We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex space admitting an exhaustion function whose Levi form has at least two positive eigenvalues at every point outside a compact set, and this condition is essential. We also construct a Stein neighborhood basis of any compact complex curve with $C^2$-boundary in a complex space

We study a self-dual $N=1$ super vertex operator algebra and prove that the full symmetry group is Conway's largest sporadic simple group. We verify a uniqueness result that is analogous to that conjectured to characterize the Moonshine vertex operator algebra (VOA). The action of the automorphism group is sufficiently transparent that one can derive explicit expressions for all the McKay-Thompson series. A corollary of the construction is that the perfect double cover of the Conway group may be characterized as a point-stabilizer in a spin module for the Spin group associated to a $24$-dimensional Euclidean space

We introduce the space $\Pi (G)$ of equivalence classes of $\pi$-points of a finite group scheme $G$ and associate a subspace $\Pi (G{)}_M$ to any $G$-module $M$. Our results extend to arbitrary finite group schemes $G$ over arbitrary fields $k$ of positive characteristic and to arbitrarily large $G$-modules, the basic results about “cohomological support varieties” and their interpretation in terms of representation theory. In particular, we prove that the projectivity of any (possibly infinite-dimensional) $G$-module can be detected by its restriction along $\pi$-points of $G$. Unlike the cohomological support variety of a $G$-module $M$, the invariant $M\mapsto \Pi (G{)}_M$ satisfies good properties for all modules, thereby enabling us to determine the thick, tensor-ideal subcategories of the stable module category of finite-dimensional $G$-modules. Finally, using the stable module category of $G$, we provide $\Pi (G)$ with the structure of a ringed space which we show to be isomorphic to the scheme $\mathrm{Proj}{H}^{•}(G,k)$

For each connected complex reductive group G, we find a family of new examples of complex quasi-Hamiltonian G-spaces with G-valued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on principal G-bundles over a disk, and they generalise the conjugacy class example of Alekseev, Malkin, and Meinrenken [3] (which appears in the simple pole case). Using the “fusion product” in the theory, this gives a finite-dimensional construction of the natural symplectic structures on the spaces of monodromy/Stokes data of meromorphic connections over arbitrary genus Riemann surfaces, together with a new proof of the symplectic nature of isomonodromic deformations of such connections

We provide correction to the formulation and the proof of Theorem 3.4 in the article “Cluster algebras and Weil-Petersson forms” ([GSV])