Elenco seminari del ciclo di seminari
“SEMINARIO DI ALGEBRA E GEOMETRIA”

The monodromy group of an IHS manifold is one of the most important tools to investigate their geometry. In the first part of the talk, I will recall the main definitions, giving some motivation. In the second half I will focus on the OG10-type. This is the only type (among the known ones) for which the monodromy group is still a mystery. We explain how to construct new monodromy operators using two families, the O'Grady and the Laza-Saccà-Voisin ones, exhibiting an explicit subgroup of the monodromy group, that we conjecture being all. Time permitting, we will also discuss a geometric constraint to the fact that the monodromy group is smaller than the group of orientation preserving isometries.

By the Bogomolov decomposition theorem, irreducible holomorphic symplectic manifolds play a central role in the classification of compact Kähler manifolds with numerically trivial canonical bundle. Very recently, Höring and Peternell completed the proof of the existence of a singular analogue of the Bogomolov decomposition theorem. In view of this result, singular irreducible symplectic varieties (following Greb, Kebekus and Peternell) are singular analogue of irreducible holomorphic symplectic manifolds. In a joint work with Arvid Perego, still in progress, we show that all moduli spaces of sheaves on projective K3 surfaces are singular irreducible symplectic varieties. We compute their Beauville form and the Hodge decomposition of their second integral cohomology, generalizing previous results, in the smooth case, due to Mukai, O'Grady and Yoshioka.

Named by André Weil after the mathematicians Kodaira, Kummer and Kähler and the mountain K2, K3 surfaces are one of the prominent classes of complex surfaces. Their (finite) symmetries are closely related to the Mathieu group M_{24}. This group, discovered in the 19th century, is one of the finite sporadic simple groups.

In questo talk presenteremo alcuni risultati recenti riguardanti Fano 4-folds, usando particolari strutture di conic bundles che tali varietà ammettono. Nella prima parte del talk ci focalizzeremo su alcuni oggetti geometrici preliminari che ci serviranno per discutere i risultati principali. In particolare, richiameremo un invariante per tali varietà, introdotto da Casagrande, chiamato Lefschetz defect. Rivisiteremo la letteratura esistente nel caso in cui tale invariante sia maggiore o uguale a quattro e discuteremo il legame che sussiste tra il Lefschetz defect e le strutture di conic bundles delle varietà in questione. Successivamente daremo una caratterizzazione delle Fano 4-folds con Lefschetz defect uguale a 3 in termini di strutture di conic bundles da cui dedurremo risultati di classificazione per tali varietà.

In collaborazione con R. Laterveer (Strasburgo) e G. Pacienza (Nancy) diamo un criterio per verificare una congettura di Voisin sui cicli algebrici 0-dimensionali di una varietà complessa, proiettiva, liscia, di dimensione al più cinque, e di genere geometrico uno.

Subvarieties of Grassmannians (and especially Fano varieties) obtained from section of homogeneous vector bundles are far from being classified. A case of particular interest is given by the Fano varieties of K3 type (FK3), for their deep links with hyperkähler geometry. In this talk we will present some examples of recently discovered FK3 varieties, and a general procedure that allows us to spread a (Hodge) K3 structure as a component of the Hodge structure of different varieties. This is in collaboration with Giovanni Mongardi and Marcello Bernardara--Laurent Manivel.

Zeta functions are historically important objects in mathematics, and continue to motivate a wealth of work in research today. Zeta functions often arise at the crossroads between different mathematical fields, such as analysis, number theory and algebraic geometry. This talk will mainly focus on so-called "motivic zeta functions", which are of a more geometric nature. These objects can be attached to degenerating families of Calabi-Yau varieties, and provide an interesting link between arithmetic and geometric aspects of such varieties. I will give an intuitive introduction to some of the key properties of motivic zeta functions, and mention a few influential open questions. I will also present some of my own results, both old and new, in this area.

Nella prima parte del seminario richiamerò alcune nozioni di base sulle varietà abeliane e sulla varietà Jacobiana di una curva liscia e irriducibile. Motiverò inoltre lo studio delle curve di genere geometrico 2 su superfici abeliane. Nella seconda parte del seminario parlerò di un lavoro in collaborazione con A.L.Knutsen. Sia (S,L) una superficie abeliana generale con polarizzazione di tipo (d_1,d_2). Nel sistema lineare |L| sono presenti un numero finito di curve di genere geometrico 2. In analogia con il risultato di Chen riguardante curve razionali su superfici K3, è naturale chiedersi se tutte le curve di genere 2 in |L| sono nodali. Dimostreremo che questo è vero se e solo se 4 non divide d_2.

