Seminario di geometria algebrica e aritmetica di Pisa |
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I will discuss the notion of shifted symplectic structures along the stalks of constructible sheaves of derived stacks on stratified spaces. I will describe a general pushforward theorem producing relative symplectic forms and will explain explicit techniques for computing such forms. As an application I will describe a universal construction of Poisson structures on derived moduli of local systems on smooth varieties and will explain how symplectic leaves arise from fixing irregular types and local formal monodromies at infinity. This is a joint work with Dima Arinkin and Bertrand Toën.
I will discuss the Hilbert scheme of d points in affine n-space, with some examples. This space has many irreducible components for n at least 3 and is poorly understood. Nonetheless, in the limit where n goes to infinity, we show that the Hilbert scheme of d points in infinite affine space has a very simple homotopy type. In fact, it has the A^1-homotopy type of the infinite Grassmannian BGL(d-1). Many questions remain. (Joint with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, Maria Yakerson.)
I will discuss joint work with Daniel Bragg on the identification of derived invariants of smooth projective varieties in characteristic p, especially using information from crystalline cohomology.
A scheme X is a topological space -- which we denote by |X| -- and a sheaf of rings on the open subsets of |X|. We study the following natural but seldom considered questions. How to read off properties of X from |X|? Does |X| alone determine X? Joint work with Max Lieblich, Martin Olsson, and Will Sawin.
Il seminario di David Rydh è stato annullato. Al suo posto parlerà Mattia Talpo.
I will report on some joint work with Piotr Achinger, about a “Betti realization” functor for varieties over the formal punctured disk Spec C((t)), i.e. defined by polynomials with coefficients in the field of formal Laurent series in one variable over the complex numbers. We give two constructions producing the same result, and one of them is via “good models” over the power series ring C[[t]] and the “Kato-Nakayama” construction in logarithmic geometry, that I will review during the talk.
We use the theory of cohomological invariants for algebraic stacks to completely describe the Brauer group of
the moduli stacks H_g of genus g hyperellitic curves over fields of characteristic zero, and the prime-to-char(k) part in positive characteristic. It turns out that the (non-trivial part of the) group is generated by cyclic algebras, by an element coming from a map to the classifying stack of étale algebras of degree 2g+2, and when g is odd by the Brauer-Severi fibration induced by taking the quotient of the universal curve by the hyperelliptic involution. This paints a richer picture than in the case of elliptic curves, where all non-trivial elements come from cyclic algebras.
This is joint work with Andrea di Lorenzo.
We investigate the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field, under the assumption that the surface is geometrically CH_0-trivial. By this we mean that over any algebraically closed field extension, the degree map on zero-dimensional Chow group of the surface is an isomorphism. This applies to Enriques surfaces. When the Néron-Severi group has no torsion, we recover earlier results of A. Pirutka. This is joint work with Federico Scavia.
Line bundles L on projective toric varieties can be understood as formal
differences of convex polyhedra in the character lattice. We
show how it is possible to use this language for understanding the
cohomology of L by studying the set-theoretic difference .
(This is joint work with Jarek Buczinski, Lars Kastner, David Ploog, and
Anna-Lena Winz.)
We give a geometric representation theory proof of a mild version of the Beauville-Voisin Conjecture for Hilbert schemes of K3 surfaces, namely the injectivity of the cycle map restricted to the subring of Chow generated by tautological classes. Our approach involves lifting formulas of Lehn and Li-Qin-Wang from cohomology to Chow groups, and using them to solve the problem by invoking the irreducibility criteria of Virasoro algebra modules, due to Feigin-Fuchs. Joint work with Davesh Maulik.
The present talk is 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.
I will discuss some ongoing work aiming at studying the action of the motivic Galois group on fundamental groups of algebraic varieties conveniently completed.
In this talk, I will give an overview of the construction of a triangulated category of motives for log smooth log schemes
over a field k, based on the notion of finite log correspondence, in analogy to Voevodsky’s DM(k). The affine line
is replaced in this context by the “cube” (P^1, \infty), i.e. the log scheme P^1_1 with log
structure coming from the divisor at infinity, as one does in the theory of motives with modulus à la Kahn-Saito-Yamazaki.
