We introduce a pairing on local intersection cohomology groups of variations of pure Hodge structure, which we call the asymptotic height pairing. Our original application of this pairing was to answer a question on the Ceresa cycle posed by R. Hain and D. Reed. (This question has since been answered independently by Hain.) Here we show that a certain analytic line bundle, called the biextension line bundle, and defined in terms of normal functions, always extends to any smooth partial compactification of the base. We then show that the asymptotic height pairing on intersection cohomology governs the extension of the natural metric on this line bundle studied by Hain and Reed (as well as, more recently, by several other authors). We also prove a positivity property of the asymptotic height pairing, which generalizes the results of a recent preprint of J. Burgos Gil, D. Holmes and R. de Jong, along with a continuity property of the pairing in the normal function case. Moreover, we show that the asymptotic height pairing arises in a natural way from certain Mumford–Grothendieck biextensions associated to normal functions.

We present a new tool for the calculation of Denef and Loeser’s motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on Hrushovski and Kazhdan’s theory of motivic volumes of semialgebraic sets. As applications, we prove a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and give a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by Lê Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson–Thomas theory.

For a nondegenerate integral quadratic form $F(x_1,\dots ,x_d)$ in $d\ge 5$ variables, we prove an optimal strong approximation theorem. Let $\Omega$ be a fixed compact subset of the affine quadric $F(x_1,\dots ,x_d)=1$ over the real numbers. Take a small ball $B$ of radius $0<r<1$ inside $\Omega$, and an integer $m$. Further assume that $N$ is a given integer which satisfies $N{\gg }_{\delta ,\Omega }(r^{-1}m{)}^{4+\delta }$ for any $\delta >0$. Finally assume that an integral vector $({\lambda }_1,\dots ,{\lambda }_d)$ mod $m$ is given. Then we show that there exists an integral solution $\mathbf{x}=(x_1,\dots ,x_d)$ of $F(\mathbf{x})=N$ such that $x_i\equiv {\lambda }_i\mathrm{mod}m$ and $\frac{\mathbf{x}}{\sqrt{N}}\in B$, provided that all the local conditions are satisfied. We also show that $4$ is the best possible exponent. Moreover, for a nondegenerate integral quadratic form in four variables, we prove the same result if $N$ is odd and $N{\gg }_{\delta ,\Omega }(r^{-1}m{)}^{6+ϵ}$. Based on our numerical experiments on the diameter of LPS Ramanujan graphs and the expected square-root cancellation in a particular sum that appears in Remark 6.8, we conjecture that the theorem holds for any quadratic form in four variables with the optimal exponent $4$.

We prove that any manifold diffeomorphic to ${\mathbb{S}}^3$ and endowed with a generic metric contains at least two embedded minimal $2$-spheres. The existence of at least one minimal $2$-sphere was obtained by Simon and Smith in 1983. Our approach combines ideas from min–max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in $3$-manifolds. We apply our methods to solve a problem posed by S. T. Yau in 1987 on whether the planar $2$-spheres are the only minimal spheres in ellipsoids centered about the origin in ${\mathbb{R}}^4$. Finally, considering the example of degenerating ellipsoids, we show that the assumptions in the multiplicity $1$ conjecture and the equidistribution of widths conjecture are in a certain sense sharp.