In this paper, restricting ourselves to the case of $q=1$, we reformulate Arakawa's theta lifting as a theta correspondence in the adelic setting and determine a commutation relation of Hecke operators satisfied by the lifting. As an application, we show that the theta lift of an elliptic Hecke eigenform is also a Hecke...

5. #### Smooth Fano polytopes can not be inductively constructed

Øbro, Mikkel
We examine a concrete smooth Fano 5-polytope $P$ with 8 vertices with the following properties: There does not exist a smooth Fano 5-polytope $Q$ with 7 vertices such that $P$ contains $Q$, and there does not exist a smooth Fano 5-polytope $R$ with 9 vertices such that $R$ contains $P$. As the polytope $P$ is not pseudo-symmetric, it is a counter example to a conjecture proposed by Sato.

6. #### Twisted Kummer and Kummer-Artin-Schreier theories

Suwa, Noriyuki
We discuss an analogue of the Kummer and Kummer-Artin-Schreier theories, twisting by a quadratic extension. The argument is developed not only over a field but also over a ring, as generally as possible.

7. #### Inflection points and double tangents on anti-convex curves in the real projective plane

Thorbergsson, Gudlugur; Umehara, Masaaki
A simple closed curve in the real projective plane is called anti-convex if for each point on the curve, there exists a line which is transversal to the curve and meets the curve only at that given point. Our main purpose is to prove an identity for anti-convex curves that relates the number of independent (true) inflection points and the number of independent double tangents on the curve. This formula is a refinement of the classical Möbius theorem. We also show that there are three inflection points on a given anti-convex curve such that the tangent lines at these three...

8. #### On Galois groups of abelian extensions over maximal cyclotomic fields

We shall consider the maximal cyclotomic extension of a finite algebraic number field and its two abelian extensions, the maximal abelian extension and the maximal abelian extension with certain restricted ramification. We shall investigate the structure of these Galois groups with the action of the cyclotomic Galois group.

9. #### Wedge product of positive currents and balanced manifolds

Alessandrini, Lucia; Bassanelli, Giovanni
We define on a manifold $X$ a wedge product $S \wedge T$ of a closed positive (1,1)-current $S$, smooth outside a proper analytic subset $Y$ of $X$, and a positive pluriharmonic $(k,k)$-current $T$, when $k$ is less than the codimension of $Y$. Using this tool, we prove that if $M$ is a compact complex manifold of dimension $n \geq 3$, which is Kähler outside an irreducible curve, then $M$ carries a balanced metric.

10. #### The structure of weakly stable constant mean curvature hypersurfaces

Cheng, Xu; Cheung, Leung-fu; Zhou, Detang
We study the global behavior of weakly stable constant mean curvature hypersurfaces in a Riemannian manifold by using harmonic function theory. In particular, a complete oriented weakly stable minimal hypersurface in the Euclidean space must have only one end. Any complete noncompact weakly stable hypersurface with constant mean curvature $H$ in the 4 and 5 dimensional hyperbolic spaces has only one end under some restrictions on $H$.

11. #### Mixed Hodge structures on log smooth degenerations

Fujisawa, Taro
We introduce the notion of a log smooth degeneration, which is a logarithmic analogue of the central fiber of some kind of degenerations of complex manifolds over polydiscs. Under suitable conditions, we construct a natural cohomological mixed Hodge complex on the reduction of a compact log smooth degeneration. In particular, we obtain mixed Hodge structures on the log de Rham cohomologies and $E_1$-degeneration of the log Hodge to de Rham spectral sequence for a certain kind of compact reduced log smooth degenerations.

12. #### Finite time dead-core rate for the heat equation with a strong absorption

Guo, Jong-Shenq; Wu, Chin-Chin
We study the solution of the heat equation with a strong absorption. It is well-known that the solution develops a dead-core in finite time for a large class of initial data. It is also known that the exact dead-core rate is faster than the corresponding self-similar rate. By using the idea of matching, we formally derive the exact dead-core rates under a dynamical theory assumption. Moreover, we also construct some special solutions for the corresponding Cauchy problem satisfying this dynamical theory assumption. These solutions provide some examples with certain given polynomial rates.

13. #### Interpolation of Markoff transformations on the Fricke surface

Sasaki, Takeshi; Yoshida, Masaaki
By the Fricke surfaces, we mean the cubic surfaces defined by the equation $p^2+q^2+r^2-pqr-k=0$ in the Euclidean 3-space with the coordinates $(p,q,r)$ parametrized by constant $k$. When $k=0$, it is naturally isomorphic to the moduli of once-punctured tori. It was Markoff who found the transformations, called Markoff transformations, acting on the Fricke surface. The transformation is typically given by $(p,q,r)\mapsto (r,q,rq-p)$ acting on $\boldsymbol{R}^3$ that keeps the surface invariant. In this paper we propose a way of interpolating the action of Markoff transformation. As a result, we show that one portion of the Fricke surface with $k=4$ admits a ${\rm GL}(2,\boldsymbol{R})$-action extending the Markoff transformations.

14. #### Involutions on numerical Campedelli surfaces

Calabri, Alberto; Lopes, Margarida Mendes; Pardini, Rita
Numerical Campedelli surfaces are minimal surfaces of general type with vanishing geometric genus and canonical divisor with self-intersection 2. Although they have been studied by several authors,their complete classification is not known. ¶ In this paper we classify numerical Campedelli surfaces with an involution, i.e., an automorphism of order 2. First we show that an involution on a numerical Campedelli surface $S$ has either four or six isolated fixed points, and the bicanonical map of $S$ is composed with the involution if and only if the involution has six isolated fixed points. Then we study in detail each of the possible cases, describing also several examples.

15. #### On stability of holomorphic maps of compact complex manifolds

Namba, Makoto

16. #### On the converse theorem for integral stability in functional differential equations

Furumochi, Tetsuo

17. #### The Gauss map of Kaehler immersions

Nishikawa, Seiki

18. #### Conformally flat Riemannian manifolds admitting a transitive group of isometries, II

Takagi, Hitoshi

19. #### Deformations of Riemannian metrics on 3-dimensional manifolds

Tanno, Shükichi

20. #### Decomposable operators in continuous fields of Hilbert spaces

Takemoto, Hideo

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