arXiv subjects
Publications and source records attributed to Hendrik Süß.
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 study linear operators preserving the property of being a volume polynomial. More, precisely we show that a linear operator preserves this property if the associated symbol is itself a volume polynomial. This can be seen as an analogue to theorems by Borcea-Brändén and Brändén-Huh for stable polynomials and Lorentzian polynomials, respectively.
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.
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.
We consider two classes of non-toric log del Pezzo $\mathbb{C}^*$-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.
We introduce and study a notion of dually Lorentzian polynomials, and show that if $s$ is non-zero and dually Lorentzian then the operator \[s(\partial_{x_1},\ldots,\partial_{x_n}):\mathbb R[x_1,\ldots,x_n] \to \mathbb R[x_1,\ldots,x_n]\] 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.
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 consider Fano manifolds admitting an algebraic torus action with general orbit of codimension one. Using a recent result of Datar and Szekelyhidi, we effectively determine the existence of Kahler-Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kahler-Einstein Fano threefolds, and Fano threefolds admitting a non-trivial Kahler-Ricci soliton.
We study the normalized volume of toric singularities. As it turns out, there is a close relation to the notion of (non-symmetric) Mahler volume from convex geometry. This observation allows us to use standard tools from convex geometry, such as the Blaschke-Santaló inequality and Radon's theorem to prove non-trivial facts about the normalized volume in the toric setting. For example, we prove that for every $ε> 0$ there are only finitely many $\mathbb{Q}$-Gorenstein toric singularities with normalized volume at least $ε$. From this result, it directly follows that there are also only finitely many toric Sasaki-Einstein manifolds of volume at least $ε$ in each dimension. Additionally, we show that the normalized volume of every toric singularity is bounded from above by that of the rational double point of the same dimension. Finally, we discuss certain bounds of the normalized volume in terms of topological invariants of resolutions of the singularity. We establish two upper bounds in terms of the Euler characteristic and of the first Chern class, respectively. We show that a lower bound, which was conjectured earlier by He, Seong, and Yau, is closely related to the non-symmetric Mahler conjecture in convex geometry.
We give new proofs of the K-polystability of two smooth Fano threefolds. One of them is a~smooth divisor in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$, which is unique up to isomorphism. Another one is the~blow up of the complete intersection $$ \Big\{x_0x_3+x_1x_4+x_2x_5=x_0^2+ωx_1^2+ω^2x_2^2+\big(x_3^2+ωx_4^2+ω^2x_5^2\big)+\big(x_0x_3+ωx_1x_4+ω^2x_2x_5\big)\Big\}\subset\mathbb{P}^5 $$ in the conic cut out by $x_0=x_1=x_2=0$, where $ω$ is a~primitive cube root of unity.
We study the Fano scheme of $k$-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 $6\times 6$ Pfaffian and $4\times 4$ permanent, as well as giving a new proof that the product and tensor ranks of the $3\times 3$ determinant equal five. Based on our results, we formulate several conjectures.
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 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.
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.
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.
Following the work of Altmann and Hausen we give a combinatorial description in terms for smooth Fano threefolds admitting a 2-torus action. We show that a whole variety of properties and invariants can be read off from this description. As an application we prove and disprove the existence of Kahler-Einstein metrics for some of these Fano threefolds, calculate their Cox rings and some of their toric canonical degenerations.
We show that there are no irregular Sasaki-Einstein structures on rational homology 5-spheres. On the other hand, using K-stability we prove the existence of continuous families of non-toric irregular Sasaki-Einstein structures on odd connected sums of $S^2 \times S^3$.
We give a classification of all pairs (X,v) of Gorenstein del Pezzo surfaces X and vector fields v which are K-stable in the sense of Berman-Nystrom and therefore are expected to admit a Kahler-Ricci solition. Moreover, we provide some new examples of Fano threefolds admitting a Kahler-Ricci soliton.