Events

Rasul Shafikov, Western University

A celebrated theorem of Duval and Sibony characterizes rationally convex real submanifolds in complex Euclidean spaces as those isotropic with respect to a Kahler form. I will discuss how the results in symplectic geometry can be used to obtain some new results in the approximation theory.

MC 5417

Thomas Brazelton, University of Pennsylvania

Classical enumerative geometry asks geometric questions of the form "how many?" and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of "equivariant enumerative geometry", which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the sum of regular representations of the orbits of solutions to an equivariant enumerative problem are conserved. We leverage this to compute the S4 orbits of the 27 lines on any symmetric cubic surface.

MC 5417

Javier González Anaya, University of California at Riverside

We discuss the Mori dream space (MDS) property for blow-ups of toric surfaces defined by rational plane triangles at a general point. We consider a parameter space of all such triangles and show how it can be used to prove the MDS property for these varieties. Furthermore, this parameter space helps explain most known results in the area and has also led to surprising new results, including examples of such surfaces with a semi-open effective cone. This is joint work with José Luis González and Kalle Karu.

MC 5403

Scott Wilson, City University of New York

A difficult open problem is to determine if there are topological obstructions to complex structures on smooth manifolds (of even dimension greater than or equal to six) beyond the fairly well-understood obstructions to almost complex structures.  In this talk, I will explain that certain types of complex structures are homotopically obstructed in these dimensions, where the “types” are organized by the structure of the underlying bicomplex of differential forms.  To establish this, I’ll describe some numerical inequalities for complex manifolds of the form “Topology is less than or equal to Complex-analysis”. The topology invariants roughly measure the failure of the algebra of differential forms to be equivalent to its cohomology, and the complex-analytic invariants measure the “wildness” of the bi-complex of differential forms. Explicit examples will be given. This is joint work with Jonas Stelzig.

MC 5417

Luca Di Cerbo, University of Florida

A well-known conjecture of Dennis Sullivan (Abel Prize 2022) asserts that a hyperbolic n-manifold cannot admit a complex structure. This conjecture is known to be true in dimension four (Toledo, Carlson, LeBrun). In this talk, I will outline a new proof of the fact that a hyperbolic 4-manifold cannot support a complex structure. This new proof has some nice features, and it generalizes to show that all extended graph 4-manifolds with at least one pure real-hyperbolic piece cannot support a complex structure.  This is joint work with M. Albanese.

MC 5417

Lisa Marquand, Stony Brook University

Compact Hyperkähler manifolds are one of the building blocks of Kähler manifolds with trivial first Chern class, but very few examples are known. One strategy for potentially finding new examples is to look at finite groups of symplectic automorphisms of the known examples and study the fixed loci or quotient. In this talk, we will obtain a classification of birational symplectic involutions of manifolds of OG10 type. We do this from three vantage points: via involutions of the Leech lattice, via involutions of cubic fourfolds, and lattice enumeration via a modified Kneser’s neighbor algorithm. In particular, we exhibit geometric realizations of these involutions in three cases. If time permits, we will mention ongoing work to identify the fixed loci in one of these examples.

MC 5417

Alexander Yampolsky, V.N. Karazin National University, Kharkiv, Ukraine

A vector field ξ on a Riemannian manifold (M,g) defines a mapping ξ:  M→TM (or ξ: M→T₁M in case of |ξ|= 1). Endowing TM with the Sasaki metric gives rise to the Riemannian metric on ξ(M) ⊂ TM or ξ(M) ⊂ T₁M, respectively. This idea allows to assign the geometric properties from the geometry of submanifolds to the vector field. So, one can talk about the intrinsic or extrinsic geometry of vector fields.

The most developed idea in a given setting is the idea of harmonic and minimal unit vector fields. The report outlines the geometry of unit vector fields focused on examples of minimal, harmonic and totally geodesic properties. Mean curvature of the Reeb vector field on (α,β) - trans-Sasakian manifold will be presented as well as its minimality and total geodesity conditions. The properties of invariant unit vector fields on the oscillator Lie group will be considered in more detail.

MC 5417

Aleksandar Milivojevic, Max Planck Institute for Mathematics, Bonn

By thinking of the six-sphere S6 as the unit sphere in the imaginary octonions, one detects a real projective seven-space RP7 in the space of all almost complex structures on S6. On the other hand, using the Haefliger-Sullivan rational homotopy theoretic model for the space of sections of a fiber bundle applied to the twistor space construction, one can abstractly calculate that the rational homology of the space of (orientation-compatible) almost complex structures on S6 agrees with that of RP7. Sullivan asked whether the inclusion of the octonionic RP7 into the space of all almost complex structures is a homotopy equivalence. We show that it is not, though it is a rational homology equivalence that induces an isomorphism on fundamental groups. We can further describe the homotopy fiber of this inclusion. 

On a related note, over six-manifolds, almost complex structures correspond to embedded half-dimensional submanifolds of the twistor space, and hence one obtains numerical invariants via their homological intersection. Time permitting, we compute these numbers concretely over the six-sphere and other six-manifolds, and comment on their relation to integrability. 

This is joint work with Bora Ferlengez and Gustavo Granja.

MC 5417

Andrei Teleman, Aix-Marseille University

Let \(X\) be a connected, compact complex manifold, and \(S\subset X\) be a separating real hypersurface. \(X\) decomposes as a union of compact complex manifolds with boundary \(\bar X^\pm\) with \(\bar X^+\cap \bar X^-=S\). Let \(\mathcal{M}\) be the moduli space of \(S\)-framed holomorphic bundles on \(X\), i.e. of pairs \((E,\theta)\) (of fixed topological type) consisting of a holomorphic bundle \(E\) on \(X\) endowed with a differentiable trivialization \(\theta\) on \(S\). This moduli space is the main object of a joint research project with Matei Toma.

The problem addressed in my talk: compare, via the obvious restriction maps, the moduli space \(\mathcal{M}\) with the corresponding Donaldson moduli spaces \(\mathcal{M}^\pm\) of boundary framed holomorphic bundles on \(\bar X^\pm\). The restrictions to \(\bar X^\pm\) of an \(S\)-framed holomorphic bundle \((E,\theta)\) are boundary framed formally holomorphic bundles \((E^\pm,\theta^\pm)\) which induce, via \(\theta^\pm\), the same tangential Cauchy-Riemann operators on the trivial bundle on \(S\). Therefore one obtains a natural map from \(\mathcal{M}\) into the fiber product \(\mathcal{M}^-\times_\mathcal{C}\mathcal{M}^+\) over the space \(\mathcal{C}\) of Cauchy-Riemann operators on the trivial bundle on \(S\).

Our result states: this map is bijective. Note that, by theorems due to S. Donaldson and Z. Xi, the moduli spaces \(\mathcal{M}^\pm\) can be identified with moduli spaces of boundary framed Hermitian Yang-Mills connections.

This seminar will be held both online and in person:

• Room: MC 5417
• Zoom link: https://uwaterloo.zoom.us/j/96883292635?pwd=KytGYnEvRmhyTTV1NC9Gc2dnT05oQT09

Gwyneth Moreland, Harvard University

We compute some higher (co)dimension nef and effective cones of the Hilbert scheme of 3 points in \(\mathbb{P}^3\). This involves studying the orbits of the PGL action on the Hilbert scheme, as well as extending Mallavibarrena and Sols' bases for the Chow groups of Hilbert schemes of points on \(\mathbb{P}^2\) to the case of the Hilbert scheme of 3 points in \(\mathbb{P}^3\). This work builds on results of Ryan and Stathis.

MC 5417