Complex multiplication, rationality and mirror symmetry for Abelian varieties

We show that complex multiplication on Abelian varieties is equivalent to the existence of a constant rational Kähler metric. We give a sufficient condition for a mirror of an Abelian variety of CM-type to be of CM-type. We also study the relationship between complex multiplication and rationality of a toroidal lattice vertex algebra.     

Lefschetz Motives and the Tate Conjecture

A Lefschetz class on a smooth projective variety is an element of the Q-algebra generated by divisor classes. We show that it is possible to define Q-linear Tannakian categories of abelian motives using the Lefschetz classes as correspondences, and we compute the fundamental groups of the categories. As an application, we prove that the Hodge conjecture for complex abelian varieties of CM-type implies the Tate conjecture for all Abelian varieties over finite fields, thereby reducing the latter to a problem in complex analysis.     

Existence of the primitive Weierstrass gap sequences on curves of genus 9

We show that for any possible Weierstrass gap sequenceL on a curve of genus 9 with twice the smallest positive non-gap > the largest gap there exists a pointed non-singular curve (C, P) over an algebraically closed field of characteristic 0 such that the gap sequence atP isL.     

Embeddings of Spherical Homogeneous Spaces

We review in these notes the theory of equivariant embeddings of spherical homogeneous spaces. Given a spherical homogeneous space G/H, the normal equivariant embeddings of G/H are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties and which encode several geometric properties of the corresponding variety.     

Lectures on Wonderful Varieties

These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varieties, and some examples.     

On the Regularity of Powers of Ideal Sheaves

We use the geometry of the secant variety to an embedded smooth curve to prove some vanishing and regularity theorems for powers of ideal sheaves.     

N-dimensional rings with an isolated singular point having nonzero K?n

We construct examples of normal affine k-algebras of dimension N with an isolated singular point and nonzero K _–N , giving counter-examples to a conjecture of Weibel.     

Quantum Cohomology of Grassmannians

We give elementary proofs of the main theorems about the (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, the presentation, and a recent theorem of Fulton and Woodward about the minimal q-power which appears in a product of two Schubert classes.     

Generic lattice ideals

A concept of genericity is introduced for lattice ideals (and hence for ideals defining toric varieties) which ensures nicely structured homological behavior. For a generic lattice ideal we construct its minimal free resolution and we show that it is induced from the Scarf resolution of any reverse lexicographic initial ideal.

     

Measures of Simultaneous Approximation for Quasi-Periods of Abelian Varieties

We examine various extensions of a series of theorems proved by Chudnovsky in the 1980s on the algebraic independence (transcendence degree 2) of certain quantities involving integrals of the first and second kind on elliptic curves; these extensions include generalizations to abelian varieties of arbitrary dimensions, quantitative refinements in terms of measures of simultaneous approximation, as well as some attempt at unifying the aforementioned theorems. In the process we develop tools that might prove useful in other contexts, revolving around explicit “algebraic” theta functions on the one hand, and Eisenstein's theorem and G-functions on the other hand.