It is classical to study the geometry of projective varieties
over algebraically closed fields through the properties of various
positive cones of divisors or curves. Several counterexamples have
shifted attention from the higher (co)dimensional...
What are the possible Hodge numbers of a smooth complex
projective variety? We construct enough varieties to show that many
of the Hodge numbers can take all possible values satisfying the
constraints given by Hodge theory. For example, there are...
I will explain how infinite sequences of flops give rise to some
interesting phenomena: first, an infinite set of smooth projective
varieties that have equivalent derived categories but are not
isomorphic; second, a pseudoeffective divisor for which...
Beauville and Voisin proved that decomposable cycles
(intersections of divisors) on a projective K3 surface span a
1-dimensional subspace of the (infinite-dimensional) group of
0-cycles modulo rational equivalence. I will address the
following...
Report on R. Virk's arXiv:1406.4855v3. This is a fun, short and
simple note with variations on the well-known theme by G. Laumon
that the Euler characteristics with and without compact supports
coincide.
I will discuss some of the topology of the fibers of proper
toric maps and a combinatorial invariant that comes out of this
picture. Joint with Luca Migliorini and Mircea Mustata.
I will outline a construction of "tropical current", a positive
closed current associated to a tropical variety. I will state basic
properties of tropical currents, and discuss how tropical currents
are related to a version of Hodge conjecture for...
The relevant preprints are: arXiv:1405.5154 "The Fano variety of
lines and rationality problem for a cubic hypersurface", Sergey
Galkin, Evgeny Shinder arXiv:1405.4902 "On two rationality
conjectures for cubic fourfolds", Nicolas Addington
