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The witten equation, mirror symmetry and quantum singularity theory
, 2009
"... For any quasihomogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to GromovWitten theory and generalizes the theory of rspin curves, which corresponds ..."
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Cited by 53 (2 self)
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For any quasihomogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to GromovWitten theory and generalizes the theory of rspin curves, which corresponds to the simple singularity Ar−1. The main results are that we resolve two outstanding conjectures of Witten. The first conjecture is that ADEsingularities are selfdual; and the second conjecture is that the total potential functions of ADEsingularities satisfy corresponding ADEintegrable hierarchies. Other cases of integrable hierarchies are also discussed.
Tautological relations and the rspin Witten conjecture
"... In [23, 24], Y.P. Lee introduced a notion of universal relation for formal Gromov–Witten potentials. Universal relations are connected to tautological relations in the cohomology ring of the moduli space Mg,n of stable curves. Y.P. Lee conjectured that the two sets of relations coincide and proved ..."
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Cited by 43 (11 self)
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In [23, 24], Y.P. Lee introduced a notion of universal relation for formal Gromov–Witten potentials. Universal relations are connected to tautological relations in the cohomology ring of the moduli space Mg,n of stable curves. Y.P. Lee conjectured that the two sets of relations coincide and proved the inclusion (tautological relations) ⊂ (universal relations) modulo certain results announced by C. Teleman. He also proposed an algorithm that, conjecturally, computes all universal/tautological relations. Here we give a geometric interpretation of Y.P. Lee’s algorithm. This leads to a much simpler proof of the fact that every tautological relation gives rise to a universal relation. We also show that Y.P. Lee’s algorithm computes the tautological relations correctly if and only if the Gorenstein conjecture on the tautological cohomology ring of Mg,n is true. These results are first steps in the task of establishing an equivalence between formal and geometric Gromov–Witten theories. In particular, it implies that in any semisimple Gromov–Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the formal and the geometric Gromov–Witten potentials coincide.
Algebraic orbifold quantum products
"... The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan’s GromovWitten Theory of orbifolds. This work will be described in detail in a subsequent paper. The presentation here is generally based on lectures given by two of us at the Orbi ..."
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Cited by 41 (1 self)
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The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan’s GromovWitten Theory of orbifolds. This work will be described in detail in a subsequent paper. The presentation here is generally based on lectures given by two of us at the Orbifold Workshop
ORBIFOLD QUANTUM RIEMANNROCH, LEFSCHETZ AND SERRE
, 2009
"... Given a vector bundle F on a smooth DeligneMumford stack X and an invertible multiplicative characteristic class c, we define orbifold GromovWitten invariants of X twisted by F and c. We prove a “quantum RiemannRoch theorem” (Theorem 4.2.1) which expresses the generating function of the twisted i ..."
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Cited by 30 (9 self)
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Given a vector bundle F on a smooth DeligneMumford stack X and an invertible multiplicative characteristic class c, we define orbifold GromovWitten invariants of X twisted by F and c. We prove a “quantum RiemannRoch theorem” (Theorem 4.2.1) which expresses the generating function of the twisted invariants in terms of the generating function of the untwisted invariants. A quantum Lefschetz hyperplane theorem is derived from this by specializing to genus zero. As an application, we determine the relationship between genus0 orbifold GromovWitten invariants of X and that of a complete intersection, under additional assumptions. This provides a way to verify mirror symmetry predictions for some complete intersection orbifolds.
Moduli of roots of line bundles on curves
, 2007
"... We treat the problem of completing the moduli space for roots of line bundles on curves. Special attention is devoted to higher spin curves within the universal Picard scheme. Two new different constructions, both using line bundles on nodal curves as ..."
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Cited by 28 (2 self)
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We treat the problem of completing the moduli space for roots of line bundles on curves. Special attention is devoted to higher spin curves within the universal Picard scheme. Two new different constructions, both using line bundles on nodal curves as
Stable twisted curves and their rspin structures
, 2007
"... The object of this paper is the notion of rspin structure: a line bundle whose rth power is isomorphic to the canonical bundle. Over the moduli functor Mg of smooth genusg curves, rspin structures form a finite torsor under the group of rtorsion line bundles. Over the moduli functor Mg of stable ..."
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Cited by 21 (7 self)
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The object of this paper is the notion of rspin structure: a line bundle whose rth power is isomorphic to the canonical bundle. Over the moduli functor Mg of smooth genusg curves, rspin structures form a finite torsor under the group of rtorsion line bundles. Over the moduli functor Mg of stable curves, rspin structures form an étale stack, but the finiteness and the torsor structure are lost. In the present work, we show how this bad picture can be definitely improved simply by placing the problem in the category of Abramovich and Vistoli’s twisted curves. First, we find that within such category there exist several different compactifications of Mg; each one corresponds to a different multiindex ⃗ l = (l0, l1,...) identifying a notion of stability: ⃗ lstability. Then, we determine the suitable choices of ⃗ l for which rspin structures form a finite torsor over the moduli of ⃗ lstable curves.
