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We study the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. In particular we do this for cyclic branched covers of fibred knots in integer homology $3$-spheres and…

Among (conformal) quantum field theories, the rational conformal field theories are singled out by the fact that their correlators can be constructed from a modular tensor category C with a distinguished object, a symmetric special…

We show that a singular Riemannian foliation of codimension two on a compact simply-connected Riemannian $(n+2)$-manifold, with regular leaves homeomorphic to the $n$-torus, is given by a smooth effective $n$-torus action. This solves in…

In this article we establish new positivity properties for direct images of twisted pluricanonical bundle of an algebraic fiber space. As a corollary we obtain an algebraicity criteria for holomorphic foliations which partly confirms a…

Let $\FF$ be a codimension one foliation on a closed manifold $M$ which admits a transverse dimension one Riemannian foliation. Then any continuous leafwise harmonic functions are shown to be constant on leaves.

Casual structure can take the form of cone bundles on a manifold, more general local preorders on a topological space, or simplicial orientations implicit in a simplicial set. This note takes a triangulation of a conal manifold M to mean an…

We shall consider minimal analytic compactifications of the affine plane with singularities. In previous work, Kojima and Takahashi proved that any minimal analytic compactification of the affine plane, which has at worse log canonical…

It is shown that codimension one parabolic foliations of complex manifolds are holomorphic. This is proved using the fact that codimension one foliations of complex manifolds are necessarily locally Monge-Amp\`ere foliations and that…

We classify families of free rational curves on all smooth Fano threefolds over the complex numbers. In particular, we prove the family of very free rational curves representing any fixed numerical curve class is either irreducible or…

PhD dissertation consists in three lines of investigation involving rational elliptic surfaces, namely 1) a study of conic bundles on these surfaces; 2) an investigation of the possible intersection numbers of two sections and 3) a theorem…

Let (M, F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F . If every such function is constant on leaves we…

We give a classification of pairs (F, f) where F is a holomorphic foliation on a projective surface and f is a non-invertible dominant rational map preserving F. We prove that both the map and the foliation are integrable in a suitable…

Let $X$ be a compact K\"ahler manifold and $\alpha$ be a class in the Dolbeault cohomology class of bidegree $(1, 1)$ on $X$. When the numerical dimension of $\alpha$ is one and $\alpha$ admits at least two smooth semi-positive…

We discuss the strong rational connectedness of smooth rationally connected surfaces. We prove in lots of cases, including the smooth locus of a log del Pezzo surface, the rational connectedness indeed implies the strong rational…

We describe the moduli space of rational curves on smooth Fano varieties of coindex 3. For varieties of dimension 5 or greater, we prove the moduli space has a single irreducible component for each effective numerical class of curves. For…

Let $\mathcal{F}$ be a transversely orientable codimension one minimal foliation without vanishing cycles of a manifold $M$. We show that if the fundamental group of each leaf of $\mathcal{F}$ has polynomial growth of degree $k$ for some…

We give conditions for a uniruled variety of dimension at least 2 to be non-solid. This study provides further evidence to a conjecture by Abban and Okada on the solidity of Fano 3-folds. To complement our results we write explicit…

Let F be a polarized irreducible holomorphic symplectic fourfold, deformation equivalent to the Hilbert scheme parametrizing length-two zero-dimensional subschemes of a K3 surface. The homology group H^2(F,Z) is equipped with an integral…

We construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 3-braid knot $K$ in $S^3$, where $r < 2g(K)-1$ and $g(K)$ denotes the Seifert genus of $K$. This confirms a prediction of the…

This paper continues the study of non-general type subvarieties begun in a joint paper with M.Schneider and A.Sommese (Int.L.Math. 10, 1999). We prove uniruledness of a projective manifold containing a submanifold not of general type whose…