We show that polarized endomorphisms of rationally connected threefolds with at worst terminal singularities are equivariantly built up from those on ℚ-Fano threefolds, Gorenstein log del Pezzo surfaces and ℙ1. Similar results are obtained for polarized endomorphisms of uniruled threefolds and fourfolds. As a consequence, we show that every smooth Fano threefold with a polarized endomorphism of degree greater than one is rational.

This paper is about the reduced group $C^{\ast }$-algebras of real reductive groups, and about Hilbert $C^{\ast }$-modules over these $C^{\ast }$-algebras. We shall do three things. First, we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced $C^{\ast }$-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced $C^{\ast }$-algebra to determine the structure of the Hilbert $C^{\ast }$-bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in a sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction in the context of tempered unitary Hilbert space representations.

Allcock, Carlson and Toledo defined a period map for cubic threefolds which takes values in a ball quotient of dimension 10. A theorem of Voisin implies that this is an open embedding. We determine its image and show that on the algebraic level this amounts to identification of the algebra of $\operatorname{SL}(5,\mathbb{C})$-invariant polynomials on the representation space $\operatorname{Sym}^3(\mathbb{C}^5)^*$ with an explicitly described algebra of meromorphic automorphic forms on the complex 10-ball.

We use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a Kawamata log terminal pair (X,Δ) can be detected on a birational model of the base of the (KX+Δ)-trivial reduction map. We then interpret the main conjectures of the minimal model program as a natural statement about the existence of curves on X.

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain $n \geq b_2-2$ disjoint smooth rational curves with self-intersection −2, where $b_2$ is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with $p_g = 0$ and $K^{\,2} = 8, 9$ which admit an automorphism of order 2.

For any Riemannian foliation $\cal F$ on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M, about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable spectral sequence of $\cal F$.

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on $\mathbb{C}^{2n+1}$ for any $n\in \mathbb{N}$. We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface $M$ admits a proper holomorphic Legendrian embedding $M{\hookrightarrow}\mathbb{C}^{2n+1}$, and we prove that for every compact bordered Riemann surface $M={M\unicode[STIX]{x0030A}}\,\cup \,bM$ there exists a topological embedding $M{\hookrightarrow}\mathbb{C}^{2n+1}$ whose restriction to the interior is a complete holomorphic Legendrian embedding ${M\unicode[STIX]{x0030A}}{\hookrightarrow}\mathbb{C}^{2n+1}$. As a consequence, we infer that every complex contact manifold $W$ carries relatively compact holomorphic Legendrian curves, normalized by any given bordered Riemann surface, which are complete with respect to any Riemannian metric on $W$.

Finding the so-called characteristic numbers of the complex projective plane $ \mathbb{C} {P}^{2} $ is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given $d$ and $g$ one has to find the number of degree $d$ genus $g$ curves that pass through a certain generic configuration of points and at the same time are tangent to a certain generic configuration of lines. The total number of points and lines in these two configurations is $3d- 1+ g$ so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when $g= 0$. Namely, we show that the tropical problem is well posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we express genus 0 characteristic numbers of $ \mathbb{C} {P}^{2} $ in terms of open Hurwitz numbers.

Let $\mathfrak{a}$ be a homogeneous ideal of a polynomial ring $R$ in $n$ variables over a field $\mathbb{k}$. Assume that $\mathrm{depth} (R/ \mathfrak{a})\geq t$, where $t$ is some number in $\{ 0, \ldots , n\} $. A result of Peskine and Szpiro says that if $\mathrm{char} (\mathbb{k})\gt 0$, then the local cohomology modules ${ H}_{\mathfrak{a}}^{i} (M)$ vanish for all $i\gt n- t$ and all $R$-modules $M$. In characteristic $0$, there are counterexamples to this for all $t\geq 4$. On the other hand, when $t\leq 2$, by exploiting classical results of Grothendieck, Lichtenbaum, Hartshorne and Ogus it is not difficult to extend the result to any characteristic. In this paper we settle the remaining case; specifically, we show that if $\mathrm{depth} (R/ \mathfrak{a})\geq 3$, then the local cohomology modules ${ H}_{\mathfrak{a}}^{i} (M)$ vanish for all $i\gt n- 3$ and all $R$-modules $M$, whatever the characteristic of $\mathbb{k}$ is.

Let $G$ be a simple algebraic group. Labelled trivalent graphs called webs can be used to produce invariants in tensor products of minuscule representations. For each web, we construct a configuration space of points in the affine Grassmannian. Via the geometric Satake correspondence, we relate these configuration spaces to the invariant vectors coming from webs. In the case of $G= \mathrm{SL} (3)$, non-elliptic webs yield a basis for the invariant spaces. The non-elliptic condition, which is equivalent to the condition that the dual diskoid of the web is $\mathrm{CAT} (0)$, is explained by the fact that affine buildings are $\mathrm{CAT} (0)$.