The Witten top Chern class via Ktheory
 J. Algebraic Geom
"... Abstract. The Witten top Chern class is the crucial cohomology class needed to state a conjecture by Witten relating the Gelfand–Dikiĭ hierarchies to higher spin curves. In [PV01], Polishchuk and Vaintrob provide an algebraic construction of such a class. We present a more straightforward constructi ..."
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Cited by 17 (2 self)
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Abstract. The Witten top Chern class is the crucial cohomology class needed to state a conjecture by Witten relating the Gelfand–Dikiĭ hierarchies to higher spin curves. In [PV01], Polishchuk and Vaintrob provide an algebraic construction of such a class. We present a more straightforward construction via Ktheory. In this way we shortcircuit the passage through bivariant intersection theory and the use of MacPherson’s graph construction. Furthermore, we show that the Witten top Chern class admits a natural lifting to the Ktheory ring. 1.
THE BIRATIONAL TYPE OF THE MODULI SPACE OF EVEN SPIN CURVES
, 2009
"... The moduli space Sg of smooth spin curves parameterizes pairs [C,η], where [C] ∈ Mg is a curve of genus g and η ∈ Pic g−1 (C) is a thetacharacteristic. The finite forgetful map π: Sg → Mg has degree 2 2g and Sg is a disjoint union of two connected components S + g and S − g of relative degrees 2 g ..."
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Cited by 17 (9 self)
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The moduli space Sg of smooth spin curves parameterizes pairs [C,η], where [C] ∈ Mg is a curve of genus g and η ∈ Pic g−1 (C) is a thetacharacteristic. The finite forgetful map π: Sg → Mg has degree 2 2g and Sg is a disjoint union of two connected components S + g and S − g of relative degrees 2 g−1 (2 g +1) and 2 g−1 (2 g −1) corresponding to even and odd thetacharacteristics respectively. A compactification Sg of Sg over Mg is obtained by considering the coarse moduli space of the stack of stable spin curves of genus g (cf. [C], [CCC] and [AJ]). The projection Sg → Mg extends to a finite branched covering π: Sg → Mg. In this paper we determine the Kodaira dimension of S + g: Theorem 0.1. The moduli space S + g of even spin curves is a variety of general type for g> 8 and it is uniruled for g < 8. The Kodaira dimension of S + 8 is nonnegative 1. It was classically known that S + 2 is rational. The Scorza map establishes a birational isomorphism between S + 3 and M3, cf. [DK], hence S + 3 is rational. Very recently, Takagi and Zucconi [TZ] showed that S + 4 is rational as well. Theorem 0.1 can be compared to [FL] Theorem 0.3: The moduli space Rg of Prym varieties of dimension g − 1 (that is, nontrivial square roots of OC for each [C] ∈ Mg) is of general type when g> 13 and g = 15. On the other hand Rg is unirational for g < 8. Surprisingly, the problem of determining the Kodaira dimension has a much shorter solution for S + g than for Rg and our results are complete. We describe the strategy to prove that S + g is of general type for a given g. We denote by λ = π ∗ (λ) ∈ Pic(S + g) the pullback of the Hodge class and by α0,β0 ∈ Pic(S + g) and αi,βi ∈ Pic(S + g) for 1 ≤ i ≤ [g/2] boundary divisor classes such that π ∗ (δ0) = α0 + 2β0 and π ∗ (δi) = αi + βi for 1 ≤ i ≤ [g/2] (see Section 2 for precise definitions). Using RiemannHurwitz and [HM] we find that
The geometry of the moduli space of odd spin curves
"... The set of odd thetacharacteristics on a general curve C of genus g is in bijection with the set θ(C) of theta hyperplanes H ∈ (Pg−1) ∨ everywhere tangent to the canonically embedded curve C KC → Pg−1. Even though the geometry and the intricate combinatorics of θ(C) have been studied classicall ..."
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Cited by 16 (12 self)
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The set of odd thetacharacteristics on a general curve C of genus g is in bijection with the set θ(C) of theta hyperplanes H ∈ (Pg−1) ∨ everywhere tangent to the canonically embedded curve C KC → Pg−1. Even though the geometry and the intricate combinatorics of θ(C) have been studied classically, see [Dol], [DK] for a modern account, it was only recently proved in [CS] that one can reconstruct a general curve [C] ∈ Mg from the hyperplane configuration θ(C). → Mg which is an étale Odd thetacharacteristics form a moduli space π: S − g cover of degree 2g−1 (2g − 1). The normalization of Mg in the function field of S − g gives rise to a finite covering π: S − g → Mg. Furthermore, S − g has a modular meaning being isomorphic to the coarse moduli space of the DeligneMumford stack of odd stable spin curves, cf. [C], [CCC], [AJ]. The map π is branched along the boundary of Mg and one to enjoy better positivity properties than K.