We construct a generalized Witten genus for spin$^c$ manifolds, which takes values in level 1 modular forms with integral Fourier expansion on a class of spin manifolds called string$^c$ manifolds. We also construct a mod 2 analogue of the Witten genus for $8k+2$ dimensional spin manifolds. The Landweber-Stong type vanishing theorems are proven for the generalizedWitten genus and the mod 2 Witten genus on string$^c$ and string (generalized) complete intersections in (product of) complex projective spaces respectively.

We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil$_3$ and the Lie group Sol$_3$ endowed with their standard left-invariant Riemannian metrics. If $\mathcal{S}$ is a properly immersed minimal surface in Nil$_3$ that lies on one side of some entire minimal graph $\mathcal{G}$, then $\mathcal{S}$ is the image of $\mathcal{G}$ by a vertical translation. If $\mathcal{S}$ is a properly immersed minimal surface in Sol$_3$ that lies on one side of a special plane $\mathcal{E}^t$ (see the discussion just before Theorem 1.5 for the definition of a special plane inSol$_3$), then $\mathcal{S}$ is the special plane $\mathcal{E}^u$ for some $u\in \mathbb{R}$.

We study moduli spaces of O'Grady’s ten-dimensional irreducible symplectic manifolds. These moduli spaces are covers of modular varieties of dimension 21, namely quotients of hermitian symmetric domains by a suitable arithmetic group. The interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group. This makes it different from both the K3 and the $K3^{[n]}$ case, which are of dimension 19 and 20 respectively.

We describe a notion of ampleness for line bundles on orbifolds with cyclic quotient singularities that is related to embeddings in weighted projective space, and prove a global asymptotic expansion for a weighted Bergman kernel associated to such a line bundle.

We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (non-reductive) automorphisms of weighted projective space then formally gives a moduli space of orbifolds. We show how to express this as a reductive quotient and so a GIT problem, thus defining a notion of stability for orbifolds.

We then prove an orbifold version of Donaldson’s theorem: the existence of an orbifold Kähler metric of constant scalar curvature implies K-semistability.

By extending the notion of slope stability to orbifolds, we therefore get an explicit obstruction to the existence of constant scalar curvature orbifold Kähler metrics. We describe the manifold applications of this orbifold result, and show how many previously known results (Troyanov, Ghigi-Kollár, Rollin-Singer, the AdSCFT Sasaki-Einstein obstructions of Gauntlett-Martelli-Sparks- Yau) fit into this framework.

We show that the attenuated geodesic ray transform on two dimensional simple surfaces is injective. Moreover, we give a stability estimate and develop a reconstruction procedure.