2025 CMS Winter Meeting
Toronto, Dec 5 - 8, 2025
Recent Developments in Complex Analysis and Geometry
Org: Alexander Brudnyi (University of Calgary), Rasul Shafikov (Western University) and Mahishanka Withanachchi (University of Calgary)
[PDF]
Org: Alexander Brudnyi (University of Calgary), Rasul Shafikov (Western University) and Mahishanka Withanachchi (University of Calgary)
[PDF]
- HARSHITH ALAGANDALA, Western University
Local Polynomial Convexity at Hyperbolic CR-singularity in $M^n \subset \mathbb{C}^n$ [PDF]
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Let $M$ be a smooth manifold of dimension $n$ embedded in $\mathbb{C}^n$.
If $T_pM \subset T_p\mathbb{C}^n$ is a totally real subspace for $p\in M$, then $M$ is locally polynomially convex at $p$.
For a generic embedding $M$,
we are interested in assessing polynomial convexity of $M$ at a CR-singularity, i.e., at a point $p\in M$ where $T_pM$ is not totally real.
An order one CR-singularity in $M$ can be broadly classified as elliptic and hyperbolic.
It is known that elliptic points give obstruction to polynomial convexity.
In the case $n=2$, $M^2 \subset \mathbb{C}^2$ is locally polynomially convex at a hyperbolic complex point.
We investigate local polynomial convexity of $M^n \subset \mathbb{C}^n$ at hyperbolic points
in higher dimension.
- ROBERTO ALBESIANO, Waterloo
- TATYANA BARRON, UWO
- ILIA BINDER, Toronto
- BLAKE BOUDREAUX, Arkansas
- ISABELLE CALENDAR, Université Gustave Eiffel
- DEBRAJ CHAKRABARTI, Central Michigan University
Restricted type estimates and the Bergman Projection [PDF]
- TATYANA BARRON, UWO
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We obtain (weighted) restricted-type estimates for the Bergman projection operator on monomial polyhedra, a class of domains generalizing the Hartogs triangle. A restricted-type estimate is an estimate in the $L^p$-norm on an operator, which however holds only on characteristic functions. From these restricted-type estimates, we recapture $L^p$-boundedness results of the Bergman projection on these domains. On some monomial polyhedra, we show that the Bergman projection could fail to be of weak type $(q_*,q_*)$ , where $q_*$ denotes the right end-point of the interval of $L^p$-boundedness of the Bergman projection.
- DAN COMAN, Syracuse University
Tian’s theorem for Grassmannian embeddings and degeneracy sets of random sections [PDF]
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Let $(X,\omega)$ be a compact Kähler manifold, $(L,h^L)$ be a positive line bundle, and $(E,h^E)$ be a Hermitian holomorphic vector bundle of rank $r$ on $X$. We show that the pullback by the Kodaira embedding associated to $L^p\otimes E$ of the $k$-th Chern class of the dual of the universal bundle over the Grassmannian converges as $p\to\infty$ to the $k$-th power of the Chern form $c_1(L,h^L)$, for $0\leq k\leq r$. As a consequence we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers $L^p\otimes E$ is $c_1(L,h^L)^r$. Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections. This is joint work with Turgay Bayraktar, Bingxiao Liu and George Marinescu.
- JESSE HULSE, Manitoba
- LUKA MERNIK, Florida Polytechnic University
- PIERRE-OLIVIER, UQTR
- ANDY RAICH, University of Arkansas
Tower multitype and compactness of the dbar-Neumann operator in complex manifolds [PDF]
- LUKA MERNIK, Florida Polytechnic University
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In this talk, I will discuss a new approach to establishing compactness of the $\bar\partial$-Neumann operator on $(p,q)$-forms on not necessarily pseudoconvex domains in complex manifolds. I will introduce a construction called the tower multitype and use it to build a stratification of the boundary. The stratification implies Property $(P_q)$ which in turn implies compactness of the $\bar\partial$-Neumann operator. I will also provide examples.
The result is joint work with Professor Dmitri Zaitsev of Trinity College Dublin.
- DROR VAROLIN, Stony Brook
- LIS VIVAS, Ohio State
- YUNUS ZEYTUNCU, University of Michigan-Dearborn
Spectral Theory of the Kohn Laplacian on Quotient Manifoldsv [PDF]
- LIS VIVAS, Ohio State
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In this talk, we study the spectrum of the Kohn Laplacian on quotient manifolds. In particular, we relate the asymptotic properties of the Kohn Laplacian's eigenvalues on sphere quotients to the group action.