We point out a link between two surfaces which have appeared recently in the literature: the surface of cuboids and the Schoen surface. Both give rise to a surface with $q=4$, whose canonical map is 2-to-1 onto a complete intersection of 4 quadrics in $\mathbb{P}^6$ with 48 nodes.

Recently M. Mustaţă and V. Srinivas related a natural conjecture about the Frobenius action on the cohomology of the structure sheaf after reduction to characteristic $p \gt 0$ with another conjecture connecting multiplier ideals and test ideals. We generalize this relation to the case of singular ambient varieties. Additionally, we connect these results to a conjecture relating $F$-injective and Du Bois singularities. Finally, using an unpublished result of Gabber, we also show that $F$-injective and Du Bois singularities have a common definition in terms of smooth hypercovers.

This paper constructs all the diptych varieties with $de\le4$ (see [BR1], Main Theorem 3.3). Our construction involves several new classes of Gorenstein almost homogeneous spaces for $\operatorname{GL}(2)\times\mathbb{G}_m^r$, in particular two infinite series arising from the algebra of apolarity.

We first provide details for the proof of Fujita's second theorem for Kähler fibre spaces over a curve, asserting that the direct image $V$ of the relative dualizing sheaf splits as the direct sum $ V = A \oplus Q$, where $A$ is ample and $Q$ is unitary flat. Our main result then answers in the negative the question posed by Fujita whether $V$ is semiample. In fact, $V$ is semiample if and only if $Q$ is associated to a representation of the fundamental group of $B$ having finite image. Our examples are based on hypergeometric integrals.

The dual complex of a singularity is defined, up-to homotopy, using resolutions of singularities. In many cases, for instance for isolated singularities, we identify and study a “minimal” representative of the homotopy class that is well defined up-to piecewise linear homeomorphism. This is derived from a more global result concerning dual complexes of dlt pairs. As an application, we also show that the dual complex of a log terminal singularity as well as the one of a simple normal crossing degeneration of a family of rationally connected manifolds are contractible.

We prove some injectivity theorems. Our proof depends on the theory of mixed Hodge structures on cohomology with compact support. Our injectivity theorems would play crucial roles in the minimal model theory for higher-dimensional algebraic varieties. We also treat some applications.

We discuss the relationship among various conjectures in the minimal model theory including the finite generation conjecture of the log canonical rings and the abundance conjecture. In particular, we show that the finite generation conjecture of the log canonical rings for log canonical pairs can be reduced to that of the log canonical rings for purely log terminal pairs of log general type.

Given a family $(F,h) \to X \times S$ of Hermite-Einstein bundles on a compact Kähler manifold $(X,g)$ we consider the higher direct image sheaves $R^q p_* \mathcal{O}(F)$ on $S$, where $p: X \times S \to S$ is the projection. On the complement of an analytic subset these sheaves are locally free and carry a natural metric, induced by the $L_2$ inner product of harmonic forms on the fibers. We compute the curvature of this metric which has a simpler form for families with fixed determinant and families of endomorphism bundles. Furthermore, we discuss the metric for moduli spaces of stable vector bundles.

Let $G$ be a finite subgroup of $\operatorname{SL}(n,\mathbb{C})$, then the quotient $\mathbb{C}^n/G$ has a Gorenstein canonical singularity. If $n=2\ \text{or}\ 3$, it is known that there exist crepant resolutions of the quotient singularity. In higher dimension, there are many results which assume existence of crepant resolutions. However, few examples of crepant resolutions are known. In this paper, we will show several trials to obtain crepant resolutions and give a conjecture on existence of crepant resolutions.

We establish effective uniform degree bounds for the generalized Gauss map images of an embedded projective variety $X \subset \mathbb P^N$ in terms of numerical invariants such as $\dim X,$ $\deg X$ and $N$. This can be seen as a generalization of a classical Castelnuovo type bound.

