We use equivariant localization techniques to give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In particular, we identify a candidate target for the elliptic Bismut–Chern character.

We study the non‑klt locus of singularities of pairs. We show that given a pair $(X,B)$ and a projective morphism $X \to Z$ with connected fibres such that $-(K_X + B)$ is nef over $Z$, the non‑klt locus of $(X,B)$ has at most two connected components near each fibre of $X \to Z$. This was conjectured by Hacon and Han.

In a different direction we answer a question of Mark Gross on connectedness of the non‑klt loci of certain pairs. This is motivated by constructions in Mirror Symmetry.

We present a series of results concerning the interplay between the scalar curvature of a manifold and the mean curvature of its boundary. In particular, we give a complete topological characterization of those compact $3$-manifolds that support Riemannian metrics of positive scalar curvature and mean-convex boundary and, in any such case, we prove that the associated moduli space of metrics is path-connected. The methods we employ are flexible enough to allow the construction of continuous paths of positive scalar curvature metrics with minimal boundary, and to derive similar conclusions in that context as well. Our work relies on a combination of earlier fundamental contributions by Gromov–Lawson and Schoen–Yau, on the smoothing procedure designed by Miao, and on the interplay of Perelman’s Ricci flow with surgery and conformal deformation techniques introduced by Codá Marques in dealing with the closed case.

Painlevé VI equation is well known as a class of nonlinear ODEs whose solutions govern the isomonodromic deformation of a second order linear Fuchsian ODE with five regular singularities on $\mathbb{CP}^1$. We propose the question how the monodromy effects the behavior of solutions. In this paper, the second of a series devoted to this question, we study Painlevé VI equation with parameters $(\frac{(2n+1)^2}{8}, \frac{-1}{8}, \frac{1}{8}, \frac{3}{8})$ with $n \in N$. A solution $\lambda(t)$ is called unitarizable if the monodromy group of the associated linear ODE of $\lambda(t)$ is conjugated to a subgroup of the unitary group $SU(2)$. For example, any algebraic solution is unitarizable. In Part I $\href{https://doi.org/10.1016/j.geomphys.2017.01.016 }{[7]}$ we studied the special case $n = 1$. In this paper we show that

(i) Any unitarizable solution has no poles in $\mathbb{R}\setminus{0, 1}$ for all $n \in \mathbb{N}$.

(ii) For $n = 2$, there are only $3$ unitarizable solutions without poles in $\mathbb{C} \setminus \{0, 1 \}$.

(iii) For n$ = 2$, every single-valued branch of any unitarizable solution in $\mathbb{C} \setminus (-\infty, 1]$ has at most $5$ poles (this upper bound is optimal).

These results are obtained by applying the PDE result of a singular Liouville equation on flat tori and the pre-modular form $Z^{(m)}_{r,s} (\tau)$ introduced in [15, $\href{https://doi.org/10.48550/arXiv.1502.03297}{29}$]. The study of the zero structure of $Z^{(m)}_{r,s} (\tau)$ is of independent interest.

The deformed Hermitian–Yang–Mills equation is a complex Hessian equation on compact Kähler manifolds that corresponds to the special Lagrangian equation in the context of the Strominger–Yau–Zaslow mirror symmetry $\href{https://doi.org/10.1016/0550-3213(96)00434-8 }{[\textrm{SYZ96}]}$. Recently, Chen $\href{ https://doi.org/10.1007/s00222-021-01035-3 }{[\textrm{Che21}]}$ proved that the existence of the solution is equivalent to a uniform stability condition in terms of holomorphic intersection numbers along test families. In this paper, we establish an analogous stability result not involving a uniform constant in accordance with a recent work on the $J$-equation by Song $\href{https://doi.org/10.48550/arXiv.2012.07956}{[\textrm{Son}20]}$, which makes further progress toward Collins–Jacob–Yau’s original conjecture $\href{https://dx.doi.org/10.4310/CJM.2020.v8.n2.a4 }{[\textrm{CJY15}]}$ in the supercritical phase case. In particular, we confirm this conjecture for projective manifolds in the supercritical phase case.

We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving a backwards scattering problem for the Einstein vacuum equations with characteristic data prescribed on the event horizon and (in the limit) at null infinity. The class admits the full “functional” degrees of freedom for the vacuum equations, and thus our solutions will in general possess no geometric or algebraic symmetries. It is essential, however, for the construction that the scattering data (and the resulting solution spacetime) converge to stationarity exponentially fast, in advanced and retarded time, their rate of decay intimately related to the surface gravity of the event horizon. This can be traced back to the celebrated redshift effect, which in the context of backwards evolution is seen as a blueshift.

We introduce the concept of improvability of the dominant energy scalar, and we derive strong consequences of non-improvability. In particular, we prove that a non-improvable initial data set without local symmetries must sit inside a null perfect fluid spacetime carrying a global Killing vector field. We also show that the dominant energy scalar is always almost improvable in a precise sense. Using these main results, we provide a characterization of Bartnik mass minimizing initial data sets which makes substantial progress toward Bartnik’s stationary conjecture.

Along the way we observe that in dimensions greater than eight there exist pp-wave counterexamples (without the optimal decay rate for asymptotically flatness) to the equality case of the spacetime positive mass theorem. As a consequence, we find counterexamples to Bartnik’s stationary and strict positivity conjectures in those dimensions.

We construct valuations on the space of finite-valued convex functions using integration of differential forms over the differential cycle associated to a convex function. We describe the kernel of this procedure and show that the intersection of this space of smooth valuations with the space of all continuous dually epi-translation invariant valuations on convex functions is dense in the latter. As an application, we obtain a description of $1$-omogeneous, continuous, dually epi-translation invariant valuations that are invariant with respect to a compact subgroup operating transitively on the unit sphere.