By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface over an algebraically closed field, a variety whose geometric generic fiber is a projective homogeneous space admits a rational point if and only if the elementary obstruction vanishes.

We show that the Craighero–Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero–Gattazzo surface is the only explicit example of a smooth simply connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface we use an algebraic reduction mod $p$ technique and deformation theory.

Let $G$ be a semi-simple algebraic group over an algebraically closed field $k$ , whose characteristic is positive and does not divide the order of the Weyl group of $G$ , and let $\breve{G}$ be its Langlands dual group over $k$ . Let $C$ be a smooth projective curve over $k$ of genus at least two. Denote by $\operatorname{Bun}_{G}$ the moduli stack of $G$ -bundles on $C$ and $\operatorname{LocSys}_{\breve{G}}$ the moduli stack of $\breve{G}$ -local systems on $C$ . Let $D_{\operatorname{Bun}_{G}}$ be the sheaf of crystalline differential operators on $\operatorname{Bun}_{G}$ . In this paper we construct an equivalence between the bounded derived category $D^{b}(\operatorname{QCoh}(\operatorname{LocSys}_{\breve{G}}^{0}))$ of quasi-coherent sheaves on some open subset $\operatorname{LocSys}_{\breve{G}}^{0}\subset \operatorname{LocSys}_{\breve{G}}$ and bounded derived category $D^{b}(D_{\operatorname{Bun}_{G}}^{0}\text{-}\text{mod})$ of modules over some localization $D_{\operatorname{Bun}_{G}}^{0}$ of $D_{\operatorname{Bun}_{G}}$ . This generalizes the work of Bezrukavnikov and Braverman in the $\operatorname{GL}_{n}$ case.

In this paper, we investigate the mixed Hodge structures of the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$ -stable parabolic Higgs bundles and the moduli space of $\boldsymbol{\unicode[STIX]{x1D6FC}}$ -stable regular singular parabolic connections. We show that the mixed Hodge polynomials are independent of the choice of generic eigenvalues and the mixed Hodge structures of these moduli spaces are pure. Moreover, by the Riemann–Hilbert correspondence, the Poincaré polynomials of character varieties are independent of the choice of generic eigenvalues.

We show that the generic fiber of a family $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$ -ruled affine surfaces always carries an $\mathbb{A}^{1}$ -fibration, possibly after a finite extension of the base $S$ . In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking $S$ , such a family actually factors through an $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$ -scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$ . For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$ -ruled surfaces over a smooth curve $B$ , the induced $\mathbb{A}^{1}$ -fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$ .

We prove that the tautological ring of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$ , the moduli space of $n$ -pointed genus two curves of compact type, does not have Poincaré duality for any $n\geqslant 8$ . This result is obtained via a more general study of the cohomology groups of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$ . We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^{k}({\mathcal{M}}_{2,n}^{\mathsf{ct}})$ for any $k$ and $n$ considered both as $\mathbb{S}_{n}$ -representation and as mixed Hodge structure/ $\ell$ -adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of $\overline{{\mathcal{M}}}_{2,n}$ is tautological for $n<20$ , and that the tautological ring of $\overline{{\mathcal{M}}}_{2,n}$ fails to have Poincaré duality for all $n\geqslant 20$ . This improves and simplifies results of the author and Orsola Tommasi.

We study moduli spaces of rational weighted stable tropical curves, and their connections with Hassett spaces. Given a vector $w$ of weights, the moduli space of tropical $w$ -stable curves can be given the structure of a balanced fan if and only if $w$ has only heavy and light entries. In this case, the tropical moduli space can be expressed as the Bergman fan of an explicit graphic matroid. The tropical moduli space can be realized as a geometric tropicalization, and as a Berkovich skeleton, its algebraic counterpart. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of unweighted spaces, and explore parallels with the algebraic world.

In this paper we prove that a smooth family of canonically polarized manifolds parametrized by a special (in the sense of Campana) quasi-projective variety is isotrivial.