During the first part of the talk I will give a gentle introduction to the theory of cohomological Hall algebras and their relevance in the study of the topology of moduli spaces, such as the Hilbert schemes of points on a smooth surface. The second part of the talk is devoted to the definition of the 2-dimensional cohomological Hall algebras of curves and surfaces. If time permits, I will discuss the construction of a categorification of these algebras. This is based on joint works with Olivier Schiffmann and Mauro Porta.

Recently Okounkov proved Nekrasov’s conjecture expressing the partition function of K-theoretic DT invariants of the Hilbert scheme of points Hilb(C^3,points) on affine 3-space as an explicit plethystic exponential. The higher rank analogue of Nekrasov’s formula is a conjecture in String Theory by Awata-Kanno. We state this conjecture and sketch how to prove it mathematically via Quot schemes. Specialising from K-theoretic to cohomological invariants, we obtain the statement of a conjecture of Szabo. This is joint work with Nadir Fasola and Sergej Monavari.

Let S be a smooth projective surface. The Hilbert schemes Hilb^n(S) of n points on S are well-understood and central objects in geometry. Less is known, however, about degenerations of such varieties. In this talk, I will present joint work with M. Gulbrandsen, K. Hulek and Z. Zhang, where we study how the Hilbert scheme degenerates "along" with the surface S. I will also discuss a few applications of our construction, and some open problems.

A Calabi-Yau (CY) pair is a pair (X, D) of a normal variety X and a reduced Z-Weil divisor D⊂X such that KX+D∼0 is a Cartier divisor linearly equivalent to zero. A Mori fibered (Mf ) CY pair is a Q-factorial (t, lc) CY pair (X, D) together with a Mori fibration f:X→S. Let (X, DX) and (Y, DY) be CY pairs. A birational map f:X to Y is volume preserving if there exists a common log resolution p:W→X, q:W→Y such that p∗(KX+DX) =q∗(KY+DY). Let (X, DX)→SXbe (Mf) CY pair. We define the special birational group of (X, DX) as the group SBir(X, DX) of volume preserving birational maps f:X to X. Our aim is to produce interesting subgroups of the Cremona group of birational self-maps of projective spaces as groups of the type SBir(Pn, DPn), where DPn is a hypersurface of degree n+ 1. Even in seemingly simple cases these groups could be quite hard to compute. We give an explicit presentation of SBir(X, DX) when X=P3 and DX is a quartic surface with divisor class group generated by the hyperplane section and whose singular locus is either empty or an A1-point. In general it seems that the worse are the singularities of the pair (X, DX) the more complicated is the group SBir(X, DX).

La teoria delle superfici K3 con involuzioni simplettiche e dei loro quozienti è ora un argomento classico ben compreso grazie ai lavori fondamentali di Nikulin, Morrison, van Geemen e Sarti. In questo seminario cercheremo di sviluppare risultati analoghi per superfici K3 con automorfismi simplettici di ordine tre: descriveremo esplicitamente l'azione indotta sul reticolo K3, isometrico al secondo gruppo di coomologia di una superficie K3, da questi particolari automorfismi; deduciamo la relazione fra le famiglie che ammettono questi automorfismi e quelle date dai loro quozienti. Se il tempo lo permetterà, daremo alcune applicazioni: una relativa alle strutture di Shioda-Inose, e un'altra nella costruzione di torri infinite di superfici K3 isogenee. Questo è un lavoro in collaborazione con Alice Garbagnati

In his celebrated article from 1989 Woronowicz introduced a covariant differential calculus on Hopf algebras, generalizing the Cartan calculus on Lie groups. The aim of this talk is to recall his construction and to explain how other notions of differential geometry, mainly connections, curvature and torsion, generalize to this setting. In the second half of the presentation we specialize to noncommutative spaces with triangular Hopf algebra symmetry and provide existence and uniqueness of a Levi-Civita connection for any non-degenerate metric. The latter observation is due to the speaker. As our main example we mention Drinfel'd twist deformation quantization of Riemannian geometry on Poisson manifolds.

