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34
Mirror Symmetry and Rational Curves on Quintic Threefolds: A Guide for Mathematicians
 J. AM. MATH. SOC
, 1993
"... We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new qexpansion principle for functions on the ..."
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Cited by 87 (9 self)
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We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new qexpansion principle for functions on the moduli space of CalabiYau manifolds, and the “mirror symmetry” phenomenon recently observed by string theorists.
Gorenstein threefold singularities with small resolutions via invariant theory of Weyl groups
 J. of Alg. Geom
, 1992
"... A fundamental new type of birational modification which first occurs in dimension three is the simple flip. This is a birational map Y �� � Y + which induces an isomorphism (Y −C) ∼ = (Y + −C +), where C and C + are smooth rational curves such that KY · C < 0 and KY + · C+> 0. (Y and Y + should be ..."
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Cited by 38 (5 self)
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A fundamental new type of birational modification which first occurs in dimension three is the simple flip. This is a birational map Y �� � Y + which induces an isomorphism (Y −C) ∼ = (Y + −C +), where C and C + are smooth rational curves such that KY · C < 0 and KY + · C+> 0. (Y and Y + should be allowed to have “terminal ” singularities.) Mori’s celebrated theorem [22] shows that these flips exist when numerically expected. A closely related type of modification is the simple flop. This has a similar definition, except that Y and Y + should be Gorenstein, with KY · C = KY + · C+ = 0. (This is more than an analogy: every flip has a branched double cover which is a flop, and this construction was used in Mori’s proof.) For both flips and flops, the curves C and C + can be contracted to points (in Y and Y +, respectively), yielding the same normal variety X. The birational map Y �� � Y + can thus be described in terms of the two contraction morphisms π: Y → X and
Multiple covers and the integrality conjecture for rational curves in CalabiYau threefolds
 J. Algebraic Geom
"... Abstract. We study the contribution of multiple covers of an irreducible rational curve C in a CalabiYau threefold Y to the genus 0 GromovWitten invariants in the following cases. 1. If the curve C has one node and satisfies a certain genericity condition, we prove that the contribution of multipl ..."
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Cited by 16 (3 self)
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Abstract. We study the contribution of multiple covers of an irreducible rational curve C in a CalabiYau threefold Y to the genus 0 GromovWitten invariants in the following cases. 1. If the curve C has one node and satisfies a certain genericity condition, we prove that the contribution of multiple covers of degree d is given by ∑ 1 n3 nd 2. For a smoothly embedded contractable curve C ⊂ Y we define schemes Ci for 1 ≤ i ≤ l where Ci is supported on C and has multiplicity i, the number l ∈ {1,...,6} being Kollár’s invariant “length”. We prove that the contribution of multiple covers of C of degree d is given by ∑ kd/n n nd 3 where ki is the multiplicity of Ci in its Hilbert scheme (and ki = 0 if i> l). In the latter case we also get a formula for arbitrary genus (Theorem 1.5). These results show that the curve C contributes an integer amount to the socalled instanton numbers that are defined recursively in terms of the GromovWitten invariants and are conjectured to be integers. 1. Motivation
Crepant resolution of trihedral singularities
 Proceedings of the Japan Academy 70
, 1994
"... The purpose of this paper is to construct a crepant resolution of quotient singularities by finite subgroups of SL(3, C) of certain type, and prove that each Euler number of the minimal model is equal to the number of conjugacy classes. The problem of finding a nice resolution of quotient singularit ..."
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Cited by 14 (0 self)
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The purpose of this paper is to construct a crepant resolution of quotient singularities by finite subgroups of SL(3, C) of certain type, and prove that each Euler number of the minimal model is equal to the number of conjugacy classes. The problem of finding a nice resolution of quotient singularities by finite subgroups
Genus zero GopakumarVafa invariants of contractible curves
"... Abstract. A version of the DonaldsonThomas invariants of a CalabiYau threefold is proposed as a conjectural mathematical definition of the GopakumarVafa invariants. These invariants have a local version, which is verified to satisfy the required properties for contractible curves. This provides a ..."
