Gromov-Witten theory of orbicurves, the space of tri-polynomials and symplectic field theory of Seifert fibrations

We compute, with symplectic field theory (SFT) techniques, the Gromov-Witten theory of ${\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}}We compute, with symplectic field theory (SFT) techniques, the Gromov-Witten theory of \mathbbP¹_a1,?,aa{\mathbb{P}^1_{\alpha_1,\ldots,\alpha_a}}, i.e., the complex projective line with a orbifold points. A natural subclass of these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of (polynomial) Frobenius manifolds and integrable systems of Hamiltonian PDEs, which extend the (dispersionless) bigraded Toda hierarchy (Carlet, The extended bigraded toda hierarchy. arXiv preprint arXiv:math-ph/0604024). We then define a Frobenius structure on the spaces of polynomials in three complex variables of the form F(x, y, z) = −xyz + P ₁(x) + P ₂(y) + P ₃(z) which contains as special cases the ones constructed on the space of Laurent polynomials (Dubrovin, Geometry of 2D topologica field theories. Integrable systems and quantum groups, Springer Lecture Notes in Mathematics 1620:120–348, 1996; Milanov and Tseng, The space of Laurent polynomials, \mathbbP¹{\mathbb{P}^1}-orbifolds, and integrable hierarchies. preprint arXiv:math/0607012v3 [math.AG]). We prove a mirror theorem stating that these Frobenius structures are isomorphic to the ones found before for polynomial \mathbbP¹{\mathbb{P}^1}-orbifolds. Finally we link rational SFT of Seifert fibrations over \mathbbP¹_a,b,c{\mathbb{P}^1_{a,b,c}} with orbifold Gromov-Witten invariants of the base, extending a known result (Bourgeois, A Morse-Bott approach to contact homology. Ph.D. dissertation, Stanford University, 2002) valid in the smooth case.     

Topological conformal field theories and Calabi-Yau categories

This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-Witten type invariants depend on a Calabi-Yau A_∞ category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate A_∞ version of the derived category of coherent sheaves on a Calabi-Yau variety, this constructs the B model at all genera. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the usual Gromov-Witten invariants are recovered. The assumptions are that open-closed Gromov-Witten theory can be constructed for X, and that the natural map from the Hochschild homology of the Fukaya category of X to the ordinary homology of X is an isomorphism.     

Rings of Formal Power Series in an Infinite Set of Indeterminates

Let α be an infinite cardinal number, Λ be an index set of cardinality > α, and {X _λ}_λ∈Λ be a set of indeterminates over an integral domain D. It is well known that there are three ways of defining the ring of formal power series in {X _λ}_λ∈Λ over D, say, D[[{X _λ}]] _i for i = 1, 2, 3. In this paper, we let D[[{X _λ}]]_α = ∪ {D[[{X _λ}_λ∈Γ]]₃ | Γ ? Λ and |Γ| ≤ α}, and we then show that D[[{X _λ}]]_α is an integral domain such that D[[{X _λ}]]₂ ? D[[{X _λ}]]_α ? D[[{X _λ}]]₃. We also prove that (1) D is a Krull domain if and only if D[[{X _λ}]]_α is a Krull domain and (2) D[[{X _λ}]]_α is a unique factorization domain (UFD) (resp., π-domain) if and only if D[[X ₁,…, X _n ]] is a UFD (resp., π-domain) for every integer n ≥ 1.     

Triple Cohomology and the Galois Cohomology of Profinite Groups

Given a cotriple 𝔾 = (G, ε, δ) on a category X and a functor E:X ^Opp→A into an abelian category A, there exists the cohomology theory of Barr and Beck: Hⁿ(X, E) ε |A| (n ≥ 0, X ε |X|), ([1], p.249). Almost all the important cohomology theories in mathematics have been shown to be special instances of such a general theory (see [1], [2] and [3]). Usually E arises from an abelian group object Y in X in the following manner: it is the contravariant functor from X into the category Ab of abelian groups that associates to each object X in X the abelian group X(X, Y) of maps from X to Y. In such a situation we shall write Hⁿ(X, Y)_𝔾 instead of Hⁿ(X, E)_G. Barr and Beck [2] have shown that the Eilenberg-MacLane cohomology groups H?ⁿ(π, A), n ≥ 2, can be re-captured as follows. One considers the free group cotriple 𝔾′ on the category Gps of groups, which induces in a natural manner a cotriple 𝔾 on the category (Gps, π) of groups over a fixed group π.     

On Local Integrability Conditions of Jet Groupoids

A jet groupoid ℛ _q over a manifold X is a special Lie groupoid consisting of q-jets of local diffeomorphisms X→X. As a subbundle of J ^q (X,X), a jet groupoid can be considered as a system of nonlinear partial differential equations (PDE). This leads to the question if ℛ _q is formally integrable. On the other hand, each jet groupoid is the symmetry groupoid of a geometric object, which is a section ω of a natural bundle ℱ. Using the jet groupoids, we give a local characterisation of formal integrability for transitive jet groupoids in terms of their corresponding geometric objects. Thanks to M. Barakat and W. Plesken for discussions. The author was supported by DFG Grant Graduiertenkolleg 775.     

