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807
Towards an enumerative geometry of the moduli space of curves
 in Arithmetic and Geometry
, 1983
"... The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of arbitrary genus g. By this we mean setting up a Chow ring for the moduli space M g of curves of genus g and its compactification.M 9, defining what seem to be ..."
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Cited by 203 (0 self)
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The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of arbitrary genus g. By this we mean setting up a Chow ring for the moduli space M g of curves of genus g and its compactification.M 9, defining what seem to be
Localization of virtual classes
"... We prove a localization formula for the virtual fundamental class in the general context of C∗equivariant perfect obstruction theories. Let X be an algebraic scheme with a C∗action and a C∗equivariant perfect obstruction theory. The virtual fundamental class [X] vir in ..."
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Cited by 173 (25 self)
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We prove a localization formula for the virtual fundamental class in the general context of C∗equivariant perfect obstruction theories. Let X be an algebraic scheme with a C∗action and a C∗equivariant perfect obstruction theory. The virtual fundamental class [X] vir in
Eigenvalues, invariant factors, highest weights, and Schubert calculus
 Bull. Amer. Math. Soc. (N.S
"... Abstract. We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Gra ..."
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Cited by 109 (3 self)
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Abstract. We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices. Contents 1. Eigenvalues of sums of Hermitian and real symmetric matrices 2. Invariant factors 3. Highest weights 4. Schubert calculus
A holomorphic Casson invariant for CalabiYau 3folds, and bundles on K3 fibrations
 J. Differential Geom
, 2000
"... We briefly review the formal picture in which a CalabiYau nfold is the complex analogue of an oriented real nmanifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a CalabiYau 3fol ..."
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Cited by 105 (5 self)
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We briefly review the formal picture in which a CalabiYau nfold is the complex analogue of an oriented real nmanifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a CalabiYau 3fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of [LT], [BF] in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in P 3, and Donaldson – and GromovWitten – like invariants of Fano 3folds. It also allows us to define the holomorphic Casson invariant of a CalabiYau 3fold X, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general K3 fibration X, enabling us to compute the invariant for some ranks and Chern classes, and equate it to GromovWitten invariants of the “Mukaidual ” 3fold for others. As an example the invariant is shown to distinguish Gross ’ diffeomorphic 3folds. Finally the Mukaidual 3fold is shown to be CalabiYau and its cohomology is related to that of X. 1
Double Bruhat Cells And Total Positivity
"... this paper we extend the results of [3, 4] to the whole variety G0 . We will try to make the point that the natural framework for the study of G0 is provided by the decomposition of G into the disjoint union of double Bruhat cells G = BuB " B \Gamma vB \Gamma ; here B and B \Gamma are two opposit ..."
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Cited by 90 (18 self)
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this paper we extend the results of [3, 4] to the whole variety G0 . We will try to make the point that the natural framework for the study of G0 is provided by the decomposition of G into the disjoint union of double Bruhat cells G = BuB " B \Gamma vB \Gamma ; here B and B \Gamma are two opposite Borel subgroups in G, and u and v belong to the Weyl group W of G. We believe these double cells to be a very interesting object of study in its own right. The term "cells" might be misleading: in fact, the topology of G is in general quite nontrivial. (In some special cases, the "real part" of G was studied in [20, 21]. V. Deodhar [9] studied the intersections BuB " B \Gamma vB whose properties are very different from those of G .) We study a family of birational parametrizations of G , one for each reduced expression i of the element (u; v) in the Coxeter group W \Theta W . Every such parametrization can be thought of as a system of local coordinates in G call these coordinates the factorization parameters associated to i. They are obtained by expressing a generic element x 2 G as an element of the maximal torus H = B " B \Gamma multiplied by the product of elements of various oneparameter subgroups in G associated with simple roots and their negatives; the reduced expression i prescribes the order of factors in this product. The main technical result of this paper (Theorem 1.9) is an explicit formula for these factorization parameters as rational functions on the double Bruhat cell G . Theorem 1.9 is formulated in terms of a special family of regular functions \Delta fl ;ffi on the group G. These functions are suitably normalized matrix coefficients corresponding to pairs of extremal weights (fl; ffi ) in some fundamental representation of G. Again, we b...
Parallel spinors and connections with skewsymmetric torsion in string theory
, 2008
"... We describe all almost contact metric, almost hermitian and G2structures admitting a connection with totally skewsymmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the ∇parallel spinors. In p ..."
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Cited by 89 (5 self)
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We describe all almost contact metric, almost hermitian and G2structures admitting a connection with totally skewsymmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the ∇parallel spinors. In particular, we obtain solutions of the type II string equations in dimension n = 5, 6 and 7.
Notes On Stable Maps And Quantum Cohomology
, 1996
"... Contents 0. Introduction 1 1. Stable maps and their moduli spaces 10 2. Boundedness and a quotient approach 12 5. The construction of M g;n (X; fi) 25 6. The boundary of M 0;n (X; fi) 29 7. GromovWitten invariants 31 8. Quantum cohomology 34 9. Applications to enumerative geometry 38 10. Varia ..."
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Cited by 87 (12 self)
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Contents 0. Introduction 1 1. Stable maps and their moduli spaces 10 2. Boundedness and a quotient approach 12 5. The construction of M g;n (X; fi) 25 6. The boundary of M 0;n (X; fi) 29 7. GromovWitten invariants 31 8. Quantum cohomology 34 9. Applications to enumerative geometry 38 10. Variations 43 References 46 0. Introduction 0.1. Overview. The aim of these notes is to describe an exciting chapter in the recent development of quantum cohomology. Guided by ideas from physics (see [W]), a remarkable structure on the solutions of certain rational enumerative geometry problems has been found: the solutions are coefficients in the multiplication table of a quantum cohomology ring. Associativity of the ring yields nontrivial relations among the enumerative solutions. In many cases, these relations suffice to solve the enumerative problem. For example, let N d be the number of degree d, rational plane curves passing through 3d \Gamma 1 general points in P . Since there is a un
Hurwitz numbers and intersections on moduli spaces of curves
 Invent. Math
"... 1.1. Topological classification of ramified coverings of the sphere. For a compact connected genus g complex curve C let f: C → CP 1 be a meromorphic function. We treat this function as a ramified covering of the sphere. Two ramified coverings (C1; f1), (C2; f2) are called topologically ..."
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Cited by 74 (2 self)
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1.1. Topological classification of ramified coverings of the sphere. For a compact connected genus g complex curve C let f: C → CP 1 be a meromorphic function. We treat this function as a ramified covering of the sphere. Two ramified coverings (C1; f1), (C2; f2) are called topologically
Finite element exterior calculus, homological techniques, and applications
 ACTA NUMERICA
, 2006
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