Mathematische Zeitschrift最新文献

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be a homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in the complex case and which comes from the theory of real closed fields otherwise.

We introduce a sequence of families of lattice polarized K3 surfaces. This sequence is closely related to complex reflection groups of exceptional type. Namely, we obtain modular forms coming from the inverse correspondences of the period mappings attached to our sequence. We study a non-trivial relation between our modular forms and invariants of complex reflection groups. Especially, we consider a family concerned with the Shephard-Todd group No.34 based on arithmetic properties of lattices and algebro-geometric properties of the period mappings.

In this work, we show that complete non-compact manifolds with non-negative Ricci curvature, Euclidean volume growth and sufficiently small curvature concentration are necessarily flat Euclidean space.

We study general random dynamical systems of continuous maps on some compact metric space. Assuming a local contraction condition and proximality, we establish probabilistic limit laws such as the (functional) central limit theorem, the strong law of large numbers, and the law of the iterated logarithm. Moreover, we study exponential synchronization and synchronization on average. In the particular case of iterated function systems on ({mathbb {S}}^1), we analyze synchronization rates and describe their large deviations. In the case of (C^{1+beta })-diffeomorphisms, these deviations on random orbits are obtained from the large deviations of the expected Lyapunov exponent.

Let K be a field, and let (fin K(z)) be rational function. The preimages of a point (x_0in mathbb {P}^1(K)) under iterates of f have a natural tree structure. As a result, the Galois group of the resulting field extension of K naturally embeds into the automorphism group of this tree. In unpublished work from 2013, Pink described a certain proper subgroup (M_{ell }) that this so-called arboreal Galois group (G_{infty }) must lie in if f is quadratic and its two critical points collide at the (ell )-th iteration. After presenting a new description of (M_{ell }) and a new proof of Pink’s theorem, we state and prove necessary and sufficient conditions for (G_{infty }) to be the full group (M_{ell }).

We define admissible and weakly admissible subcategories in exact categories and prove that the former induce semi-orthogonal decompositions on the derived categories. We develop the theory of thin exact categories, an exact-category analogue of triangulated categories generated by exceptional sequences. The right and left abelian envelopes of exact categories are introduced, an example being the category of coherent sheaves on a scheme as the right envelope of the category of vector bundles. The existence of right (left) abelian envelopes is proven for exact categories with projectively (injectively) generating subcategories with weak (co)kernels. We show that highest weight categories are precisely the right/left envelopes of thin categories. Ringel duality on highest weight categories is interpreted as a duality between the right and left abelian envelopes of a thin exact category. A duality for thin exact categories compatible with Ringel duality is introduced by means of derived categories and Serre functor on them.

Given a CR manifold with non-degenerate Levi form, we show that the operators of the functional calculus for Toeplitz operators are complex Fourier integral operators of Szegő type. As an application, we establish semi-classical asymptotics for the dimension of the spectral spaces of Toeplitz operators. We then consider a CR manifold with a compact Lie group action G and we establish quantization commutes with reduction for Toeplitz operators. Moreover, we also compute semi-classical asymptotics for the dimension of the spectral spaces of G-invariant Toeplitz operators.

The famous theorem of Matsumura–Oort states that if X is a proper scheme, then the automorphism group functor (mathfrak {Aut}(X)) of X is a locally algebraic group scheme. In this paper we generalize this theorem to the category of superschemes, that is if ({mathbb {X}}) is a proper superscheme, then the automorphism group functor (mathfrak {Aut}({mathbb {X}})) of ({mathbb {X}}) is a locally algebraic group superscheme. Moreover, we also show that if (H^1(X, {mathchoice{text{ T }}{text{ T }}{text{ T }}{text{ T }}}_X)=0), where X is the geometric counterpart of ({mathbb {X}}) and ({mathchoice{text{ T }}{text{ T }}{text{ T }}{text{ T }}}_X) is the tangent sheaf of X, then (mathfrak {Aut}({mathbb {X}})) is a smooth group superscheme.

We initiate a systematic construction of real analytic Lagrangian fibrations from integer matrices. We prove that when the matrix is of full column rank, the perverse filtration associated with the Lagrangian fibration matches the mixed Hodge-theoretic weight filtration of the isolated cluster variety associated with the matrix.

We prove a theorem which implies that all Segre–Veronese varieties of multidegree ((d_1,dots ,d_k)) and format ((n_1,dots ,n_k)) with (n_1ge cdots ge n_k>0) are not defective if (d_1ge 3), (d_2ge 3) and (d_ige 2) for all (i>2). As a particular case we prove the non-defectivity of any Segre–Veronese variety with at least 2 factors and (d_ige 3) for all i, extending to the case (k>2) a theorem of Galuppi and Oneto. Our general result also shows that many Segre–Veronese varieties with 2 factors are not secant defective if they are embedded in bidegree (x, 2), (xge 4).