We provide a general description of the module T^1_X of first-order infinitesimal deformations of a not necessarily normal affine toric variety X with a special emphasis on seminormal varieties satisfying Serre's (S_2) condition. In the surface case we are able to give a very detailed and concrete picture which includes the dimension of each homogeneous component.

We introduce and study a notion of dually Lorentzian polynomials, and show that if s is non-zero and dually Lorentzian then the operator s(∂ _x_1,… ,∂ _x_n):ℝ[x_1,… ,x_n] →ℝ[x_1,… ,x_n] preserves (strictly) Lorentzian polynomials. From this we conclude that any theory that admits a mixed Alexandrov-Fenchel inequality also admits a generalized Alexandrov-Fenchel inequality involving dually Lorentzian polynomials. As such we deduce generalized Alexandrov-Fenchel inequalities for mixed discriminants, for integrals of Kähler classes, for mixed volumes, and in the theory of valuations.

We show that diagonalization, products and lower truncations preserve the property of being a denormalized volume polynomial. We also discuss an application to poset inequalities.

Volume polynomials measure the growth of Minkowski sums of convex bodies and of tensor powers of positive line bundles on projective varieties. We show that Aluffi's covolume polynomials are precisely the polynomial differential operators that preserve volume polynomials, reflecting a duality between homology and cohomology. We then present several applications to matroid theory.

We show that for every toric surface X apart from ℙ^2 and ℙ^1×ℙ^1 and every ample line bundle ℒ on X there exists an ample polarisation A for X, such that the syzygy bundle M_ℒ^⊗ d associated to the tensor power ℒ^⊗ d is not stable with respect to A for every d sufficiently large.

Toric varieties provide a rich class of examples in algebraic geometry that benefit from deep and fruitful interactions with combinatorics. This workshop highlighted recent interactions between toric geometry and mirror symmetry, matroids, deformation theory and moduli spaces, and non-commutative geometry, as well as some exciting new developments within toric geometry itself.

We show that for every toric surface apart from the projective plane and a product of two projective lines and every ample line bundle there exists a polarisation such that the syzygy bundle associated to sufficiently high powers of the line bundle is not slope stable.

In this note we investigate the Cheltsov–Rubinstein conjecture. We show that this conjecture does not hold in general and some counterexamples will be presented.

We give new proofs of the K-polystability of two smooth Fano threefolds. One of them is a smooth divisor in ℙ^1×ℙ^1×ℙ^2 of degree (1, 1, 1), which is unique up to isomorphism. Another one is the blow up of the complete intersection {x_0x_3+x_1x_4+x_2x_5=x_0^2+ω x_1^2+ω ^2x_2^2+ (x_3^2+ω x_4^2+ω ^2x_5^2 )+ (x_0x_3+ω x_1x_4+ω ^2x_2x_5 )}⊂ℙ^5 in the conic cut out by x_0=x_1=x_2=0 , where ω is a primitive cube root of unity.

We consider two classes of non-toric log del Pezzo ℂ^*-surfaces: on the one side the 1/3-log canonical ones and on the other side those of Picard number one and Gorenstein index at most 65. In each of the two classes we figure out the surfaces admitting a Kähler-Einstein metric, a Kähler-Ricci soliton and those allowing a Sasaki-Einstein metric on the link of their anticanonical cone. We encounter examples that admit a Kähler-Ricci soliton but no Sasaki-Einstein cone link metric.

Algebraic varieties are shapes defined by polynomial equations. Smooth Fano threefolds are a fundamental subclass that can be thought of as higher-dimensional generalizations of ordinary spheres. They belong to 105 irreducible deformation families. This book determines whether the general element of each family admits a Kähler–Einstein metric (and for many families, for all elements), addressing a question going back to Calabi 70 years ago. The book's solution exploits the relation between these metrics and the algebraic notion of K-stability. Moreover, the book presents many different techniques to prove the existence of a Kähler–Einstein metric, containing many additional relevant results such as the classification of all Kähler–Einstein smooth Fano threefolds with infinite automorphism groups and computations of delta-invariants of all smooth del Pezzo surfaces. This book will be essential reading for researchers and graduate students working on algebraic geometry and complex geometry.

In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on singularities with a torus action of complexity 1, threefold singularities, and hypersurface singularities. Given a real value v>0, we prove that the class of K-semistable threefold singularities with normalized volume at least v forms a bounded family. Analogous statements are proved in the case of n-dimensional complexity-1 and n-dimensional hypersurface singularities for arbitary n. In the general case of klt singularities, i.e. without the assumption on K-semistability, we show that, up to special degenerations, the normalized volume bounds singularities with a complexity-1 torus action. We exhibit a 3-dimensional example which shows that this last statement is optimal.

We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a smooth toric variety of Picard rank 2, there exists an ample line bundle with respect to which the tangent bundle is stable if and only if it is an iterated blow-up of projective space.

Motivated by Buchstaber's and Terzic' work on the complex Grassmannians G(2,4) and G(2,5) we describe the moment map and the orbit space of oriented Grassmannians of planes under the action of a maximal compact torus. Our main tool is the realisation of these oriented Grassmannians as smooth complex quadric hypersurfaces and the relatively simple Geometric Invariant Theory of the corresponding algebraic torus action.

Показано, что на рационально гомологических пятимерных сферах не существует нерегулярных структур Сасаки-Эйнштейна. Кроме того, с помощью $K$-стабильности доказано существование непрерывных семейств неторических нерегулярных структур Сасаки-Эйнштейна на связных суммах нечетного числа копий $S^2 \times S^3$. Библиография: 30 названий.

We determine the integral homology of the orbit space of a maximal compact torus action on the Grassmannian Gr(2,5). Our approach uses the well-known Geometric Invariant Theory of the maximal algebraic torus action on this Grassmannian.

We show that there are no irregular Sasaki-Einstein structures on rational homology 5-spheres. On the other hand, using -stability we prove the existence of continuous families of nontoric irregular Sasaki- Einstein structures on odd connected sums of . Bibliography: 30 titles.

We study the Fano scheme of [Formula: see text]-planes contained in the hypersurface cut out by a generic sum of products of linear forms. In particular, we show that under certain hypotheses, linear subspaces of sufficiently high dimension must be contained in a coordinate hyperplane. We use our results on these Fano schemes to obtain a lower bound for the product rank of a linear form. This provides a new lower bound for the product ranks of the [Formula: see text] Pfaffian and [Formula: see text] permanent, as well as giving a new proof that the product and tensor ranks of the [Formula: see text] determinant equal five. Based on our results, we formulate several conjectures.

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.

We give a classification of all pairs \((X,\xi )\) of Gorenstein del Pezzo surfaces X and vector fields \(\xi \) which are K-stable in the sense of Berman–Witt–Nyström and therefore are expected to admit a Kähler–Ricci solition. Moreover, we provide some new examples of Fano threefolds admitting a Kähler–Ricci soliton.