Let $X$ be a smooth complex projective algebraic variety. Given a line bundle $\mathcal{L}$ over $X$ and an integer $r \gt 1$, one defines the stack $\sqrt[r]{\mathcal{L} / X}$ of $r$-th roots of $\mathcal{L}$. Motivated by Gromov-Witten theoretic questions, in this paper we analyze the structure of moduli stacks of genus $0$ twisted stable maps to $\sqrt[r]{\mathcal{L} / X}$. Our main results are explicit constructions of moduli stacks of genus $0$ twisted stable maps to $\sqrt[r]{\mathcal{L} / X}$ starting from moduli stacks of genus $0$ stable maps to $X$. As a consequence, we prove an exact formula expressing genus $0$ Gromov-Witten invariants of $\sqrt[r]{\mathcal{L} / X}$ in terms of those of $X$.

We consider the evolution of compact surfaces by fully nonlinear, parabolic curvature flows for which the normal speed is given by a smooth, degree one homogeneous function of the principal curvatures of the evolving surface. Under no further restrictions on the speed function, we prove that initial surfaces on which the speed is positive become weakly convex at a singularity of the flow. This generalises the corresponding result of Huisken and Sinestrari for the mean curvature flow to the largest possible class of degree one homogeneous surface flows.

On a (pseudo-)Riemannian manifold $(\mathcal{M}, g)$, some fields of endomorphisms, i.e. sections of $\mathrm{End}(T \mathcal{M})$, may be parallel for $g$. They form an associative algebra $\mathfrak{e}$, which is also the commutant of the holonomy group of $g$. As any associative algebra, $\mathfrak{e}$ is the sum of its radical and of a semi-simple algebra $\mathfrak{s}$. This $\mathfrak{s}$ may be of eight different types; see C. Boubel, The algebra of the parallel endomorphisms of a pseudo-Riemannian metric: semi-simple part. Then, for any self adjoint nilpotent element $N$ of the commutant of such an $\mathfrak{s}$ in $\mathrm{End}(T \mathcal{M})$, the set of germs of metrics such that $\mathfrak{e} \supset \mathfrak{s} \cup \{ N \}$ is non-empty. We parametrize it. Generically, the holonomy algebra of those metrics is the full commutant $\mathfrak{o}(g)^{\mathfrak{s} \cup \{ N \} }$ and then, apart from some “degenerate” cases, $\mathfrak{e} = \mathfrak{s} \oplus (N)$, where $(N)$ is the ideal spanned by $N$. To prove it, we introduce an analogy with complex differential calculus, the ring $\mathbb{R}[X]/(X^n)$ replacing the field $\mathbb{C}$. This treats the case where the radical of $\mathfrak{e}$ is principal and consists of self adjoint elements. We add a glimpse in the case where this radical is not principal.

We consider the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci form. This is an evolution equation first studied by M. Gill, and coincides with the Kähler-Ricci flow if the initial metric is Kähler. We find the maximal existence time for the flow in terms of the initial data. We investigate the behavior of the flow on complex surfaces when the initial metric is Gauduchon, on complex manifolds with negative first Chern class, and on some Hopf manifolds. Finally, we discuss a new estimate for the complex Monge-Ampère equation on Hermitian manifolds.

In this erratum, we give corrected versions of Conjecture 1.1 and Proposition 5.3 of our paper SYZ mirror symmetry for toric Calabi-Yau manifolds.