Givental's theorem for complete intersections in smooth toric varieties is generalized to Fano varieties. The Gromov-Witten invariants are found for Fano varieties of dimension ≥3 that are complete intersections in weighted projective spaces or singular toric varieties. A generalized Riemann-Roch equation is also obtained for such varieties. As a consequence, the counting matrices of smooth Fano threefolds with Picard group  and anticanonical degrees 2, 8, and 16 are calculated. Bibliography: 29 titles.

Quantum cohomology of smooth complete intersections in weighted projective spaces and in singular toric varieties

By an additive action on an algebraic variety $X$ of dimension $n$ we mean a regular action $\mathbb{G}_a^n \times X \to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. We prove that if a complete toric variety $X$ admits an additive action, then it admits an additive action normalized by the acting torus. Normalized additive actions on a toric variety $X$ are in bijection with complete collections of Demazure roots of the fan of $X$. Moreover, any two normalized additive actions on $X$ are isomorphic.

Additive actions on toric varieties

We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. The current version of the text includes the introduction and the section on equivariant embeddings into the projective space $\mathbb{P}^n$. Comments and suggestions are very welcome.

Equivariant completions of affine spaces

By an additive action on an algebraic variety $X$ we mean a regular effective action $\mathbb{G}_a^n\times X\to X$ with an open orbit of the commutative unipotent group $\mathbb{G}_a^n$. In this paper, we give a uniqueness criterion for additive action on a complete toric variety.

On uniqueness of additive actions on complete toric varieties

We prove that the boundaries of the Milnor fibers of smoothings of non-isolated reduced complex surface singularities are graph manifolds. Moreover, we give a method, inspired by previous work of Nemethi and Szilard, to compute associated plumbing graphs.

https://tel.archives-ouvertes.fr/tel-02180967/document

Topology of smoothings of non-isolated singularities of complex surfaces

/pdf/review-tadao-oda-convex-bodies-and-algebraic-geometry-an-x3ejn676qy.pdf

Review: Tadao Oda, Convex bodies and algebraic geometry: An introduction to toric varieties

/pdf/book-review-convex-bodies-and-algebraic-geometry-an-355sgtvpes.pdf

Miles Reid

Papers

Review: Tadao Oda, Convex bodies and algebraic geometry: An introduction to toric varieties

Book Review: Convex bodies and algebraic geometry: An introduction to toric varieties