We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.

Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

We recover the Newton diagram (modulo a natural ambiguity) from the link for any surface hypersurface singularity with non-degenerate Newton principal part whose link is a rational homology sphere. As a corollary, we show that the link determines the embedded topological type, the Milnor fibration, and the multiplicity of such a germ. This proves (an even stronger version of) Zariski's conjecture about the multiplicity for such a singularity.

This is the second in a sequence of papers on the geometry of spaces of rational curves of degree e on a general hypersurface $X \subset\mathbb{P}^n$ of degree d. In Part I (J. reine angew. Math. 571 (2004), 73–106) it is proved that, if $d<({n+1})/{2}$, then for each e the space of rational curves is irreducible, reduced and has the expected dimension. In this paper it is proved that, if $d^2 + d + 1 \leq n$, then for each e the space of rational curves is a rationally connected variety; in particular it has negative Kodaira dimension.

We extend the group-theoretic construction of local models of Pappas and Zhu [Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math. 194 (2013), 147–254] to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when $p\geqslant 5$. We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.

We prove that the direct image complex for the $D$-twisted $\text{SL}_{n}$ Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for $\text{GL}_{n}$ is due to Chaudouard and Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove $\unicode[STIX]{x1D6FF}$-regularity results for some auxiliary weak abelian fibrations.

We prove the vanishing of the equivariant holomorphic torsion forms associated to the holomorphic de Rham complex. We also consider corresponding torsion forms associated to complexes in hypercohomology, for which we give a local formula. Rigidity of certain genera explains part of these formulas. The Lefschetz operators in Hodge theory play a prominent role in the proofs.

Let k be a number field with ring of integers $\mathfrak{O}_k$, and let $\Gamma=A_4$ be the tetrahedral group. For each tame Galois extension N/k with group isomorphic to $\Gamma$, the ring of integers $\mathfrak{O}_{N}$ of N determines a class in the locally free class group $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$. We show that the set of classes in $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ realized in this way is the kernel of the augmentation homomorphism from $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ to the ideal class group $\mathrm{Cl}(\mathfrak{O}_k)$. This refines a result of Godin and Sodaïgui (J. Number Theory98 (2003), 320–328) on Galois module structure over a maximal order in $k[\Gamma]$. To the best of our knowledge, our result gives the first case where the set of realizable classes in $\mathrm{Cl}(\mathfrak{O}_k[\Gamma])$ has been determined for a nonabelian group $\Gamma$ and an arbitrary number field k.

We define the notion of a morphism of generalized semi-stable type, which is a generalization of the notion of a semistable degeneration over a curve. We partially generalize Steenbrink‘s results on the limit of Hodge structures to the case of such a morphism. As an application we prove the E1-degeneration of the relative Hodge–De Rham spectral sequence for this case.

The Chevalley involution of a connected, reductive algebraic group over an algebraically closed field takes every semisimple element to a conjugate of its inverse, and this involution is unique up to conjugacy. In the case of the reals we prove the existence of a real Chevalley involution, which is defined over $\mathbb{R}$, takes every semisimple element of $G(\mathbb{R})$ to a $G(\mathbb{R})$-conjugate of its inverse, and is unique up to conjugacy by $G(\mathbb{R})$. We derive some consequences, including an analysis of groups for which every irreducible representation is self-dual, and a calculation of the Frobenius Schur indicator for such groups.

Let p>q and let G be the group U(p, q) or Spin0(p, q). Let P=LN be the maximal parabolic subgroup of G with Levi subgroup $L\cong M\times U$ where$\openup2(M,U)=\left \{ \matrix{ ({\rm GL} _q({\mathbb C}),{\rm U}(\,p-q)),\hfill \mbox { if } G={\rm U}(\,p,q), \hfill\cr ({\rm GL} ^+_q({\mathbb R}),{\rm Spin}(\,p-q)), \mbox { if } G={\rm Spin}_0(\,p,q). }\right.$Let χ be a one-dimensional character of M and τμ an irreducible representation of U with highest weight μ. Let $\pi_{\chi,\mu}$ be the representation of P which is trivial on N and $\pi_{\chi,\mu}|_L=\chi\boxtimes \tau ^\mu$. Let $I_{p,q}$ be the Harish-Chandra module of the induced representation ${\rm Ind}_{P}^{G} \pi_{\chi,\mu}$. In this paper, we shall determine (i) the reducibility of $I_{p,q}$, (ii) the K-types of all the irreducible subquotients of $I_{p,q}$ when it is reducible, where K is the maximal compact subgroup of G, (iii) the module diagram of $I_{p,q}$ (from which one can read off the composition structure), and (iv) the unitarity of $I_{p,q}$ and its subquotients. Except in the cases $q=p-1$ and $q=1$, $I_{p,q}$ is not K-multiplicity free.

