Log Canonical Divisors Research Articles

An elementary semi-ampleness result for log canonical divisors

If the log canonical divisor on a projective variety with only Kawamata log terminal singularities is numerically equivalent to some semi-ample Q-divisor, then it is semi-ample.

Stability of tri-canonical curves of genus two

We completely classify tri-canonically embedded curves of genus two that are Chow semistable, and identify the moduli space of them with the compact moduli space of binary sextics. This moduli space is the log canonical model for the pair \(\big(\overline{M}_2,\alpha\Delta_0+\frac{1+\alpha}{2}\Delta_1+\frac{1}{2}\Xi\big)\) for 7/10 \(< \alpha \le\) 9/11 whose log canonical divisor pulls back to \(K_{\overline{M}_2}+\alpha\delta\) on the moduli stack

A Note on the Ampleness of Numerically Positive Log Canonical and Anti-Log Canonical Divisors

In this short note, we consider the conjecture that the log canonical divisor (resp.\ the anti-log canonical divisor) $K_X + \Delta$ (resp.\ $-(K_X + \Delta)$) on a pair $(X, \Delta)$ consisting of a complex projective manifold $X$ and a reduced simply normal crossing divisor $\Delta$ on $X$ is ample if it is numerically positive. More precisely, we prove the conjecture for $K_X + \Delta$ with $\Delta \neq 0$ in dimension $4$ and for $-(K_X + \Delta)$ with $\Delta \neq 0$ in dimension $3$ or $4$.

On log canonical divisors that are log quasi-numerically positive

Abstract Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.

Higher direct images of log canonical divisors

In this paper, we investigate higher direct images of log canonical divisors. After we reformulate Kollar's torsion-free theorem, we treat the relationship between higher direct images of log canonical divisors and the canonical extensions of Hodge filtration of gradedly polarized variations of mixed Hodge structures. As a corollary, we obtain a logarithmic version of Fujita--Kawamata's semi-positivity theorem. The final section is an appendix, which is a result of Morihiko Saito.

Hodge decomposition of Alexander invariants

Multivariable Alexander invariants of algebraic links calculated in terms of algebro-geometric invariants (polytopes and ideals of quasiadjunction). The relations with log-canonical divisors, the multiplier ideals and a semicontinuity property of polytopes of quasiadjunction are discussed.

On numerically effective log canonical divisors

Let (X, Δ) be a 4‐dimensional log variety which is proper over the field of complex numbers and with only divisorial log terminal singularities. The log canonical divisor KX + Δ is semiample, if it is numerically effective (NEF) and the Iitaka dimension κ(X, KX + Δ) is strictly positive. For the proof, we use Fujino′s abundance theorem for semi‐log canonical threefolds.

On a conjecture of Shokurov: characterization of toric varieties

We verify a special case of V. V. Shokurov's conjecture about characterization of toric varieties. More precisely, we consider three-dimensional log varieties with only purely log terminal singularities and numerically trivial log canonical divisor. In this situation we prove an inequality connecting the rank of the group of Weil divisors modulo algebraic equivalence and the sum of coefficients of the boundary. We describe such varieties for which the equality holds and show that all of them are toric.

Subadjunction of log canonical divisors, II

We extend a subadjunction formula of log canonical divisors as in [Kawamata, Contemp. Math. 207 (1997), 79-88] to the case when the codimension of the minimal center is arbitrary by using the positivity of the Hodge bundles.