Let $X$ be a Fano manifold of Picard number 1, different from projective space. We study the question whether the space $\mathrm{Hom}^s(Y,X)$ of surjective morphisms from a projective manifold $Y$ to $X$ is homogeneous under the automorphism group $\mathrm{Aut}_o(X)$. An affirmative answer is given in [4] under the assumption that $X$ has a minimal dominating family $\mathcal{K}$ of rational curves whose variety of minimal rational tangents $\mathcal{C}_x$ at a general point $x \in X$ is non-linear or finite. In this paper, we study the case where $\mathcal{C}_x$ is linear of arbitrary dimension, which covers the cases unsettled in [4]. In this case, we will define a reduced divisor $B^{\mathcal{K}} \subset X$ and an irreducible subvariety $M^{\mathcal{K}} \subset \mathrm{Chow}(X)$ naturally associated to $\mathcal{K}$. We give a sufficient condition in terms of $\mathbf{B}^{\mathcal{K}}$ and $M^{\mathcal{K}}$ for the homogeneity of $\mathrm{Hom}^s(Y,X)$. This condition is satisfied if $\mathcal{C}_x$ is finite and our result generalizes [4]. A new ingredient, which is of independent interest, is a similar rigidity result for surjective morphisms to projective space in logarithmic setting.

We examine Lefschetz pencils of a certain hypersurface in ${\bf P}^{3}$ over an algebraically closed field of characteristic $p > 2$, and determine the group structure of sections of the fiber spaces derived from the pencils. Using the structure of a Lefschetz pencil, we give a geometric proof of the unirationality of Fermat surfaces of degree $p^a + 1$ with a positive integer $a$ which was first poved by Shioda [10]. As byproducts, we also see that on the hypersurface there exists a $(q^{3} + q^{2} + q + 1)_{q + 1}$-symmetric configuration (resp. a $((q^{3} + 1)(q^{2} + 1)_{q + 1}, (q^{3} + 1)(q + 1)_{q^{2} + 1}$)-configuration) made up of the rational points over ${\bf F}_{q}$ (resp. over ${\bf F}_{q^{2}}$) and the lines over ${\bf F}_{q}$ (resp. over ${\bf F}_{q^{2}}$) with $q = p^{a}$.

We prove that the automorphism group of a Calabi-Yau threefold with Picard number three is either finite, or isomorphic to the infinite cyclic group up to finite kernel and cokernel.

Let $ X $ be an irreducible symplectic manifold and $ L $ a divisor on $ X $. Assume that $ L $ is isotropic with respect to the Beauville-Bogomolov quadratic form. We define the rational Lagrangian locus and the movable locus on the universal deformation space of the pair $ (X,L) $. We prove that the rational Lagrangian locus is empty or coincides with the movable locus of the universal deformation space.

We prove a conjecture of Starr [11] that the conclusion of bend and break holds for proper algebraic spaces over an algebraically closed field of characteristic zero. The proof uses the minimal model program, rather than reduction modulo $p$ and it even applies to spaces with kawamata log terminal singularities.

We establish here the surjectivity of the Albanese map corresponding to compact Kähler manifold whose anti-canonical bundle is nef. This answers a question raised by J.-P. Demailly, Th. Peternell and M. Schneider. One of the main techniques we are using is the positivity of the relative canonical bundle associated to Kähler fiber spaces, obtained via the “psh variation” of fiber-wise twisted Kähler-Einstein metrics.

Given a derived equivalence of orbifolds associated to projective varieties with (not necessarily Gorenstein) quotient singularities, we deduce consequences related to the behavior of orbifold Hodge numbers and the Picard variety, extending what is known in the smooth case.

We propose a conjectural stronger version of Bogomolov-Gieseker inequality for stable sheaves on quintic 3-folds. Our conjecture is derived from an attempt to construct a Bridgeland stability condition on graded matrix factorizations, which should correspond to the Gepner point via mirror symmetry and Orlov equivalence. We prove our conjecture in the rank two case.

We discuss how the motivic integration will be generalized to wild Deligne-Mumford stacks, that is, stabilizers may have order divisible by the characteristic of the base or residue field. We pose several conjectures on this topic. We also present some possible applications concerning stringy invariants, resolution of singularities, and weighted counts of extensions of local fields.