On the construction of tame supercuspidal representations

Let $F$ be a non-archimedean local field of residual characteristic $p \neq 2$ . Let $G$ be a (connected) reductive group over $F$ that splits over a tamely ramified field extension of $F$ . We revisit Yu's construction of smooth complex representations of $G(F)$ from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in Yu [Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), 579–622], whose proofs relied on a typo in a reference.

On the classification of non-big Ulrich vector bundles on surfaces and threefolds

In this paper, we classify Ulrich vector bundles that are not big on smooth complex surfaces and threefolds.

On the geometric André–Oort conjecture for variations of Hodge structures

Let 𝕍 {{\mathbb{V}}} be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S. In this paper, we show that the union of the non-factor special subvarieties for ( S , 𝕍 ) {(S,{\mathbb{V}})} , which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S. This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures.

A comparison of positivity in complex and tropical toric geometry

Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the cone of invariant closed positive currents on the complex toric variety with closed positive currents on the tropicalization. In a subsequent paper, this correspondence will be used to develop a Bedford–Taylor theory of plurisubharmonic functions on the tropicalization.

Non-gravitational Effects of the Metric Field over Complex Manifolds

Objective of this work is to study whether some of the known non-gravitational phenomena can be explained as motion on a straight line as gravity is treated within General Relativity. To do that, we explore a metric field on a complexified manifold with holomorphic coordinates. Specifically we look into the behaviour of geodesics on such a smooth complex manifold and the path traced out by its real component. This yields a family of equations of motions in real coordinates which is shown to have deviations from usual geodesic equation and in that way expands the geodesic to capture contributions from additional fields and interactions beyond the mere gravitational ones as a function of the metric field.

On representations of reductive p-adic groups over ℚ-algebras

In this paper we study certain category of smooth modules for reductive p-adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a ℚ-algebra. We prove some fundamental results in these settings, and as an example we give a classification of admissible unramified irreducible representations using the reduction to the complex case.