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Cited by 11 (0 self)
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Abstract. A version of the DonaldsonThomas invariants of a CalabiYau threefold is proposed as a conjectural mathematical definition of the GopakumarVafa invariants. These invariants have a local version, which is verified to satisfy the required properties for contractible curves. This provides a new viewpoint on the computation of the local GromovWitten invariants of contractible curves by Bryan, Leung, and the author. 1 Introduction. Let X be a CalabiYau threefold, β ∈ H2(X,Z), and g a nonnegative integer. The GopakumarVafa invariants n g β were first introduced as an integervalued index arising from Dbranes and M2branes wrapping holomorphic curves in
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Cited by 8 (7 self)
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Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
Gorenstein quotient singularities of monomial type in dimension three
, 1994
"... The purpose of this paper is to construct a crepant resolution of quotient singularities by finite subgroups of SL(3, C) of monomial type (type (B),(C) and (D) in [5]), and prove that the Euler number of the resolution is equal to the number ..."
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Cited by 6 (0 self)
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The purpose of this paper is to construct a crepant resolution of quotient singularities by finite subgroups of SL(3, C) of monomial type (type (B),(C) and (D) in [5]), and prove that the Euler number of the resolution is equal to the number
Primitive contractions of Calabi–Yau threefolds
"... Abstract. We construct examples of primitive contractions of Calabi–Yau threefolds with exceptional locus being P 1 × P 1, P 2, and smooth del Pezzo surfaces of degrees ≤ 5. We describe the images of these primitive contractions and find their smoothing families. In particular we give a method to co ..."
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Cited by 5 (1 self)
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Abstract. We construct examples of primitive contractions of Calabi–Yau threefolds with exceptional locus being P 1 × P 1, P 2, and smooth del Pezzo surfaces of degrees ≤ 5. We describe the images of these primitive contractions and find their smoothing families. In particular we give a method to compute the Hodge numbers of the generic fibers of the smoothings of each Qfactorial Calabi–Yau threefold with one isolated singularity obtained after a primitive contraction of type II. As an application we get examples of natural conifold transitions between some families of Calabi–Yau threefolds. 1.
Holomorphic forms on threefolds
, 2003
"... Abstract. Two conjectures relating the Kodaira dimension of a smooth projective variety and existence of number of nowhere vanishing 1forms on the variety are proposed and verified in dimension 3. §1 introduction Let X be a smooth projective variety over the complex number field C. Let ω ∈ H0 (X, Ω ..."
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Cited by 4 (0 self)
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Abstract. Two conjectures relating the Kodaira dimension of a smooth projective variety and existence of number of nowhere vanishing 1forms on the variety are proposed and verified in dimension 3. §1 introduction Let X be a smooth projective variety over the complex number field C. Let ω ∈ H0 (X, Ωi X) be a nontrivial global holomorphic differential iform. We would like to understand the nature of zero locus of ω in terms of the birational geometry of X. On one hand algebraic varieties are birationally classified according to their Kodaira dimensions using high multiples of holomorphic form of top degreethe canonical bundle and Mori’s theory of extremal contractions and flips are birational operations done to part of the base locus of (multiples of) canonical bundle, on the other hand very little is known about the impact of zero locus of holomorphic forms of lower degree has on the birational geometry of the underlying variety. The first question one may ask is: What makes ω to have zero locus? It is proved in [Z] that for any global holomorphic 1form 0 ̸ = ω ∈ H 0 (X, ΩX), the zero locus Z(ω) is not empty provided that the canonical bundle KX is ample. It is natural to suspect that the same conclusion should hold for varieties of general type. Indeed we propose two conjectures:
Finite representations of a quiver arising from string theory, eprint math.AG/0507316
"... Abstract. Inspired by Cachazo, Katz and Vafa (“Geometric transitions and N = 1 quiver theories ” (hepth/0108120)), we examine representations of “N = 1 quivers ” arising from string theory. We derive some mathematical consequences of the physics, and show that these results are a natural extension ..."
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Cited by 4 (0 self)
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Abstract. Inspired by Cachazo, Katz and Vafa (“Geometric transitions and N = 1 quiver theories ” (hepth/0108120)), we examine representations of “N = 1 quivers ” arising from string theory. We derive some mathematical consequences of the physics, and show that these results are a natural extension of Gabriel’s ADE theorem. Extending the usual ADE case that relates quiver representations to curves on surfaces, we relate these new quiver representations to curves on threefolds. 1.