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The Hodge Conjecture, posits a profound connection between the topology and algebraic geometry of complex algebraic varieties. It asserts that Hodge cycles, specific elements in the cohomology of a K\"ahler variety with rational properties,…

We show that finite-dimensional Lie algebras over a field of characteristic zero such that their high-degree cohomology in any finite-dimensional non-trivial irreducible module vanishes, are, essentially, direct sums of semisimple and…

We prove the Noether-Lefschetz conjecture on the moduli space of quasi-polarized K3 surfaces. This is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal…

We propose a novel constructive framework for approaching the Hodge Conjecture via explicit degenerations. Building on limiting mixed Hodge structures (LMHS), we formulate a criterion under which a rational class of type (p, p) on a smooth…

We establish a genericity property in the representation theory of a flat family of finite-dimensional algebras in the sense of Cline-Parshal-Scott. More precisely, we show that the decomposition matrices as introduced by Geck and Rouquier…

The Lefschetz fixed point theorem provides a powerful obstruction to the existence of minimal homeomorphisms on well-behaved spaces such as finite CW-complexes. We show that these obstructions do not hold for more general spaces. More…

We study properties concerning decomposition in cohomology by means of generalized-complex structures. This notion includes the $\mathcal{C}^\infty$-pure-and-fullness introduced by Li and Zhang in the complex case and the Hard Lefschetz…

We extend the group theoretic construction of local models of Pappas and Zhu 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…

This thesis is comprised of three chapters. The first chapter deals with bounded complexes of Gorenstein projective and Gorenstein injective modules. Deploying methods of relative homological algebra, we approximate such complexes with…

mu-constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms…

Faltings' approach in $p$-adic Hodge theory can be schematically divided into two main steps: firstly, a local reduction of the computation of the $p$-adic \'etale cohomology of a smooth variety over a $p$-adic local field to a Galois…

Let $K$ be the fraction field of a two-dimensional henselian, excellent, equi-characteristic local domain. We prove a local-global principle for Galois cohomology with finite coefficients over $K$. We use classical machinery from \'etale…

While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity,…

Using an elementary argument, we prove new fixed point theorems for classical elliptic complexes. We obtain new results for conformal relations and coisotropic intersections. We obtain theorems for the average intersections of families of…

We show that the cobordism class of a polarization of Hodge module defines a natural transformation from the Grothendieck group of Hodge modules to the cobordism group of self-dual bounded complexes with real coefficients and constructible…

For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the…

For any two degrees coprime to the rank, we construct a family of ring isomorphisms parameterized by GSp(2g) between the cohomology of the moduli spaces of stable Higgs bundles which preserve the perverse filtrations. As consequences, we…

In the paper we show that for a normal-crossings degeneration $Z$ over the ring of integers of a local field with $X$ as generic fibre, the local monodromy operator and its powers determine invariant cocycle classes under the decomposition…

We introduce two novel complementary notions of the Lefschetz number for a functor from a finite acyclic category to itself and we prove a Lefschetz fixed-object theorem and a Lefschetz fixed-morphism theorem. In order to do so, we use the…

This paper aims at developing a "local--global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the…