Unitary anti-self-dual connections on Asymptotically Locally Flat (ALF) hyperkähler spaces are constructed in terms of data organized in a bow. Bows generalize quivers, and the relevant bow gives rise to the underlying ALF space as the moduli space of its particular representation—the small representation. Any other representation of that bow gives rise to anti-self-dual connections on that ALF space.

We prove that each resulting connection has finite action, i.e. it is an instanton. Moreover, we derive the asymptotic form of such a connection and compute its topological class.

The smooth (resp. metric and complex) Nielsen Realization Problem for K3 surfaces $M$ asks: when can a finite group $G$ of mapping classes of $M$ be realized by a finite group of diffeomorphisms (resp. isometries of a Ricci-flat metric, or automorphisms of a complex structure)? We solve the metric and complex versions of Nielsen Realization, and we solve the smooth version for involutions. Unlike the case of 2-manifolds, some $G$ are realizable and some are not, and the answer depends on the category of structure preserved. In particular, Dehn twists are not realizable by finite order diffeomorphisms. We introduce a computable invariant $L_G$ that determines in many cases whether $G$ is realizable or not, and apply this invariant to construct an $S_4$ action by isometries of some Ricci-flat metric on $M$ that preserves no complex structure.

We also show that the subgroups of $\mathrm{Diff}(M)$ of a given prime order $p$ which fix pointwise some positive-definite 3-plane in $H^2(M; \mathbb{R})$ and preserve some complex structure on $M$ form a single conjugacy class in $\mathrm{Diff}(M)$ (it is known that then $p \in \{2, 3, 5, 7\}$).

We establish a positive characteristic analogue of intersection cohomology theory for variations of Hodge structure. It includes: a) the de Rham-Higgs comparison theorem for the intersection de Rham complex; b) the $E_1$-degeneration theorem for the intersection de Rham complex of a periodic de Rham bundle; c) the Kodaira–Saito vanishing theorem for the intersection cohomology groups of a periodic Higgs bundle. These results generalize the decomposition theorem of Deligne–Illusie [DI] and the de Rham–Higgs theorem of Ogus–Vologodsky [OV], the $E_1$-degeneration theorem of Deligne-Illusie [DI], Illusie [I90], Faltings [Fa] and the Kodaira-Saito vanishing theorem of Arapura [Ar]. As an application, we give an algebraic proof of the $E_1$-degeneration theorem due to Cattani-Kaplan–Schmid [CKS] and Kashiwara–Kawai [KK], and the vanishing theorem of Saito [Sa] for VHSs of geometric origin.

We obtain sharp upper and lower bounds for the diameter of Ricci-flat Kähler metrics on polarized Calabi–Yau degeneration families, as conjectured by Kontsevich–Soibelman.

L$\def\Q{\overline{\mathbb{Q}}}\def\V{\mathbb{V}}$Let $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected base $S$, with everything defined over $\Q$. Denote by $\V =R^{2i} f_\ast \mathbb{Z}(i)$ the associated integral variation of Hodge structure on the degree $2i$ cohomology. We consider the following question: when can a fiber $\mathbb{V}_s$ above an algebraic point $s \in S(\Q)$ be isomorphic to a transcendental fiber $\mathbb{V}_{s^\prime}$ with $s^\prime \in S(\mathbb{C}) \setminus S(\Q)$? When $V$ induces a quasi-finite period map $\varphi : S \to \mathbb{D}$, conjectures in Hodge theory predict that such isomorphisms cannot exist. We introduce new differential-algebraic techniques to show this is true for all points $s \in S(\Q)$ outside of an explicit proper closed algebraic subset of $S$. As a corollary, we establish the existence of a canonical $\Q$-algebraic model for normalizations of period images.

Motivated by various possible generalizations of Taubes’s $SW = Gr$ theorem [T] to Floer-theoretic setting, we prove certain variants of Taubes’s convergence theorem in [T] (the first part of his proof of $SW = Gr$). In place of the closed symplectic $4$-manifold considered in [T], this article considers non-compact manifolds with cylindrical ends, equipped with a self-dual harmonic $2$-form with non-degenerate zeroes. This extends and simplifies some central technical ingredients of the author’s prior work in [LT] and [KLT5]. Other expected applications include: extending the $HM = PFH$ theorem in [LT] and the $HM = HF$ theorem in [KLT1]–[KLT5] to TQFTs on both sides [L1]; definitions of large-perturbation Seiberg–Witten analogs of Heegaard Floer theory’s link Floer homologies and link cobordism invariants.

In this paper, we prove that the round cylinders are rigid in the space of Ricci shrinkers. Namely, any Ricci shrinker that is sufficiently close to $S^{n-1} \times \mathbb{R}$ in the pointed-Gromov-Hausdorff topology must itself be isometric to $S^{n-1} \times \mathbb{R}$.