Manuscripta Mathematica最新文献

Let X be a complex smooth Fano variety of dimension n. In this paper, we give a classification of such X when the pseudoindex is equal to (dfrac{dim X+1}{2}) and the Picard number greater than one.

We investigate the reduced unramified Witt group of the product of two smooth projective conics (X_1), (X_2) over a field. In some particular cases, this group denoted as (W(X_1,X_2)) turns out to be very small (zero or ({mathbb {Z}}/2{mathbb {Z}})). On the other hand, certain examples when it is infinite are constructed. We give sufficient conditions providing nontriviality of (W(X_1,X_2)) in terms of 2-fold Pfister forms (pi _1), (pi _2) associated with the conics. These conditions and constructions of the corresponding nonzero elements in (W(X_1,X_2)) depend on ({text {ind}}(pi _1+pi _2)). Also we study the question of triviality (nontriviality) of this group with respect to extensions of the ground field.

We study an eigenvalue problem for the Laplacian on a compact Kähler manifold. Considering the k-th eigenvalue (lambda _{k}) as a functional on the space of Kähler metrics with fixed volume on a compact complex manifold, we introduce the notion of (lambda _{k})-extremal Kähler metric. We deduce a condition for a Kähler metric to be (lambda _{k})-extremal. As examples, we consider product Kähler manifolds, compact isotropy irreducible homogeneous Kähler manifolds and flat complex tori.

We construct log canonical pairs (X, B) with B a nonzero reduced divisor and (K_X+B) ample that have the smallest known volume. We conjecture that our examples have the smallest volume in each dimension. The conjecture is true in dimension 2, by Liu and Shokurov. The examples are weighted projective hypersurfaces that are not quasi-smooth. We also develop an example for a related extremal problem. Esser constructed a klt Calabi–Yau variety which conjecturally has the smallest mld in each dimension (for example, mld 1/13 in dimension 2 and 1/311 in dimension 3). However, the example was only worked out completely in dimensions at most 18. We now prove the desired properties of Esser’s example in all dimensions (in particular, determining its mld).

We construct some integral elements in the motivic cohomology of the Hesse cubic curves and express their regulators in terms of generalized hypergeometric functions and Kampé de Fériet hypergeometric functions. By using these hypergeometric expressions, we obtain numerical examples of the Bloch-Beilinson conjecture on special values of L-functions.

Let X be an orthogonal Shimura variety, and let (mathcal {C}^{textrm{ort}}_{r}(X)) be the cone generated by the cohomology classes of orthogonal Shimura subvarieties in X of dimension r. We investigate the asymptotic properties of the generating rays of (mathcal {C}^{textrm{ort}}_{r}(X)) for large values of r. They accumulate towards rays generated by wedge products of the Kähler class of X and the fundamental class of an orthogonal Shimura subvariety. We also compare (mathcal {C}^{textrm{ort}}_{r}(X)) with the cone generated by the special cycles of dimension r. The main ingredient to achieve the results above is the equidistribution of orthogonal Shimura subvarieties.

Some classes of cubic fourfolds are birational to fibrations over ({mathbb {P}}^2), where the fibers are rational surfaces. This is the case for cubics containing a plane (resp. an elliptic ruled surface), where the fibers are quadric surfaces (resp. del Pezzo sextic surfaces). It is known that the rationality of these cubic hypersurfaces is related to the rationality of these surfaces over the function field of ({mathbb {P}}^2) and to the existence of rational (multi)sections of the fibrations. We study, in the moduli space of cubic fourfolds, the intersection of the divisor ({mathcal {C}}_{8}) (resp. ({mathcal {C}}_{18})) with ({mathcal {C}}_{14}), ({mathcal {C}}_{26}) and ({mathcal {C}}_{38}), whose elements are known to be rational cubic fourfolds. We provide descriptions of the irreducible components of these intersections and give new explicit examples of rational cubics fibered in (quartic, quintic) del Pezzo surfaces or in quadric surfaces over ({mathbb {P}}^2). We also investigate the existence of rational sections for these fibrations. Under some mild assumptions on the singularities of the fibers, these properties can be translated in terms of Brauer classes on certain surfaces.

We generalize the work of Lindenstrauss and Venkatesh establishing Weyl’s Law for cusp forms from the spherical spectrum to arbitrary archimedean type. Weyl’s law for the spherical spectrum gives an asymptotic formula for the number of cusp forms that are bi-(K_{infty }) invariant in terms of eigenvalue T of the Laplacian. We prove that an analogous asymptotic holds for cusp forms with archimedean type (tau ), where the main term is multiplied by (dim {tau }). While in the spherical case, the surjectivity of the Satake Map was used, in the more general case that is not available and we use Arthur’s Paley–Wiener theorem and multipliers.

We study line bundles on smooth toric Deligne-Mumford stacks ({mathbb {P}}_{mathbf {Sigma }}) of arbitrary dimension. We give a sufficient condition for when infinitely many line bundles on ({mathbb {P}}_{mathbf {Sigma }}) have trivial cohomology. In dimension three, this sufficient condition is also a necessary condition under the technical assumption that (mathbf {Sigma }) has no more than one pair of collinear rays.

Let X be an adic space locally of finite type over a complete non-archimedean field k, and denote ({textbf {Cov}}_{X}^{textrm{oc}}) (resp. ({textbf {Cov}}_{X}^{textrm{adm}})) the category of étale coverings of X that are locally for the Berkovich overconvergent topology (resp. for the admissible topology) disjoint union of finite étale coverings. There is a natural inclusion ({textbf {Cov}}_{X}^{textrm{oc}}subseteq {textbf {Cov}}_{X}^{textrm{adm}}). Whether or not this inclusion is strict is a question initially asked by de Jong. Some partial answers have been given in the recents works of Achinger, Lara and Youcis in the finite or equal characteristic 0 cases. The present note shows that this inclusion can be strict when k is of mixed characteristic (0, p) and p-closed. As a consequence, the natural morphism of Noohi groups (pi _1^{mathrm {dJ, , adm}}(mathcal {C}, overline{x})rightarrow pi _1^{mathrm {dJ, ,oc}}(mathcal {C},overline{x}) ) is not an isomorphism in general.