Low-dimensional Complex and Conformal Geometry

Abstract: Vaisman manifolds form a special class of Hermitian, non-Kähler manifolds, which can be obtained as deformations of positive elliptic bundles over Kähler orbifolds. I will present different aspects of this type of deformations and in particular the relation to the algebraic reductions of Vaisman manifolds. 

Abstract: We will prove that every  compact LCK manifold of algebraic codimension one is an elliptic principal bundle over a smooth projective manifold, up to proper modifications and  generically finite maps.

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Abstract: Using the recent advances in the structure of algebraic groups over imperfect fields, we propose a generalization of Serre's Conjecture I and of results that revolve around it. In particular, we prove that the first Galois cohomology set of any unirational algebraic group is always trivial if the cohomological dimension of the field is at most 1 in Kato's sense. (Joint work with Alexandre Lourdeaux.)

Abstract: Motivic cohomology is a cohomology theory that can be defined internally within Grothendieck's category of motives. Voevodsky developed this theory for smooth varieties, demonstrating its profound connections to algebraic cycles and algebraic K-theory. However, its behaviour in mixed-characteristic remains less well understood. Building on recent advancements by Bachmann, Elmanto, Morrow, and Bouis, in a joint work with Bouis, we demonstrate a purity result over deeply-ramified bases in mixed-characteristic. I will discuss an application of this result in p-adic Hodge theory.

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Abstract: We establish asymptotic formulas for the second moment of central values of L-functions associated with r-th order Hecke characters. Our analysis covers two distinct families: characters attached to square-free ideals and those attached to r-th power-free ideals. In the latter case, the asymptotic formula includes secondary-order terms. As an application we obtain explicit, positive proportions for the nonvanishing of central values of these L-functions.

Our approach relies on the method of multiple Dirichlet series to achieve the necessary analytic continuation for the generating series for the second moment. This is joint work with Adrian Diaconu, Bogdan Ion and Vicentiu Pasol.

Abstract: We study a limited asymptotic expansion of the logarithmic derivative of the Selberg zeta function corresponding to the spin Dirac operator on compact surfaces for families of hyperbolic metrics degenerating towards complete hyperbolic metrics with cusps. The main tools in this investigation are a Selberg trace formula for the Dirac operator and an adapted pseudodifferential calculus that includes both the family of Dirac operators on the family of compact surfaces and the Dirac operator on the limit non-compact surface, together with their resolvents. This talk is based on joint work with Sergiu Moroianu and Rareș Stan.

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Abstract: We define a Chern--Simons invariant of connections on stably trivial vector bundles over smooth manifolds, taking values in 3-forms modulo closed forms with integral cohomology class. This invariant is additive for connections defined on a direct sum of bundles, under a certain block-diagonality condition on the curvature. We deduce an obstruction for conformally immersing lens spaces in a translation manifold of dimension 4.

Abstract: On a Riemannian manifold M, a symmetric Killing tensor is a section of Sym(TM) that is constant along any geodesic. It is said to be decomposable if it can be expressed as a polynomial in the Riemannian metric and Killing vector fields. In 2020, Del Barco and Moroianu showed that on a certain family of nilpotent Lie groups, there exist invariant Killing 2-tensors that are not decomposable. In this talk, we completely characterize the symmetric Killing p-tensors defined on a family of solvable Lie groups, specifically the almost abelian ones, and show that they are decomposable. This talk is based on joint work with Viviana del Barco and Andrei Moroianu.

Abstract: A Lie algebra is called almost abelian if it has an abelian ideal of codimension 1. On almost abelian Lie algebras endowed with an integrable complex structure, we will characterize the existence of special Hermitian structures in terms of the bracket structure on the Lie algebra. More precisely, we will consider special Hermitian metrics, such as Kähler, balanced, pluriclosed and Gauduchon ones, and some structures generalizing them, known as p-Kähler and p-pluriclosed structures.

This is based on a joint work with Andrei Moroianu.

Abstract: We study compact locally conformally Kähler (lcK) manifolds which are Calabi--Yau, in the sense that $c_1^{BC}(X)=0$.  First of all, we prove that all the known lcK manifolds which are Calabi--Yau are Vaisman. Then we prove that an lcK Chern--Ricci flat metric that is Gauduchon is necessarily Vaisman.  Finally, specializing to Calabi--Yau solvmanifolds with left-invariant complex structure, we prove that a left-invariant metric is lcK if and only if it is Vaisman. Therefore, they are finite quotients of the Kodaira manifold.