In this talk I will present an extension of polar duality, called framed duality, beyond the class of Fano toric varieties. The key idea is thinking of polar duality as a duality between toric varieties ``framed'' by their anti-canonical divisor and then allowing a more general ``framing'', in principle just given by an effective divisor. When restricted to a general section of the framing linear system, this construction gives back a Batyrev-type mirror symmetry between toric hypersurfaces (and complete intersections). This process, when restricted to Calabi-Yau complete intersections reduces precisely to Batyrev-Borisov duality, when considered for negative Kodaira dimension, produces Landau-Ginzburg mirror models closely related to those proposed by Givental, and, when considered for positive Kodaira dimension, suggests interesting improvements of Hori-Vafa mirror models, so getting a unified approach to Mirror Symmetry of toric complete intersections.

La prima parte del seminario sarà dedicata ad un'ampia introduzione ai cubic fourfolds - cioé le ipersuperfici cubiche di grado 3 nello spazio proiettivo di dimensione 5. Ne descrivero' la geometria, la teoria d'intersezione, e come si deformano in famiglie (o come si dice comunemente: lo spazio di moduli). Continuero' approfondendo la teoria dell'intersezione dei divisori (i.e. sottovarietà di codimensione 1) di cubiche speciali nello spazio di moduli. Questi divisori parametrizzano le cubiche che contengono più superfici che le generiche cubiche. Daremo delle condizioni necessarie affinché (fino a) 20 di questi divisori si intersechino, e descriveremo le superfici K3 associate a queste cubiche - nel senso della teoria di Hodge. Applicheremo questa costruzione per costruire nuove famiglie di cubiche con Chow motive di dimensione finita e di tipo abeliano. Infine, considereremo alcune varietà di HyperKähler associate alle cubiche (la varietà di Fano delle rette contenute nel 4fold, il LLSvS 8fold, ecc.) e mostreremo che in alcuni casi i nostri precedenti risultati implicano che anche queste varietà di HK hanno un Chow motive finito dimensionale.

It is conjectured that many birational transformations, called $K$-inequalities, have a categorical counterpart in terms of an embedding of derived categories. In the special case of simple $K$-equivalence (or more generally $K$-equivalence), a derived equivalence is expected: we propose a method to prove the conjecture for a wide class of simple $K$-equivalences. This method relies on the construction of roofs of projective bundles introduced by Kanemitsu. Roofs are special Fano varieties of Picard number two admitting two projective bundle structures, and they are related to the construction of pairs of Calabi—Yau varieties: we prove that a $K$-equivalent pair is derived equivalent if the associated pair of Calabi—Yau varieties is derived equivalent, and we apply this technique on several cases. The proofs are based on the manipulation of semiorthogonal decompositions by mutations of exceptional objects.

First, we give an introduction to the notion of moduli stack of a dg category. Then we explain what shifted symplectic structures are and how they are connected to Calabi-Yau structures on dg categories. More concretely, we will show that the cotangent complex to the moduli stack of a dg category A is isomorphic to the moduli stack of the *Calabi-Yau completion* of A, answering a conjecture of Keller-Yeung. This is joint work with Damien Calaque and Tristan Bozec arxiv.org/abs/2006.01069

Dopo aver richiamato alcune nozioni sui fibrati lineari big, si parlerà di fibrati vettoriali big sia su superfici K3 sia sul loro schema di Hilbert di punti. In particolare, verranno descritte alcune famiglie di fibrati big, stabili e globalmente generati.

The generalized Kummer variety K_n of an abelian surface A is the fibre of the natural map Hilb^{n+1}A->Sym^{n+1}A->A. Debarre described a Lagrangian fibration on K_n whose fibres are the kernels of JacC->A, where C are curves in a fixed linear system in A. In this talk we consider the dual of the Debarre system, constructed in a similar way to the duality between SL- and PGL-Hitchin systems described by Hausel and Thaddeus. We conjecture that these dual fibrations are mirror symmetric, in the sense that their (stringy) Hodge numbers are equal, and we verify this in a few cases. In fact, there is another isotrivial Lagrangian fibration on K_n. We can describe its dual fibration and verify the mirror symmetry relation in many more cases.

We will discuss an explicit graph formula, in terms of boundary strata classes, for the wall-crossing of universal (over Mbargn) Brill-Noether classes. More precisely, fix two stability conditions and for universal compactified Jacobians that are on different sides of a wall in the stability space. Then we can compare the two universal Brill-Noether classes on the two compactified Jacobians by pulling one of them back along the (rational) identity map. The calculation involves constructing a resolution by means of subsequent blow-ups. If time permits, we will discuss the significance of our formula and potential applications. This is joint with Alex Abreu.