We study the topology of a space $\Delta _ {g} $ parametrizing stable tropical curves of
genus $ g $ with volume $1 $, showing that its reduced rational homology is canonically …

As announced in Gross and Siebert (in Algebraic geometry: Salt Lake City 2015,
Proceedings of Symposia in Pure Mathematics, vol 97, no 2. AMS, Providence, pp 199–230 …

We study the asymptotic behavior of volume forms on a degenerating family of compact
complex manifolds. Under rather general conditions, we prove that the volume forms …

Let be a pair, and let be a contraction with nef over S. A conjecture, known as the Shokurov–
Kollár connectedness principle, predicts that has at most two connected components, where …

We show that a large class of maximally degenerating families of $ n $-dimensional
polarized varieties comes with a canonical basis of sections of powers of the ample line …

We associate a ring R to a log Calabi-Yau pair (X, D) or a degeneration of Calabi-Yau
manifolds X-> B. The vector space underlying R is determined by the tropicalization of (X, D) …

The absolute regularity of a Fano variety, denoted by reg^(X), is the largest dimension of the
dual complex of a log Calabi–Yau structure on X. The absolute coregularity is defined to be …

Using the Minimal model program, any degeneration of K-trivial varieties can be arranged to
be in a Kulikov type form, ie with trivial relative canonical divisor and mild singularities. In the …

We construct non-archimedean SYZ (Strominger–Yau–Zaslow) fibrations for maximally
degenerate Calabi–Yau varieties, and we show that they are affinoid torus fibrations away …

We study the relation between the coregularity, the index of log Calabi-Yau pairs, and the
complements of Fano varieties. We show that the index of a log Calabi-Yau pair $(X, B) $ of …