This is a joint work in progress with Doosung Park (Zurich) and Paul Arne Ostvaer (Oslo).
In the context of mirror symmetry, the moduli space of complex Calabi-Yau varieties acquires canonical local coordinates near a "large complex structure limit point". In characteristic p geometry, the formal deformation space of an ordinary Calabi-Yau variety tends to have such canonical coordinates ("Serre-Tate parameters") as well. As observed e.g. by Jan Stienstra, these situations are formally very similar, and one would like to compare the two when both make sense. A framework for doing this could be supplied by a version of Serre-Tate theory for log Calabi-Yau varieties. In my talk, I will describe a first step in this direction; a construction of canonical liftings modulo p^2 of certain log schemes. I will link this to a question of Keel describing global moduli of maximal log Calabi-Yau pairs.
We define a non-commutative version of the punctured formal
neighbourhood in algebraic geometry, and apply this construction to give
decompositions of the (derived) stacks Coh^-(X), Perf(X) and Bun_G(X)
along a non-linear flag of sub-stacks of X. This is joint work with B. Hennion and M. Porta.
Time permitting, I will also sketch an application to the construction of Hecke operators associated
to a non-linear flag on a surface.
This is a report on joint work with Bagnarol and Perroni, available at arxiv:1907.00826. For any locally free coherent sheaf on a fixed smooth projective curve, we study the class, in the Grothendieck ring of varieties, of the Quot scheme that parametrizes zero-dimensional quotients of the sheaf. We prove that this class depends only on the rank of the sheaf and on the length of the quotients. As an application, we obtain an explicit formula that expresses it in terms of the symmetric products of the curve.
If time allows, we will discuss further work of Andrea Ricolfi extending the result from smooth curves to arbitrary smooth projective manifolds, and its application to Bagnarol's thesis (2019) on the Hodge motive of genus zero stable maps to Grassmannians.
25 years ago Vafa and Witten predicted generating functions for the
Euler numbers of the moduli spaces of sheaves on algebraic surfaces.
In this talk I review joint work with Martijn Kool to interpret and
check these predictions in terms of virtual Euler numbers, and to extend
them to finer invariants like chi_y genus. Time permitting I will also
mention recent results on Chern numbers of tautological sheaves,
and Verlinde type formulas.
Semi-log-canonical surfaces with ample canonical divisor are called stable. Their moduli
space is a natural compactification (the KSBA compactification) of the Gieseker moduli space of canonical models of
surfaces of general type.
Among the singularities that are allowed in stable surfaces, we have cyclic
quotient singularities 1/m(1; q) and a special role is played by those with m =dn^2, q=dna-1 gcd(n; a) = 1.
These singularities together with all du Val singularities are called
T-singularities.
We give bounds on such singularities and describe some constructions.
The moduli space of stable surfaces is a modular compactification of the
Gieseker moduli space of (canonical models of) surfaces of general type but
has components consisting solely of non-smoothable surfaces.
I will construct a smoothing of a particular reducible surface X with K^2 = 1
and p_g = 2, and thus show that the family of such surfaces is indeed
contained in the closure of the smooth locus.
X is the union of a singular K3 surface and a singular Enriques surface,
glued along an elliptic curve.
Every abelian variety A over a field k admits a universal
extension by an affine k-group scheme. The talk will present
a construction of this universal affine extension (first due to
Serre when k is algebraically closed of characteristic zero)
and discuss its structure, with applications to the category
of homogeneous vector bundles over A.
In the talk I will introduce the crystalline site of a
variety in positive characteristic, discuss the notion of crystals and
isocrystals over it and define its crystalline fundamental group. I
will then discuss Berthelot's conjecture and its relation with the
exactness of the homotopy sequence of a fibration.
Let X be a Fano threefold, and let S be a smooth anticanonical surface (hence a K3) lying in X. Any moduli space of simple vector bundles on S carries a holomorphic symplectic structure. Following an idea of Tyurin, I will show that in some cases those vector bundles which come from X form a Lagrangian subvariety of the moduli space. Most of the talk will be devoted to concrete examples of this situation.
For the abstract click here.