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487
The intermediate Jacobian of the cubic threefold
 Ann. of Math
, 1972
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Cited by 132 (0 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
Torification and factorization of birational maps
"... Abstract. Building on work of the fourth author in [68], we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero ..."
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Cited by 88 (1 self)
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Abstract. Building on work of the fourth author in [68], we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero
Singular hermitian metrics on positive line bundles
"... The notion of a singular hermitian metric on a holomorphic line bundle is introduced as a tool for the study of various algebraic questions. One of the main interests of such metrics is the corresponding L2 vanishing theorem for ∂ cohomology, which gives a useful criterion for the existence of sect ..."
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Cited by 73 (16 self)
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The notion of a singular hermitian metric on a holomorphic line bundle is introduced as a tool for the study of various algebraic questions. One of the main interests of such metrics is the corresponding L2 vanishing theorem for ∂ cohomology, which gives a useful criterion for the existence of sections. In this context, numerically effective line bundles and line bundles with maximum Kodaira dimension are characterized by means of positivity properties of the curvature in the sense of currents. The coefficients of isolated logarithmic poles of a plurisubharmonic singular metric are shown to have a simple interpretation in terms of the constant ε of Seshadri’s ampleness criterion. Finally, we use singular metrics and approximations of the curvature current to prove a new asymptotic estimate for the dimension of cohomology groups with values in high multiples O(kL) of a line bundle L with maximum Kodaira dimension.
The Local Prop Anabelian Geometry of Curves
 RIMS Preprint 1097
, 1996
"... Let X be a connected scheme. Then one can associate (after Grothendieck) to X its algebraic fundamental group π1(X). This group π1(X) is a profinite group which is uniquely determined (up to inner automorphisms) by the property that the category of ..."
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Cited by 70 (38 self)
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Let X be a connected scheme. Then one can associate (after Grothendieck) to X its algebraic fundamental group π1(X). This group π1(X) is a profinite group which is uniquely determined (up to inner automorphisms) by the property that the category of
Semicontinuity of complex singularity exponents and KählerEinstein metrics on Fano orbifolds
, 2000
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FRational Rings Have Rational Singularities
 Amer. Jour. Math
, 1997
"... . It is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational. A key point in the proof is a characterization of Frational local rings as those CohenMacaulay local rings (R; m) in which ..."
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Cited by 49 (10 self)
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. It is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational. A key point in the proof is a characterization of Frational local rings as those CohenMacaulay local rings (R; m) in which the local cohomology module H d m (R) (where d is the dimension of R) have no submodules stable under the natural action of the Frobenius map. An analog for finitely generated algebras over a field of characteristic zero is developed, which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over C , without reference to a desingularization. With the development of the theory of tight closure by M. Hochster and C. Huneke [HH1], a natural question arose. What information does this powerful new tool provide about the structure of the singularities of an algebraic variety? The main theorem of this paper is the following: Theorem 3.1. If a...
Algebraic analysis for nonidentifiable learning machines
 Neural Computation
"... This paper clarifies the relation between the learning curve and the algebraic geometrical structure of a nonidentifiable learning machine such as a multilayer neural network whose true parameter set is an analytic set with singular points. By using a concept in algebraic analysis, we rigorously pr ..."
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Cited by 47 (16 self)
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This paper clarifies the relation between the learning curve and the algebraic geometrical structure of a nonidentifiable learning machine such as a multilayer neural network whose true parameter set is an analytic set with singular points. By using a concept in algebraic analysis, we rigorously prove that the Bayesian stochastic complexity or the free energy is asymptotically equal to λ1 log n − (m1 − 1) log log n+constant, where n is the number of training samples and λ1 and m1 are the rational number and the natural number which are determined as the birational invariant values of the singularities in the parameter space. Also we show an algorithm to calculate λ1 and m1 based on the resolution of singularities in algebraic geometry. In regular statistical models, 2λ1 is equal to the number of parameters and m1 = 1, whereas in nonregular models such as multilayer networks, 2λ1 is not larger than the number of parameters and m1 ≥ 1. Since the increase of the stochastic complexity is equal to the learning curve or the generalization error, the nonidentifiable learning machines are the better models than the regular ones if the Bayesian ensemble learning is applied. 1 1
Essential dimensions of algebraic groups and a resolution theorem for Gvarieties
 Canad. J. Math
"... Abstract. Let G be an algebraic group and let X be a generically free Gvariety. We show that X can be transformed, by a sequence of blowups with smooth Gequivariant centers, into a Gvariety X ′ with the following property: the stabilizer of every point of X ′ is isomorphic to a semidirect product ..."
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Cited by 40 (15 self)
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Abstract. Let G be an algebraic group and let X be a generically free Gvariety. We show that X can be transformed, by a sequence of blowups with smooth Gequivariant centers, into a Gvariety X ′ with the following property: the stabilizer of every point of X ′ is isomorphic to a semidirect product U> ⊳ A of a unipotent group U and a diagonalizable group A. As an application of this and related results, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus
Topology of Real Algebraic Sets
 Mathematical Sciences Research Institute Publications
, 1992
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