We prove that every quasi-projective semi log canonical pair has a natural quasi-log structure with several good properties. It implies that various vanishing theorems, torsion-free theorem, and the cone and contraction theorem hold for semi log canonical pairs.

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Fundamental theorems for semi log canonical pairs

We discuss the variations of mixed Hodge structure for cohomology with compact support of quasi-projective simple normal crossing pairs. We show that they are graded polarizable admissible variations of mixed Hodge structure. Then we prove a generalization of the Fujita–Kawamata semi-positivity theorem.

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Variations of Mixed Hodge Structure and Semipositivity Theorems

We discuss the variations of mixed Hodge structure for cohomology with compact support of quasi-projective simple normal crossing pairs. We show that they are graded polarizable admissible variations of mixed Hodge structure. Then we prove a generalization of the Fujita--Kawamata semi-positivity theorem.

Variations of mixed Hodge structure and semi-positivity theorems

We prove some semipositivity theorems for singular varieties coming from graded polarizable admissible variations of mixed Hodge structure. As an application, we obtain that the moduli functor of stable varieties is semipositive in the sense of Kollar. This completes Kollar's projectivity criterion for the moduli spaces of higher-dimensional stable varieties.

Semipositivity theorems for moduli problems

Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X, D) is said to be semi-simple normal crossings (semi-snc) at \({a \in X}\) if X is simple normal crossings at a (i.e., a simple normal crossings hypersurface, with respect to a local embedding in a smooth ambient variety), and D is induced by the restriction to X of a hypersurface that is simple normal crossings with respect to X. We construct a composition of blowings-up \({f:\tilde{X}\rightarrow {X}}\) such that the transformed pair \({(\tilde{X}, \tilde{D})}\) is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal crossings locus of (X, D). The result answers a question of Kollar.

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Resolution of singularities of pairs preserving semi-simple normal crossings

Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X,D) is said to be semi-simple normal crossings (semi-snc) at a point a of X if X is simple normal crossings at a (i.e., a simple normal crossings hypersurface, with respect to a local embedding in a smooth ambient variety), and D is induced by the restriction to X of a hypersurface that is simple normal crossings with respect to X. We construct a composition of blowings-up f: X' -> X such that the transformed pair (X',D') is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal crossings locus of (X,D). The result answers a question of Kollar.

It is shown that, for any reduced algebraic variety in characteristic zero, one can resolve all but simple normal crossings (snc) singularities by a finite sequence of blowings-up with smooth centres which, at every step, avoids points where the transformed variety together with the exceptional divisor has only snc singularities. The proof follows the philosophy of arXiv:1107.5595 that the desingularization invariant can be used together with natural geometric information to compute local normal forms of singularities.

Desingularization by blowings-up avoiding simple normal crossings

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The subject is partial resolution of singularities. Given an algebraic variety X (not necessarily equidimensional) in characteristic zero (or, more generally, a pair (X, D), where D is a divisor on X), we construct a functorial desingularization of all but stable simple normal crossings (stable-snc) singularities, by smooth blowings-up that preserve such singularities. A variety has stable simple normal crossings at a point if, locally, its irreducible components are smooth and transverse in some smooth embedding variety. We also show that our main assertion is false for more general simple normal crossings singularities.

Franklin Vera Pacheco

Papers

Resolution of singularities of pairs preserving semi-simple normal crossings

Resolution of singularities of pairs preserving semi-simple normal crossings

Desingularization by blowings-up avoiding simple normal crossings

Desingularization by blowings-up avoiding simple normal crossings

Desingularization preserving stable simple normal crossings