The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the surface *S*={(*z*1,*z*2,*z*3)∈C3:(*z*1)5+(*z*2)3+(*z*3)2=0} with the 5-dimensional sphere S*ε*5={(*z*1,*z*2,*z*3)∈C3:|*z*1|2+|*z*2|2+|*z*3|2=*ε*2}. An algebraic link in the Poincaré sphere is the intersection of a germ of a complex analytic curve in (*S*, 0) with the sphere S*ε*5 of radius *ε* small enough. Here we discuss to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. We show that, if the strict transform of a curve in (*S*, 0) does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding *E*8-diagram, then its Alexander polynomial determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere is determined by the Poincaré series of the filtration defined by the corresponding curve valuations. (They coincide with each other for a reducible curve singularity and differ by the factor (1−*t*) for an irreducible one.) We show that, under conditions similar to those for curves, the Poincaré series of a collection of divisorial valuations determines the combinatorial type of the minimal resolution of the collection.

We show that for maximal Cohen–Macaulay modules over the homogeneous coordinate ring of a smooth Calabi–Yau varieties *X*, the computation of Betti numbers can be reduced to computations of dimensions of certain HomHom spaces in the bounded derived category Db(X). In the simplest case of a smooth elliptic curve *E* embedded in P2 as a smooth cubic, we get explicit values for Betti numbers. The description of the automorphism group of the derived category Db(E) in terms of the spherical twist functors of Seidel and Thomas plays a major role in our approach. We show that there are only four possible shapes of the Betti tables up to shifts in internal degree, and two possible shapes up to shifts in internal degree and taking syzygies.

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.

We provide a new criterion for flexibility of affine cones over varieties covered by flexible affine varieties. We apply this criterion to prove flexibility of affine cones over secant varieties of Segre–Veronese embeddings and over certain Fano threefolds. We further prove flexibility of total coordinate spaces of Cox rings of del Pezzo surfaces.

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.

We continue, generalize and expand our study of linear degenerations of flag varieties from Cerulli Irelli et al. (Math Z 287(1–2):615–654, 2017). We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degenerate flag variety. We compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus. We also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus. Finally, we prove an analogue of the Borel–Weil theorem for the flat irreducible locus.

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.