Advances in mathematics

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

Communications on pure and applied mathematics

We prove the existence of asymptotically cylindrical (ACyl) Calabi–Yau 3–folds starting with (almost) any deformation family of smooth weak Fano 3–folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi–Yau 3–folds; previously only a few hundred ACyl Calabi–Yau 3–folds were known. We pay particular attention to a subclass of weak Fano 3–folds that we call semi-Fano 3–folds. Semi-Fano 3–folds satisfy stronger cohomology vanishing theorems and enjoy certain topological properties not satisfied by general weak Fano 3–folds, but are far more numerous than genuine Fano 3–folds. Also, unlike Fanos they often contain ℙ1s with normal bundle O(−1) ⊕O(−1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi–Yau 3–folds. We introduce some general methods to compute the basic topological invariants of ACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds, and study a small number of representative examples in detail. Similar methods allow the computation of the topology in many other examples. All the features of the ACyl Calabi–Yau 3–folds studied here find application in [arXiv:1207.4470] where we construct many new compact G2–manifolds using Kovalev’s twisted connected sum construction. ACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds are particularly well-adapted for this purpose.

/pdf/asymptotically-cylindrical-calabi-yau-3-folds-from-weak-fano-1ly6wsd2u0.pdf

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

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 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.

K-Stability for Fano Manifolds with Torus Action of Complexity One

https://pure.mpg.de/pubman/item/item_3121492_1/component/file_3121493/Ilten_toric_oa_2013.pdf

Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models

We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension 1. 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 nontrivial Kahler–Ricci soliton.

/pdf/k-stability-for-fano-manifolds-with-torus-action-of-44hyoyzqf8.pdf

K-stability for Fano manifolds with torus action of complexity $1$

We describe polarized complexity-one T-varieties combinatorially in terms of so-called divisorial polytopes, and show how geometric properties of such a variety can be read off the corresponding divisorial polytope. We compare our description with other possible descriptions of polarized complexity-one T-varieties. We also describe how to explicitly find generators of affine complexity-one T-varieties.

Nathan Ilten

Papers

The Geometry of T-Varieties

K-Stability for Fano Manifolds with Torus Action of Complexity One

Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models

K-stability for Fano manifolds with torus action of complexity $1$

Polarized Complexity-One T-Varieties