We prove Pursell–Shanks type results for the Lie algebra $\mathcal{D}(M)$ of all linear differential operators of a smooth manifold M, for its Lie subalgebra $\mathcal{D}^1(M)$ of all linear first-order differential operators of M and for the Poisson algebra S(M) = Pol(T*M) of all polynomial functions on T*M, the symbols of the operators in $\mathcal{D}(M)$. Chiefly, however, we provide explicit formulas completely describing the automorphisms of the Lie algebras $\mathcal{D}^1(M)$, S(M) and $\mathcal{D}(M)$.

If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $Y$ is an L-space if and only if $\unicode[STIX]{x1D70B}_{1}(Y)$ is not left-orderable.

Let $\mathbb{C}$ be a complete non-archimedean-valued algebraically closed field of characteristic p > 0 and consider the punctured unit disc $\dot{D} \subset \mathbb{C}$. Let q be a power of p and consider the arithmetic Frobenius automorphism $\sigma_{\dot{D}}: x \mapsto x^{q^{-1}}$. A $\sigma$-bundle is a vector bundle $\mathcal{F}$ on $\dot{D}$ together with an isomorphism $\tau_\mathcal{F}: \sigma^\ast_{\dot{D}} \mathcal{F} \stackrel{\sim}{\longrightarrow} \mathcal{F}$. The aim of this article is to develop the basic theory of these objects and to classify them. It is shown that every $\sigma$-bundle is isomorphic to a direct sum of indecomposable $\sigma$-bundles $\mathcal{F}_{d,r}$ which depend only on rational numbers d/r. This result has close analogies with the classification of rational Dieudonné modules and of vector bundles on the projective line or on an elliptic curve. It has interesting consequences concerning the uniformizability of Anderson's t-motives that will be treated in a future paper.

We construct non-archimedean SYZ (Strominger–Yau–Zaslow) fibrations for maximally degenerate Calabi–Yau varieties, and we show that they are affinoid torus fibrations away from a codimension-two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt (divisorially log terminal) models along one-dimensional strata.

With analytic applications in mind, in particular beyond endoscopy, we initiate the study of the elliptic part of the trace formula. Incorporating the approximate functional equation into the elliptic part, we control the analytic behavior of the volumes of tori that appear in the elliptic part. Furthermore, by carefully choosing the truncation parameter in the approximate functional equation, we smooth out the singularities of orbital integrals. Finally, by an application of Poisson summation we rewrite the elliptic part so that it is ready to be used in analytic applications, and in particular in beyond endoscopy. As a by product we also isolate the contributions of special representations as pointed out in [Beyond endoscopy, in Contributions to automorphic forms, geometry and number theory (Johns Hopkins University Press, Baltimore, MD, 2004), 611–697].

This article can be divided into two loosely connected parts. The first part is devoted to proving a singular version of the logarithmic Kodaira–Akizuki–Nakano vanishing theorem of Esnault and Viehweg in the style of Navarro-Aznar et al. This in turn is used to prove other vanishing theorems. In the second part, these vanishing theorems are used to prove an Arakelov–Parshin type boundedness result for families of canonically polarized varieties with rational Gorenstein singularities.

We express the cohomology of the complement of a real subspace arrangement of diagonal linear subspaces in terms of the Betti numbers of a minimal free resolution. This leads to formulas for the cohomology in some cases, and also to a cohomology vanishing theorem valid for all arrangements.

We consider the Prym map from the space of double coverings of a curve of genus g with r branch points to the moduli space of abelian varieties. We prove that 𝒫:ℛg,r→𝒜δg−1+r/2 is generically injective if We also show that a very general Prym variety of dimension at least 4 is not isogenous to a Jacobian.

Let Λ be a finite-dimensional hereditary algebra over a finite field k, ${\mathcal H}$(Λ) and ${\mathcal C}$(Λ) be, respectively, the Hall algebra and the composition algebra of Λ, ${\mathcal P}$ be the isomorphism classes of finite dimensional Λ-modules and I the isomorphism classes of simple Λ-modules. We define δ$_α$ and $_α$ δ, α in ${\mathcal P}$, to be the right and left derivations of ${\mathcal H}$(Λ) respectively. By using these derivations and the action of the braid group on the set of exceptional sequences of Λ-mod, we provide an effective algorithm of calculating the root vectors of real Schur roots. This means that we get an inductive method to express u$_λ$ as the combinations of elements u$_i$ in the Hall algebra, where i ∈ I and λ in ${\mathcal P}$ is any exceptional Λ-module. Because of the canonical isomorphism between the Drinfeld–Jimbo quantum group and the generic composition algebra, our algorithm is applicable directly to quantum groups. In particular, all the root vectors are obtained in this way in the finite type cases.

In this paper we compute the cohomology with compact supports of a Siegel threefold as a virtual module over the product of the Galois group of$\bar{\bb Q}$ over ${\bb Q}$ and the Hecke algebra. We use a method which has been developed by Ihara, Langlands and Kottwitz: comparison of the Grothendieck—Lefschetz formula and the Arthur—Selberg trace formula.