IPM Algebraic Geometry Seminar
IPM is holding a biweekly zoom Algebraic Geometry seminar for Winter/Spring 2024.
Where: Zoom
Meeting ID: 908 611 6889
Passcode: 13440 × the number of lines on a cubic surface
Seminar mailing list: google group
Poster: here
Past Talks
Speaker: Razieh Ahmadian, Shahid Beheshti University (Iran)
Title: Hironaka's Question F and its Simplification
Abstract: A special case of Hironaka's QUESTION F, named F', asks about the strong
factorization of birational maps between reduced nonsingular algebraic schemes, which is still open. Suppose
that $\varphi : X\dashrightarrow Y$ is such a map, and let $U\subset X$ be the open subset where $\varphi$ is an isomorphism. This problem asks if there exists a diagram
$$
\xymatrix{ & Z \ar[dl]_{\varphi_{1}}\ar[dr]^{\varphi_{2}}\\
X \ar[rr]^{\varphi} & & Y}
$$
where the morphisms $\varphi_{1}$ and $\varphi_{2}$ are sequences of blow-ups of non-singular centers disjoint
from $U$. In this talk, we will discuss how strong factorization can be simplified by providing a complete answer to
the problem of toroidalization of morphisms, while we introduce the strong Oda conjecture.
Date: May 1, 2024
Speaker: Victoria Hoskins, Radboud University Nijmegen (Netherlands)
Title: Motives of stacks of bundles and sheaves on curves
Abstract: The geometry of moduli spaces and stacks of vector bundles on curves
have been intensively studied from different perspectives; for example,
via point counting over finite fields by Harder and Narasimhan, and
gauge theoretically by Atiyah and Bott over the complex numbers.
Following Grothendieck’s vision that a motive of an algebraic variety
should capture many of its cohomological invariants, Voevodsky
introduced a triangulated category of motives which partially realises
this idea. After describing some properties of this category, I will
present a formula for the motive of the moduli stack of vector bundles
on a smooth projective curve; this formula is compatible with classical
computations of invariants of this stack due to Harder, Atiyah--Bott and
Behrend--Dhillon. The proof involves rigidifying this stack using
Flag-Quot schemes parametrising Hecke modifications as well as a motivic
version of an argument of Laumon and Heinloth on the cohomology of small
maps, which is closely related to the Grothendieck-Springer resolution.
I will explain how to extend this to a formula for the stack of coherent
sheaves and, if there is time, I will give an overview of other motivic
descriptions of closely related moduli spaces. This is joint work with
Simon Pepin Lehalleur.
Date: April 17, 2024
Video link
Passcode: Sr4!XT6f
Speaker: Simon Pepin Lehalleur, University of Amsterdam (Netherlands)
Title: Cohomology and motives of moduli spaces of Higgs bundles and motivic mirror symmetry
Abstract: Higgs bundles are vector bundles equipped with an
additional "twisted endomorphism". Introduced by Nigel Hitchin in a context of mathematical physics,
they have turned to be central objects in differential and algebraic geometry. In particular, moduli spaces
of Higgs bundles have a very rich geometry that is both related to the geometry of moduli of vector bundles
but also has additional symplectic features. I will introduce these moduli spaces and discuss some of what
is known about their cohomology and their motivic invariants. There has been a lot of recent progress
in this direction and I will try to describe the main threads. I will conclude with a discussion of my
joint work with Victoria Hoskins on a motivic version of the "cohomological mirror symmetry" conjecture of Hausel and Thaddeus for SL_n and PGL_n Higgs bundles.
Date: February 28, 2024
Video link
Passcode: $2c%g*XF