Time: 14:00-15:00, June 8; 10:30-11:30, June 9-11
Venue：E14-220
Speaker: Tao Su, BIMSA
Abstract:
Character varieties are fundamental objects in non-abelian Hodge theory, topology, and mirror symmetry. A conjecture of Katzarkov-Noll-Pandit-Simpson predicts that the dual boundary complex of any smooth character variety is a sphere. Originally motivated by the search for a geometric interpretation of the cohomological P=W conjecture, this statement provides a topological description of the geometry at infinity of character varieties.
In this minicourse, I will explain a proof of the weak geometric P=W conjecture for all very generic character varieties, together with extensions to the wild case. The lectures will also discuss the role of microlocal geometry, Legendrian links, and mirror symmetry in the study of character varieties.
The precise content may be adjusted depending on the audience and the pace of the lectures, but a tentative plan is as follows.
Lecture 1. Dual Boundary Complexes and the Geometric P=W Conjecture
I will review the background on the dual boundary complexes of algebraic varieties. Along the way I will discuss the Kontsevich-Kollár-Xu conjecture, the geometric P=W conjecture, its relation to the cohomological P=W theorem, and the current state of the subject.
Lecture 2. Wild Character Varieties via Microlocal Geometry
I will explain a microlocal reformulation of the irregular Riemann–Hilbert correspondence over curves. This realizes wild character varieties as moduli spaces of constructible sheaves on punctured Riemann surfaces with microsupport constrained by Stokes Legendrian links encircling the punctures.
Lecture 3. Cell Decomposition and Dual Boundary Complexes of Character Varieties
I will explain a cell decomposition theorem for very generic character varieties via their relation to braid varieties. The key result decomposes the character variety into locally closed pieces of the form
$(\mathbb{C}^{\times})^{d-2b}\times \mathcal A$,
where $\mathcal A$ is stably isomorphic to $\mathbb C^b$. I will then explain how this decomposition leads to a proof of the weak geometric P=W conjecture for these varieties.
Lecture 4. Mirror P=W Conjecture for Character Varieties
Time permitting, I will discuss an interaction between character varieties and mirror symmetry. Inspired by mirror symmetry, Harder-Katzarkov-Przyjalkowski formulated a mirror P=W conjecture for log Calabi-Yau varieties. Character varieties are expected to provide a particularly rich class of such examples. For $GL_n$, this conjecture is essentially equivalent to Mellit's curious Hard Lefschetz theorem, whose proof combines the holomorphic symplectic geometry of character varieties with a weak form of the cell decomposition theorem. In general, the mirror P=W conjecture involves Langlands dual character varieties related by a twisted SYZ duality, leading to a rich direction of current research.
