Abstract: The Hodge diamond of a smooth projective complex variety exhibits fundamental symmetries, arising from Poincaré duality and the purity of Hodge structures. In the case of a singular projective variety, the complexity of the singularities is closely related to the symmetries of the analogous Hodge-Du Bois diamond. For example, the failure of the first nontrivial Poincaré duality is reflected in the defect of factoriality. Based on joint work with Mihnea Popa, I will discuss how local and global conditions on singularities influence the topology of algebraic varieties.

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Abstract: In this talk, we will define the local fundamental group of an isolated singularity as the fundamental group of its link. For algebraic singularities, Kapovich and Kollár showed that any finitely presented group can appear. Although, restrictions appear if one restricts to different families of singularities. For the case of log canonical singularities, we will show that in dimension 3 the free groups cannot appear. Finally, we will present a construction for some local fundamental groups in dimensions 3 and 4. This construction, in particular, will yield the free groups as the local fundamental group of isolated 4-dimensional log canonical singularities. This is based on joint work with Joaquín Moraga

Abstract: Log Calabi-Yau pairs can be thought of as generalizations of Calabi-Yau varieties. Among them, there exists the notion of volume preserving maps. A natural and tough problem is classifying log Calabi-Yau pairs up to volume preserving equivalence. For this task, one can make use of the coregularity, the most important volume preserving invariant of a log Calabi-Yau pair. Recently, other invariants and refinements in the classification have been studied, especially in the coregularity 0 case. In this talk, I will survey some results and comment on works in progress regarding what is known to date about the classification for the case of log Calabi-Yau pairs of the form $(\mathbb{P}^3,D)$.

Abstract: For a normal complex algebraic variety X equipped with a complex representation V of its fundamental group, a Shafarevich map f:X->Y is a map which contracts precisely those algebraic subvarieties on which V has finite monodromy. Such maps were constructed for projective X by Eyssidieux, and recently have been constructed analytically in the quasiprojective case by Brunebarbe and Deng--Yamanoi, in both cases using techniques from non-abelian Hodge theory. In joint work with Y. Brunebarbe and J. Tsimerman, we show that these maps are algebraic. This is a generalization of the Griffiths conjecture on the algebraicity of images of period maps, and the proof critically uses o-minimal GAGA. We will also explain how the same techniques can be used to prove the Shafarevich conjecture in the "linear case", which puts strong restrictions on the complex analytic varieties that arise as universal covers of algebraic varieties admitting linear representations of their fundamental groups.

Abstract: Generalizing finiteness theorems of Parshin, Arakelov, and Faltings, Deligne proved in 1987 that only finitely many Z-local systems of a fixed rank underlie a polarized variation of Hodge structure, over a fixed quasiprojective variety. Deligne conjectured that an appropriate form of this finiteness also holds in families of quasiprojective varieties. In the 1990's, Simpson refined this conjecture in the following form: the nonabelian Hodge locus is algebraic. I will discuss joint work with Salim Tayou that these conjectures are true when the algebraic monodromy group is cocompact.

Abstract: The intersection cohomology complex IC_X on a toric variety X has been well studied starting with the works of Stanley and Fieseler, and more recently, the works of de Cataldo-Migliorini-Mustata and Saito. However, it has a richer structure as a Hodge module (denoted IC^H_X) in the sense of Saito's theory, and so we have the graded de Rham complexes gr_k(DR(IC^H_X)), which are complexes of coherent sheaves carrying significant information about X. In this talk, I will describe the generating function of the cohomology sheaves of gr_k(DR(IC^H_X)) and give a precise formula relating it with the stalks of the perverse sheaf IC_X (in particular, this implies that the generating function depends only on the combinatorial data of the toric variety). Time permitting, I will also show that the generating function can be computed explicitly in an algorithmic way. This is joint work with Hyunsuk Kim.

Abstract: We present a local monodromy theorem for abelian surfaces and K3 surfaces in characteristic p. This generalizes the work of Igusa, that studies the monodromy representation associated to the p-power torsion points of the universal elliptic curve around a supersingular point of the modular curve, to the setting of orthogonal Shimura varieties.