Given a family of varieties $X\rightarrow \mathbb{P}^{n}$ over a number field, we determine conditions under which there is a Brauer–Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.

An affine symplectic singularity $X$ with a good $\mathbf{C}^{\ast }$-action is called a conical symplectic variety. In this paper we prove the following theorem. For fixed positive integers $N$ and $d$, there are only a finite number of conical symplectic varieties of dimension $2d$ with maximal weights $N$, up to an isomorphism. To prove the main theorem, we first relate a conical symplectic variety with a log Fano Kawamata log terminal (klt) pair, which has a contact structure. By the boundedness result for log Fano klt pairs with fixed Cartier index, we prove that conical symplectic varieties of a fixed dimension and with a fixed maximal weight form a bounded family. Next we prove the rigidity of conical symplectic varieties by using Poisson deformations.

The author finds a limit on the singularities that arise in geometric generic fibers of morphisms between smooth varieties of positive characteristic by studying changes in embedding dimension under inseparable field extensions. This result is then used in the context of the minimal model program to rule out the existence of smooth varieties fibered by certain nonnormal del Pezzo surfaces over bases of small dimension.

We show that, by taking normalizations over certain auxiliary good reduction integral models, one obtains integral models of toroidal and minimal compactifications of PEL-type Shimura varieties which enjoy many features of the good reduction theory studied as in the earlier works of Faltings and Chai’s and the author’s. We treat all PEL-type cases uniformly, with no assumption on the level, ramifications, and residue characteristics involved.

We classify the horizontal $\text{SL}(2)$s and $\mathbb{R}$-split polarized mixed Hodge structures on a Mumford–Tate domain.

This is part IV of our series of articles on log abelian varieties. In this part, we study the algebraic theory of proper models of log abelian varieties.

Let $G$ be a split reductive group. We introduce the moduli problem of bundle chains parametrizing framed principal $G$-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of $G$. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple $G$, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev–Manin’s spaces of weighted pointed curves and with Kausz’s compactification of $GL_{n}$.

We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral: one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard–Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of $K3$ surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the $K3$ family. We prove a conjecture by David Broadhurst which states that at a special kinematical point the Feynman integral is given by a critical value of the Hasse–Weil $L$-function of the $K3$ surface. This result is shown to be a particular case of Deligne’s conjectures relating values of $L$-functions inside the critical strip to periods.

We determine the index of five of the seven hypergeometric Calabi–Yau operators that have finite index in $\mathit{Sp}_{4}(\mathbb{Z})$ and in two cases give a complete description of the monodromy group. Furthermore, we find six nonhypergeometric Calabi–Yau operators with finite index in $\mathit{Sp}_{4}(\mathbb{Z})$, most notably a case where the index is one.

To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman–Millson, Green–Lazarsfeld, Nadel, Simpson, Dimca–Papadima, and of the second author.

This paper deals with the geometric local theta correspondence at the Iwahori level for dual reductive pairs of type II over a non-Archimedean field $F$ of characteristic $p\neq 2$ in the framework of the geometric Langlands program. First we construct and study the geometric version of the invariants of the Weil representation of the Iwahori-Hecke algebras. In the particular case of $(\mathbf{GL}_{1},\mathbf{GL}_{m})$ we give a complete geometric description of the corresponding category. The second part of the paper deals with geometric local Langlands functoriality at the Iwahori level in a general setting. Given two reductive connected groups $G$ and $H$ over $F$, and a morphism ${\check{G}}\times \text{SL}_{2}\rightarrow \check{H}$ of Langlands dual groups, we construct a bimodule over the affine extended Hecke algebras of $H$ and $G$ that should realize the geometric local Arthur–Langlands functoriality at the Iwahori level. Then, we propose a conjecture describing the geometric local theta correspondence at the Iwahori level constructed in the first part in terms of this bimodule, and we prove our conjecture for pairs $(\mathbf{GL}_{1},\mathbf{GL}_{m})$.