In this work, we have proved a number of purely geometric statements by algebraic methods. Also we have proved Sylvester’s law of Nullity and Exercise: the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullity of A. Keywords: Transformation of Groups, Nullity, Kernel, Image, Non-Singular, Symmetry Group, Shear, Compression, Elongation Reflection Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 27 – 30

Transformation of Groups

This is a continuation of an earlier work in which we proposed a problem of minimizing normalized volumes over Q-Gorenstein Kawamata log terminal singularities. Here we consider its relation with K-semistability, which is an important concept in the study of Kahler–Einstein metrics on Fano varieties. In particular, we prove that for a Q-Fano variety V, the K-semistability of (V,−KV) is equivalent to the condition that the normalized volume is minimized at the canonical valuation ordV among all C∗-invariant valuations on the cone associated to any positive Cartier multiple of −KV. In this case, we show that ordV is the unique minimizer among all C∗-invariant quasimonomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over V.

K-semistability is equivariant volume minimization

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.

Gluing Affine Torus Actions Via Divisorial Fans

We carry out a systematic study of a class of 6D F-theory models and associated Calabi-Yau threefolds that are constructed using base surfaces with a generalization of toric structure. In particular, we determine all smooth surfaces with a structure invariant under a single C∗ action (sometimes called “T-varieties” in the mathematical literature) that can act as bases for an elliptic fibration with section of a Calabi-Yau threefold. We identify 162,404 distinct bases, which include as a subset the previously studied set of strictly toric bases. Calabi-Yau threefolds constructed in this fashion include examples with previously unknown Hodge numbers. There are also bases over which the generic elliptic fibration has a Mordell-Weil group of sections with nonzero rank, corresponding to non-Higgsable U(1) factors in the 6D supergravity model; this type of structure does not arise for generic elliptic fibrations in the purely toric context.

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6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces

Let X = Spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus $ \mathbb{T} $ of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine $ {\mathbb{Z}^n} $-graded domain A, so that ∂ generates a k+-action on X that is normalized by the $ \mathbb{T} $-action.

Affine t-varieties of complexity one and locally nilpotent derivations

This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established the- ory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.

The Geometry of T-Varieties

We show how to construct certain homogeneous deformations for rational normal varieties with codimension one torus action. This can then be used to construct homogeneous deformations of any toric variety in arbitrary degree. For locally trivial deformations coming from this construction, we calculate the image of the Kodaira-Spencer map. We then show that for a smooth complete toric variety, our homogeneous deformations span the space of first-order deformations.

Deformations of Rational T-Varieties

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Deformations of rational T-varieties

This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.

Robert Vollmert

Papers

The Geometry of T-Varieties

Deformations of Rational T-Varieties

Deformations of rational T-varieties

The Geometry of T-Varieties

Upgrading and downgrading torus actions