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Homological Algebra of Mirror Symmetry

by Maxim Kontsevich - in Proceedings of the International Congress of Mathematicians , 1994
"... Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
Abstract - Cited by 523 (3 self) - Add to MetaCart
Calabi-Yau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves

Gromov-Witten classes, quantum cohomology, and enumerative geometry

by M. Kontsevich, Yu. Manin - Commun. Math. Phys , 1994
"... The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological ..."
Abstract - Cited by 474 (3 self) - Add to MetaCart
Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Application to counting rational curves on del Pezzo surfaces and projective spaces are given. Let V be a projective algebraic manifold. Methods of quantum field theory recently led to a

Finding rational solutions of rational systems of autonomous ODEs

by L. X. Châu Ngô A
"... In this paper we provide an algorithm to find explicitly rational solutions of a rational system of autonomous ordinary differential equations (ODEs) from its invariant algebraic curves. The method is based on the rational parametrization of the rational invariant algebraic curves and intensively us ..."
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In this paper we provide an algorithm to find explicitly rational solutions of a rational system of autonomous ordinary differential equations (ODEs) from its invariant algebraic curves. The method is based on the rational parametrization of the rational invariant algebraic curves and intensively

EXERCISES IN THE BIRATIONAL GEOMETRY OF ALGEBRAIC VARIETIES

by János Kollár , 2008
"... The book [KM98] gave an introduction to the birational geometry of algebraic varieties, as the subject stood in 1998. The developments of the last decade made the more advanced parts of Chapters 6 and 7 less important and the detailed treatment of surface singularities in Chapter 4 less necessary. H ..."
Abstract - Cited by 322 (1 self) - Add to MetaCart
. Although singular, this is the “best ” model to work with. There are lots of these but they appear less frequently outside algebraic geometry. There are also two types of “mongrels”: (Ruled): Birational to P 1 × (curve of genus ≥ 1). (Elliptic): These fiber over a curve with general fiber an elliptic curve

Vector bundles over an elliptic curve

by M. F. Atiyah - Proc. London Math. Soc , 1957
"... THE primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve (defined over an algebraically closed field k). The interest of the elliptic curve lies in the fact that it provides the first non-trivial case, Grothendieck (6) having shown that for a rational curve e ..."
Abstract - Cited by 293 (0 self) - Add to MetaCart
THE primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve (defined over an algebraically closed field k). The interest of the elliptic curve lies in the fact that it provides the first non-trivial case, Grothendieck (6) having shown that for a rational curve

Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism

by Pavel Etingof, Victor Ginzburg - Invent. Math
"... To any finite group Γ ⊂ Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ, of the algebra C[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of CP r, where r = number of conjugacy classes of symplectic ..."
Abstract - Cited by 280 (39 self) - Add to MetaCart
‘rational ’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let Γ = Sn, the Weyl group of g = gl n. We construct a 1-parameter deformation of the Harish-Chandra homomorphism from D(g) g, the algebra of invariant polynomial differential operators on the Lie algebra g = gl n

Rational Points on Elliptic Curves

by Robert Gross, Joseph Silverman , 1992
"... Abstract. We give a quantitative bound for the number of S-integral points on an elliptic curve over a number field K in terms of the number of primes dividing the denominator of the j-invariant, the degree [K: Q], and the number of primes in S. Let K be a number field of degree d and MK the set of ..."
Abstract - Cited by 125 (1 self) - Add to MetaCart
Abstract. We give a quantitative bound for the number of S-integral points on an elliptic curve over a number field K in terms of the number of primes dividing the denominator of the j-invariant, the degree [K: Q], and the number of primes in S. Let K be a number field of degree d and MK the set

RELATIONSHIPS AMONG BIRATIONAL INVARIANTS OF ALGEBRAIC PLANE CURVES

by Shigeru Iitaka, Gakushuin Univeristy
"... Let S be a nonsingular rational surface S andD a nonsingular curve on S. Suppose that m ¸ a ¸ 1. Then Pm;a[D] = dim jmKS + aDj + 1. ..."
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Let S be a nonsingular rational surface S andD a nonsingular curve on S. Suppose that m ¸ a ¸ 1. Then Pm;a[D] = dim jmKS + aDj + 1.

Equivariant Gromov-Witten invariants

by Alexander B. Givental - INTERNAT. MATH. RES. NOTICES , 1996
"... The objective of this paper is to describe the construction and some applications of the equivariant counterpart to the Gromov-Witten (GW) theory, i.e., intersection theory on spaces of (pseudo-) holomorphic curves in (almost) Kähler manifolds. Given a Killing action of a compact Lie group G on a co ..."
Abstract - Cited by 127 (10 self) - Add to MetaCart
symplectic and complex geometry on a compact Kähler Calabi-Yau n-fold and, respectively, complex and symplectic geometry on another Calabi-Yau n-fold, called the mirror partner of the former one. The remarkable application [8]ofthe mirror conjecture to the enumeration of rational curves on Calabi-Yau 3-folds

Algebraic and rational differential invariants

by Escuela De Matemáticas
"... These notes start with an introduction to differential invariants. They continue with an algebraic treatment of the theory. The algebraic, dif-ferential algebraic and differential geometric tools that are necessary to the development of the theory are explained in detail. We expose the recent result ..."
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results on the topic of rational and algebraic differential invari-ants. Finally we give a new algebraic version of the finiteness theorem of Lie–Tresse for the case of finite dimensional algebraic groups.
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