Let G be a reductive p-adic group. Given a compact-mod-center maximal torus S⊂G and sufficiently regular character χ of S, one can define, following Adler, Yu and others, a supercuspidal representation π(S,χ) of G. For S unramified, we determine when π(S,χ) is generic, and which generic characters it contains.

For the special orthogonal group G = SO(2n + 1) over a p-adic field, we consider a discrete series representation of a standard Levi subgroup of G. We prove that the Arthur R-group and the classical R-group of $\pi$ are isomorphic. If $\pi$ is generic, we consider the Aubert involution $\hat{\pi}$. Under the assumption that $\hat{\pi}$ is unitary, we prove that the Arthur R-group of $\hat{\pi}$ is isomorphic to the R-group of $\hat{\pi}$ defined by Ban (Ann. Sci. École Norm. Sup. 35 (2002), 673–693; J. Algebra 271 (2004), 749–767). This is done by establishing the connection between the A-parameters of $\pi$ and $\hat{\pi}$. We prove that the A-parameter of $\hat{\pi}$ is obtained from the A-parameter of $\pi$ by interchanging the two $\textit{SL}(2,\mathbb{C})$ components.

We proove an analogue of the Beilinson-Bernstein Theorem, in the case of the projective space, for the ring of arithmetical differential operators introduced by P. Berthelot.

We study the deformation theory of algebraic stacks. The 2-category of deformations using Ext groups of the cotangent complex is described. Then we show that an algebraic stack has a versal deformation under certain conditions.

A Lefschetz class on a smooth projective variety is an element of the Q-algebra generated by divisor classes. We show that it is possible to define Q-linear Tannakian categories of abelian motives using the Lefschetz classes as correspondences, and we compute the fundamental groups of the categories. As an application, we prove that the Hodge conjecture for complex abelian varieties of CM-type implies the Tate conjecture for all Abelian varieties over finite fields, thereby reducing the latter to a problem in complex analysis.

The holonomic rank of the A-hypergeometric system MA(β) is the degree of the toric ideal IA for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of non-generic parameters to construct a combinatorial formula for the rank jump of MA(β). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of MA(β) and show that the Zariski closure of each of its strata is a union of translates of linear subspaces of the parameter space. These results hold for generalized A-hypergeometric systems as well, where the semigroup ring of A is replaced by a non-trivial weakly toric module M⊆ℂ[ℤA] . We also provide a direct proof of the main result in [M. Saito, Isomorphism classes of A-hypergeometric systems, Compositio Math. 128 (2001), 323–338] regarding the isomorphism classes of MA (β) .

We present a new connection between the Hele-Shaw flow, also known as two-dimensional Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this, we prove short-time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also when starting from a smooth Jordan domain. Applying the same ideas, we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.

Finite $W$-algebras are certain associative algebras arising in Lie theory. Each $W$-algebra is constructed from a pair of a semisimple Lie algebra ${\mathfrak{g}}$ (our base field is algebraically closed and of characteristic 0) and its nilpotent element $e$. In this paper we classify finite-dimensional irreducible modules with integral central character over $W$-algebras. In more detail, in a previous paper the first author proved that the component group $A(e)$ of the centralizer of the nilpotent element under consideration acts on the set of finite-dimensional irreducible modules over the $W$-algebra and the quotient set is naturally identified with the set of primitive ideals in $U({\mathfrak{g}})$ whose associated variety is the closure of the adjoint orbit of $e$. In this paper, for a given primitive ideal with integral central character, we compute the corresponding $A(e)$-orbit. The answer is that the stabilizer of that orbit is basically a subgroup of $A(e)$ introduced by G. Lusztig. In the proof we use a variety of different ingredients: the structure theory of primitive ideals and Harish-Chandra bimodules for semisimple Lie algebras, the representation theory of $W$-algebras, the structure theory of cells and Springer representations, and multi-fusion monoidal categories.

In this note we classify the derived-tame tree algebras up to derived equivalence. A tree algebra is a basic algebra A = kQ/I whose quiver Q is a tree. The algebra A is said to be derived-tame when the repetitive category  of A is tame. We show that the tree algebra A is derived-tame precisely when its Euler form χA is non-negative. Moreover, in this case, the derived equivalence class of A is determined by the following discrete invariants: The number of vertices, the corank and the Dynkin type of χA. Representatives of these derived equivalence classes of algebras are given by the following algebras: the hereditary algebras of finite or tame type, the tubular algebras and a certain class of poset algebras, the so-called semichain-algebras which we introduce below.