Klaus Altmann

Bio: Klaus Altmann is an academic researcher from Free University of Berlin. The author has contributed to research in topics: Toric variety & Cohomology. The author has an hindex of 15, co-authored 64 publications receiving 1071 citations. Previous affiliations of Klaus Altmann include Humboldt University of Berlin.

Polyhedral divisors and algebraic torus actions

Abstract: We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our approach extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.

The versal deformation of an isolated toric Gorenstein singularity

Abstract: Given a lattice polytope Q ⊆ ℝ n , we define an affine scheme that reflects the possibilities of splitting Q into a Minkowski sum. Denoting by Y the toric Gorenstein singularity induced by Q, we construct a flat family over with Y as special fiber. In case Y has an isolated singularity, this family is versal.

Gluing Affine Torus Actions Via Divisorial Fans

Abstract: Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a “proper polyhedral divisor” introduced in earlier work, we develop the concept of a “divisorial fan” and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like $ \mathbb{C} $ *-surfaces and projectivizations of (nonsplit) vector bundles over toric varieties.

Polyhedral Divisors and Algebraic Torus Actions

Abstract: We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our theory extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.

Gluing affine torus actions via divisorial fans

Abstract: Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a ``proper polyhedral divisor'' introduced in earlier work, we develop the concept of a ``divisorial fan'' and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like C*-surfaces and projectivizations of (non-split) vector bundles over toric varieties.

Combinatorial Commutative Algebra

Abstract: Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

Non-spherical horizons, I

Abstract: We formulate an extension of Maldacena’s AdS/CFT conjectures to the case of branes located at singular points in the ambient transverse space. For singularities which occur at finite distance in the moduli space of M or F theory models with spacetime-filling branes, the conjectures identify the worldvolume theory on the p-branes with a compactification of M or IIB theory on AdSp+2 × HD−p−2. We show how the singularity determines the horizon H, and demonstrate the relationship between global symmetries on the worldvolume and gauge symmetries in the AdS model. As a first application, we study some singularities relevant to the D3-branes required in four-dimensional F -theory. For these we are able to explicitly derive the low-energy field theory on the worldvolume and compare its properties to predictions from the dual AdS model. In particular, we examine the baryon spectra of the models and the fate of the Abelian factors in the gauge group. October 1998 ∗ On leave from Department of Particle Physics, Weizmann Institute of Science, Rehovot, Israel Spacetime-filling branes have emerged as an essential feature of string and M-theory compactifications in at least three contexts: (1) new branches of the heterotic string in six dimensions with “extra” tensor multiplets, which can be represented by a Hořava– Witten-type compactification of M-theory on (S/Z2) × K3 but with extra spacetimefilling M5-branes representing the extra tensor multiplets [1,2,3]; (2) F -theory models in four dimensions (which can be regarded as compactifications of the IIB string with D7branes included) which in general require spacetime-filling D3-branes to cancel a tadpole anomaly [4]; and (3) M-theory models in three dimensions, which require spacetime-filling M2-branes to cancel a similar tadpole anomaly [4]. In each of these cases, the spacetimefilling brane meets the compactifying space at a single point, and the string or M-theory remains finite near the brane. Remarkably, this short list of branes (M5, D3, and M2) is precisely the list of branes for which a certain scaling limit is expected to lead to a “boundary” conformal field theory in the recent AdS/CFT conjectures [5,6,7]. In fact, the scaling limit can be taken even when the space transverse to the branes is curved, as in the compactification scenarios above. The details of the metric far from the location y0 of the brane in the transverse space become irrelevant; for the purposes of studying the scaling limit, the metric on the compactifying space can be approximated by some metric on its tangent space Ty0 at y0. In the scaling limit, the rescaled supergravity metric approaches a metric of the form AdSp+2 × S in which the anti-de Sitter space has been formed out of the worldvolume of the brane and the radial direction within Ty0 , and S k is the unit sphere within Ty0 . Maldacena’s conjecture proposes that the M or string theory on this space AdSp+2 × S, with N units of flux of the supergravity k-form field strength through S, is dual to a specific conformal field theory on the boundary of AdSp+2. The conjecture applies to the large N limit when a large number of these branes have been brought together; in the compactification context, fairly large values of N can be obtained by bringing together all available branes in a given model. Virtually all points in the compactifying space have identical behavior in this scaling limit. That situation changes, however, if we consider a compactifying space which itself 1 This last requirement excludes consideration of F -theory models in eight dimensions with the D7-brane being spacetime-filling. 2 The exception is the four-dimensional F -theory models, where points located along the D7branes behave differently; in particular, the string coupling becomes infinite at such points. The behavior of D3-branes at such points has recently been determined [8,9,10], and we will not

Mirror Symmetry via Logarithmic Degeneration Data I

Abstract: This paper lays the foundations of a program to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certain sorts of degenerations of Calabi-Yau manifolds a log Calabi-Yau space, which is a log structure on the degenerate fibre. The log CY space captures essentially all the information of the degeneration, and hence all mirror statements for the "large complex structure limit" given by the degeneration can already be derived from the log CY space. In this paper we begin by discussing affine manifolds with singularities. Given such an affine manifold along with a polyhedral decomposition, we show how to construct a scheme consisting of a union of toric varieties. In certain non-degenerate cases, we can also construct log structures on these schemes. Conversely, given certain sorts of degenerations, one can build an affine manifold with singularities structure on the dual intersection complex of the degeneration. Mirror symmetry is then obtained as a discrete Legendre transform on these affine manifolds, thus providing an algebro-geometrization of the Strominger-Yau-Zaslow conjecture. The deepest result of this paper shows an isomorphism between log complex moduli of a log CY space and log Kahler moduli of its mirror.