Related papers: A variant of the effective adjunction conjecture w…

We prove a variant of Manin's conjecture for Campana points on wonderful compactifications of semi-simple algebraic groups of adjoint type. We use this to provide evidence for a new conjecture on the leading constant in Manin's conjecture…

We establish adjunction and inversion of adjunction for log canonical centers of arbitrary codimension in full generality.

Let f: X \to Z be a surjective morphism of smooth complex projective varieties with connected fibers. Suppose that L is a pseudo-effective divisor on X that is f-numerically trivial. We show that there is a divisor D on Z such that L is…

We prove the K- and L-theoretic Farrell-Jones Conjecture (with coefficients in additive categories) for virtually solvable groups.

We develop new techniques for constructing model structures from a given class of cofibrations, together with a class of fibrant objects and a choice of weak equivalences between them. As a special case, we obtain a more flexible version of…

Here we investigate the property of effectivity for adjoint divisors. Among others, we prove the following results: (i) A normal projective variety $X$ with at most canonical singularities is uniruled if and only if for each very ample…

We give counterexamples to Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients.

We investigate effectiveness and ampleness of adjoint divisors of the form $aL+bK_X$, where $L$ is a suitably positive line bundle on a smooth projective variety $X$ and $a,b$ are positive integers.

We give a counterexample to the PIA (precise inversion of adjunction) conjecture for minimal log discrepancies. We also give a counterexample to the LSC conjecture for families.

We prove that the log Iitaka conjecture holds for log canonical fibrations when log canonical divisor of a sufficiently general fiber is abundant.

We first announce our recent result on adjunction and inversion of adjunction. Then we clarify the relationship between our inversion of adjunction and Hacon's inversion of adjunction for log canonical centers of arbitrary codimension.

We establish a conjecture of Mumford characterizing rationally connected complex projective manifolds in several cases.

Implementations of known reductions of the Strong Real Jacobian Conjecture (SRJC), to the case of an identity map plus cubic homogeneous or cubic linear terms, and to the case of gradient maps, are shown to preserve significant algebraic…

We study very basic slc-trivial fibrations. We show that restricting on any lc center of a very basic slc-trivial fibration, its moduli part is numerically trivial if and only if it is $\mathbb Q$-linearly trivial. We then prove that…

In this paper, we show an extension type theorem for twisted pluricanonical sections on a family of smooth projective manifolds (the twisting line bundle being pseudo-effective and having a prescribed multiplier ideal on the central fiber).

This note generalizes factorization for formulas with multiplicities and conjectures that the connection method along with this feature is computationally as powerful as resolution, also seen from a complexity point of view.

We establish new results on weighted $L^2$ extension of holomorphic top forms with values in a holomorphic line bundle, from a smooth hypersurface cut out by a holomorphic function. The weights we use are determined by certain functions…

We present an inductive strategy to show the existence of rational curves on compact Kaehler manifolds which are not minimal models but have a pseudoeffective canonical bundle. The tool for this inductive strategy is a weak subadjunction…

In this article, we first describe codimension two regular foliations with numerically trivial canonical class on complex projective manifolds whose canonical class is not numerically effective. Building on a recent algebraicity criterion…

We study weak approximation on rationally connected varieties under an assumption of strong approximation for a "simple" variety or under Schinzel's hypothesis. We also get some unconditional results.