Results 1  10
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7,552
The selfduality equations on a Riemann surface
 Proc. Lond. Math. Soc., III. Ser
, 1987
"... In this paper we shall study a special class of solutions of the selfdual YangMills equations. The original selfduality equations which arose in mathematical physics were defined on Euclidean 4space. The physically relevant solutions were the ones with finite action—the socalled 'instanton ..."
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Cited by 524 (6 self)
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In this paper we shall study a special class of solutions of the selfdual YangMills equations. The original selfduality equations which arose in mathematical physics were defined on Euclidean 4space. The physically relevant solutions were the ones with finite action—the socalled 'instantons'. The same equations may be
Homological Algebra of Mirror Symmetry
 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 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
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Cited by 529 (3 self)
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Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual CalabiYau 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 on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on CalabiYau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated
The irreducibility of the space of curves of given genus
 Publ. Math. IHES
, 1969
"... Fix an algebraically closed field k. Let Mg be the moduli space of curves of genus g over k. The main result of this note is that Mg is irreducible for every k. Of course, whether or not M s is irreducible depends only on the characteristic of k. When the characteristic s o, we can assume that k ~ ..."
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Cited by 512 (2 self)
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Fix an algebraically closed field k. Let Mg be the moduli space of curves of genus g over k. The main result of this note is that Mg is irreducible for every k. Of course, whether or not M s is irreducible depends only on the characteristic of k. When the characteristic s o, we can assume that k ~ (1, and then the result is classical. A simple proof appears in EnriquesChisini [E, vol. 3, chap. 3], based on analyzing the totality of coverings of p1 of degree n, with a fixed number d of ordinary branch points. This method has been extended to char. p by William Fulton [F], using specializations from char. o to char. p provided that p> 2g qi. Unfortunately, attempts to extend this method to all p seem to get stuck on difficult questions of wild ramification. Nowadays, the Teichmtiller theory gives a thoroughly analytic but very profound insight into this irreducibility when kC. Our approach however is closest to Severi's incomplete proof ([Se], Anhang F; the error is on pp. 344345 and seems to be quite basic) and follows a suggestion of Grothendieck for using the result in char. o to deduce the result in char. p. The basis of both Severi's and Grothendieck's ideas is to construct families of curves X, some singular, with pa(X)=g, over nonsingular parameter spaces, which in some sense contain enough singular curves to link together any two components that Mg might have. The essential thing that makes this method work now is a recent " stable reduction theorem " for abelian varieties. This result was first proved independently in char. o by Grothendieck, using methods of etale cohomology (private correspondence with J. Tate), and by Mumford, applying the easy half of Theorem (2.5), to go from curves to abelian varieties (cf. [M2]). Grothendieck has recently strengthened his method so that it applies in all characteristics (SGA 7, ~968) 9 Mumford has also given a proof using theta functions in char. ~2. The result is this: Stable Reduction Theorem. Let R be a discrete valuation ring with quotient field K. Let A be an abelian variety over K. Then there exists a finite algebraic extension L of K such
Existence of minimal models for varieties of log general type
 J. AMER. MATH. SOC
, 2008
"... We prove that the canonical ring of a smooth projective variety is finitely generated. ..."
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Cited by 386 (34 self)
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We prove that the canonical ring of a smooth projective variety is finitely generated.
Generalized CalabiYau manifolds
 Q. J. Math
"... A geometrical structure on evendimensional manifolds is defined which generalizes the notion of a CalabiYau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphisms and closed 2forms. In the special case o ..."
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Cited by 330 (3 self)
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A geometrical structure on evendimensional manifolds is defined which generalizes the notion of a CalabiYau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphisms and closed 2forms. In the special case of six dimensions we characterize them as critical points of a natural variational problem on closed forms, and prove that a local moduli space is provided by an open set in either the odd or even cohomology. 1
ElectricMagnetic duality and the geometric Langlands program
, 2006
"... The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N = 4 super YangMills theory in four dimensions. The key ingredients are electricmagnetic duality of gauge theory, mirror symmetry of sigmamodels, branes, Wilson and ’t H ..."
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Cited by 300 (26 self)
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The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N = 4 super YangMills theory in four dimensions. The key ingredients are electricmagnetic duality of gauge theory, mirror symmetry of sigmamodels, branes, Wilson and ’t
Results 1  10
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7,552