Seminario di geometria algebrica e aritmetica di Pisa |
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In this talk we will report on ongoing work aiming to establish a Dubrovin conjecture for general Frobenius
manifolds. The Dubrovin conjecture was formulated in 1998 (with a very precise statement in 2018)
as a relation between the Frobenius manifold coming from the quantum cohomology of a Fano manifold X
and the derived category of coherent sheaves of X. We will give some speculations about how to extend that relation
to one between semisimple Frobenius manifolds and some categories. We will sketch the situation in a particular example (3-point Ising model).
The Chow ring of moduli of curves is an important invariant
which is the subject of extensive investigations and conjectures, since
Mumford first introduced the topic in his pioneering work.
Chow-Witt groups are a recent refinement of the usual Chow group, and it
is then a natural question to ask whether one can say something on the
Chow-Witt groups of moduli of curves.
In this talk, I would like to present a recent computation of the
Chow-Witt ring of the moduli stack of smooth/stable elliptic curves and
explain the meaning of the generators of these rings in terms of
quadratic invariants of families of elliptic curves. Joint work with
Lorenzo Mantovani.
Recently there has been spectacular progress, due to many scholars, on the construction of moduli (called K-moduli) of Fano varieties using K-stability (which is related to the existence of Kähler-Einstein metrics). It is a natural question to understand the geometry of these (newly constructed) spaces. Although smooth Fano varieties have unobstructed deformations, in joint work with Kaloghiros we constructed the first examples of obstructed K-polystable Fano varieties by using toric geometry. These give singular points on K-moduli of Fanos. In this talk I will try to explain these constructions; as a corollary I will show that K-moduli of Fano of dimension at least 3 can have arbitrarily many local branches.
We give an outline of a (conjectural) construction of a cohomology theory for smooth and proper varieties over local fields with values in locally compact groups, satisfying a Pontrjagin duality. For certain weights, we give an ad hoc construction which satisfies such a duality unconditionally.
Dynamical Galois groups are invariants associated to dynamical systems generated by the iteration of a self-rational map of P^1. These are still very mysterious objects, and it is conjectured that abelian groups only appear in very special cases. We will show how the problem is deeply related to a dynamical property of these rational maps (namely that of being post-critically finite) and we will explain how to approach and prove certain non-trivial cases of the conjecture. This is based on joint works with A. Ostafe, C. Pagano and U. Zannier.
Irreducible holomorphic symplectic (IHS) varieties can be thought as a generalization of hyperkähler manifolds allowing singularities.
Among them we can find for example moduli spaces of sheaves on K3 and abelian surfaces, which have been recently shown to play a crucial role in nonabelian Hodge theory.
After recalling the main features of IHS varieties, I will present several results concerning their intersection cohomology and the perverse filtration associated with a Lagrangian fibration, establishing a compact analogue of the celebrated P=W conjecture by de Cataldo, Hausel and Migliorini for varieties which admit a symplectic resolution.
The talk is based on joint works with Mirko Mauri, Junliang Shen and Qizheng Yin.
Motivated by a conjecture of Kapranov-Ginzburg-Vasserot, we will explain
how to reinterpret the classical Poisson structure on the affine
Grassmannian using tools from derived symplectic (and Poisson) geometry.
Our approach is based on the study of symplectic structures on derived
stacks whose cotangent complex is not perfect, and therefore uses in a
crucial way the formalism of Tate modules. The talk is based on joint
work with Mauro Porta and Pavel Safronov.
By the work of Richard Hain, the archimedean height pairing on ordinary algebraic cycles can be interpreted as an invariant of an associated mixed Hodge structure. In this talk, we will present
a similar construction for higher cycles in the Bloch complex. Families of higher cycles produce admissible variations of mixed Hodge structure. We will describe the asymptotic behavior of the
height pairing in the case where the associated variation of mixed Hodge structure is Hodge-Tate.
This is joint work with J. Burgos Gil and S. Goswami.
The surprisingly deep analogy between compact Riemann surfaces and compact metric graphs lies at the heart of many recent developments in tropical geometry. In this talk I will give a short introduction to tropical geometry via this analogy focusing on linear systems. I will explain the process of tropicalization and discuss the realizability problem as well as some its partial solutions (and why a general solution is provably out or reach). This talk is based on joint work with Madeline Brandt, Martin Moeller and Annette Werner, as well as Dmitry Zakharov.
Seshadri constants measure local positivity of line bundles and it is an open
question if they can be irrational on algebraic surfaces. I will recall this
concept and prove that for a general point on a general hypersurface of
degree md in P(1,1,1,m) the Seshadri constant \epsilon (O_X(1), x) approaches the possibly irrational number \sqrt d as m
grows (d >1 and m>2).
This is joint work with A. Küronya.