Abstract: The moduli space of cubic surfaces and cubic threefolds have different reincarnations in algebraic geometry, and can be compactified via geometric invariant theory or via looking at the minimal compactification of their ball quotient models. These compactifications are singular, and resolved respectively by the Kirwan blowup and the toroidal compactification. We show that these smooth compactifications have equal Betti numbers, but are non-isomorphic. Based on joint works with Casalaina-Martin, Hulek, and Laza.

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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.

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.

Abstract: Rational curves play a critical role in understanding the birational geometry of varieties. Free curves are the easiest to work with, but on Fano varieties that are even mildly singular, it remains an open question whether these free rational curves exist. In this talk, we discuss free curves of higher genus. Using some ideas on stability of vector bundles, we show that any klt Fano variety has these higher-genus free curves. We then use the existence of these free curves to get some applications, including the existence of free rational curves in terminal Fano threefolds, the lengths of extremal rays of the cone of curves, and studying the fundamental group of the smooth locus of a terminal variety. This is joint work with Eric Jovinelly and Brian Lehmann.

Abstract: In this talk, I will discuss recent work on Brill-Noether theory for moduli spaces of sheaves on surfaces. I will first discuss the cohomology of the general stable sheaf on surfaces such as K3 and abelian surfaces. If time permits, I will describe some recent results on the cohomology jumping loci. This talk is based on joint work with Jack Huizenga, Howard Nuer, Neelarnab Raha and Kota Yoshioka.

Abstract: Let Y be a Calabi-Yau manifold. The Morrison Cone Conjecture is a fundamental conjecture in Algebraic Geometry on the geometry of the nef cone Nef(Y) and the movable cone Mov(Y): while these cones are in general not rational polyhedral, the cone conjecture predicts that the action of Aut(Y) on Nef(Y) admits a rational polyhedral fundamental domain, and that the action of Bir(Y) on Mov(Y) admits a rational polyhedral fundamental domain. Even though the conjecture has been settled in special cases, it is still wide open in dimension at least 3. We prove that if the cone conjecture holds for a smooth Calabi-Yau threefold Y, then it also holds for any smooth deformation of Y.

Abstract: Cubic fourfolds have been classically studied up to birational equivalence, with an eye towards rationality problems. On the other hand, the Fano variety of lines F(X) on a cubic fourfold X is a hyperkahler manifold, and the rationality/irrationality of X is conjecturely reflected in the geometry of the Fano variety of lines. We give examples of conjecturally irrational cubic fourfolds with birationally equivalent Fano varieties of lines. Two of our examples, which are special families in C_12, provide new examples of pairs of cubic fourfolds with equivalent Kuznetsov components. Further, we show the cubic fourfolds themselves are birational. Our examples were discovered by studying the group of birational transformations of the Fano varieties of lines of these cubic fourfolds. This is joint work with Corey Brooke and Sarah Frei, building on our previous work with Xuqiang Qin.