Moment-angle manifolds provide a wide class of examples of non-Kähler compact complex manifolds. A complex moment-angle manifold ZZis zero.

Let $Fl^a_\lambda$ be the PBW degeneration of the flag varieties of type $A_{n-1}$. These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$. We prove that $Fl^a_\lambda$ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\lambda$ of $Fl^a_\lambda$. The varieties $R_\lambda$ can be viewed as towers of successive $P^1$-fibrations, thus providing an analogue of the classical Bott-Samelson-Demazure-Hansen desingularization. We prove that the varieties $R_\lambda$ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel-Weil type theorem for $Fl^a_\lambda$. Using the Atiyah-Bott-Lefschetz formula for $R_\la$, we compute the $q$-characters of the highest weight $\msl_n$-modules.

The Oeljeklaus–Toma (OT-)manifolds are compact, complex, non-Kähler manifolds constructed by Oeljeklaus and Toma, and generalizing the Inoue surfaces. Their construction uses the number-theoretic data: a number field *K* and a torsion-free subgroup *U* in the group of units of the ring of integers of *K*, with rank of *U* equal to the number of real embeddings of *K*. OT-manifolds are equipped with a torsion-free flat affine connection preserving the complex structure (this structure is known as “flat affine structure”). We prove that any complex subvariety of smallest possible positive dimension in an OT-manifold is also flat affine. This is used to show that if all elements in U∖{1} are primitive in *K*, then *X* contains no proper analytic subvarieties.

Linear degenerate flag varieties are degenerations of flag varieties as quiver Grassmannians. For type A flag varieties, we obtain characterizations of flatness, irreducibility and normality of these degenerations via rank tuples. Some of them are shown to be isomorphic to Schubert varieties and can be realized as highest weight orbits of partially degenerate Lie algebras, generalizing the corresponding results on degenerate flag varieties. To study normality, cell decompositions of quiver Grassmannians are constructed in a wider context of equioriented quivers of type A.

A manifold M is locally conformally Kähler (LCK) if it admits a Kähler covering M˜ with monodromy acting by holomorphic homotheties. Let M be an LCK manifold admitting a holomorphic conformal flow of diffeomorphisms, lifted to a non-isometric homothetic flow on M˜ . We show that M admits an automorphic potential, and the monodromy group of its conformal weight bundle is Z.

We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an important application, we show that they have Kähler–Einstein metrics if they are general.

We study Tian’s α-invariant in comparison with the α1-invariant for pairs (Sd,H) consisting of a smooth surface Sd of degree *d* in the projective three-dimensional space and a hyperplane section *H*. A conjecture of Tian asserts that α(Sd,H)=α1(Sd,H). We show that this is indeed true for d=4 (the result is well known for d⩽3), and we show that α(Sd,H)<α1(Sd,H) for d⩾8 provided that Sd is general enough. We also construct examples of Sd, for d=6 and d=7, for which Tian’s conjecture fails. We provide a candidate counterexample for S5.

We show that if K is a quadratic field, and if there exists a quadratic Q-curve E/K of prime degree N, satisfying weak conditions, then any unit u of OK satisfies a congruence u^r ≡ 1 (mod N), where r = g.c.d.(N − 1, 12). If K is imaginary quadratic, we prove a congruence, modulo a divisor of N, between an algebraic Hecke character ψ ̃ and, roughly speaking, the elliptic curve. We show that this divisor then occurs in a critical value L(ψ ̃ , 2), by constructing a non-zero element in a Selmer group and applying a theorem of Kato.

A Laguerre minimal surface is an immersed surface in the Euclidean space being an extremal of the functional \int (H^2/K - 1) dA. In the present paper, we prove that the only ruled Laguerre minimal surfaces are up to isometry the surfaces R(u,v) = (Au, Bu, Cu + D cos 2u) + v (sin u, cos u, 0), where A, B, C, D are fixed real numbers. To achieve invariance under Laguerre transformations, we also derive all Laguerre minimal surfaces that are enveloped by a family of cones. The methodology is based on the isotropic model of Laguerre geometry. In this model a Laguerre minimal surface enveloped by a family of cones corresponds to a graph of a biharmonic function carrying a family of isotropic circles. We classify such functions by showing that the top view of the family of circles is a pencil.

Given a generic family $Q$ of 2-dimensional quadrics over a smooth 3-dimensional base $Y$ we consider the relative Fano scheme $M$ of lines of it. The scheme $M$ has a structure of a generically conic bundle $M \to X$ over a double covering $X \to Y$ ramified in the degeneration locus of $Q \to Y$. The double covering $X \to Y$ is singular in a finite number of points (corresponding to the points $y \in Y$ such that the quadric $Q_y$ degenerates to a union of two planes), the fibers of $M$ over such points are unions of two planes intersecting in a point. The main result of the paper is a construction of a semiorthogonal decomposition for the derived category of coherent sheaves on $M$. This decomposition has three components, the first is the derived category of a small resolution $X^+$ of singularities of the double covering $X \to Y$, the second is a twisted resolution of singularities of $X$ (given by the sheaf of even parts of Clifford algebras on $Y$), and the third is generated by a completely orthogonal exceptional collection.

We construct a compactification $M^{μss}$ of the Uhlenbeck–Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable framed sheaves. We show that there exists a projective morphism $\gamma: M^{ss}\to M^{μss}$, where $M^{μss}$ is the moduli space of $S$-equivalence classes of Gieseker-semistable framed sheaves. The space $M^{μss}$ has a natural set-theoretic stratification which allows one, via a Hitchin–Kobayashi correspondence, to compare it with the moduli spaces of framed ideal instantons.