Poisson Traces and D-Modules on Poisson Varieties

To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism ${\phi : X \to Y}To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson traces on X, i.e. distributions invariant under Hamiltonian flow. When X has finitely many symplectic leaves, we prove that M(X) is holonomic. Thus, when X is affine and has finitely many symplectic leaves, the space of Poisson traces on X is finite-dimensional. More generally, to any morphism f: X ? Y{\phi : X \to Y} and any quasicoherent sheaf of Poisson modules N on X, we attach a right D-module M_f(X,N){M_\phi(X,N)} on X, and prove that it is holonomic if X has finitely many symplectic leaves, f{\phi} is finite, and N is coherent.     

辛轨形群胚的辛邻域定理

本文给出一种几何的子轨形群胚的定义,还给出了判定子轨形群胚的依据,并证明了紧子轨形群胚的轨形管状邻域、紧辛子轨形群胚的辛邻域和紧Lagrangian子轨形群胚的Lagrangian邻域的存在性.     

On Oliver’s p-group conjecture: II

Let p be an odd prime and S a finite p-group. B. Oliver’s conjecture arises from an open problem in the theory of p-local finite groups. It is the claim that a certain characteristic subgroup \mathfrakX(S){\mathfrak{X}(S)} of S always contains the Thompson subgroup. In previous work the first two authors and M. Lilienthal recast Oliver’s conjecture as a statement about the representation theory of the factor group S/\mathfrakX(S){S/\mathfrak{X}(S)}. We now verify the conjecture for a wide variety of groups S/\mathfrakX(S){S/\mathfrak{X}(S)}.     

An algebraic index theorem for orbifolds

Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann-Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem.     

A generalization of Higgs bundles to higher dimensional varieties

Let X be a smooth n-dimensional projective variety defined over and let L be a line bundle on X. In this paper we shall construct a moduli space parametrizing -cohomology L-twisted Higgs pairs, i.e., pairs where E is a vector bundle on X and . If we take , the canonical line bundle on X, the variety is canonically identified with the cotangent bundle of the smooth locus of the moduli space of stable vector bundles on X and, as such, it has a canonical symplectic structure. We prove that, in the general case, in correspondence to the choice of a non-zero section , one can define, in a natural way, a Poisson structure on . We also analyze the relations between this Poisson structure on and the canonical symplectic structure of the cotangent bundle to the smooth locus of the moduli space of parabolic bundles over X, with parabolic structure over the divisor D defined by the section s. These results generalize to the higher dimensional case similar results proved in [Bo1] in the case of curves. Received November 4, 1997; in final form May 28, 1998     

Distance graphs with maximum chromatic number

The distance graph G(D) has the set of integers as vertices and two vertices are adjacent in G(D) if their difference is contained in the set D⊂Z. A conjecture of Zhu states that if the chromatic number of G(D) achieves its maximum value |D|+1 then the graph has a triangle. The conjecture is proven to be true if |D|?3. We prove that the chromatic number of a distance graph with D={a,b,c,d} is five only if either D={1,2,3,4k} or D={a,b,a+b,b-a}. This confirms a stronger version of Zhu's conjecture for |D|=4, namely, if the chromatic number achieves its maximum value then the graph contains K₄.     

Mader’s Conjecture On Extremely Critical Graphs

A non-complete graph G is called an (n,k)-graph if it is n-connected but G—X is not (n−|X|+1)-connected for any X ⊂V (G) with |X|≤k. Mader conjectured that for k≥3 the graph K_2k+2−(1−factor) is the unique (2k,k)-graph(up to isomorphism). Here we prove this conjecture.     

Properties of Riesz fractional derivatives for Korteweg–de Vries solitons

Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (?Δ) ^α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (−Δ) ^α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u ₀(X) with X = x − 4t, these derivatives, u _α (X) = D ^α u ₀(X), and their Hilbert transforms, v _α (X) = −HD ^α u ₀(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ₊(s, a) and ζ₋(s, a), respectively. New properties are established for u _α (X) and v _α (X). It is proved that the functions w _α (X) = u _α (X) + iv _α (X) with α > −1 are solutions of the differential equation

-\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},      14. Minimal error difference formulas      Herbert E. Salzer 《Journal of Computational and Applied Mathematics》1977,3(4):263-267 The usual formula for the rth difference of f(X), at intervals of h, may introduce an error of 2rε, where ε is the |error| in f(X). When f(X) is either an exact polynomial of the nth degree, or very closely approximated by one within a finite interval, say [?1, 1], the rth difference, at X = X0, is expressible as ∑n+1i=1 ai f(Xi), where for certain points Xi within [?1, 1], depending upon (X0, h), ∑n+1i=1 |ai| may be very much less than 2r. Nodes Xi that minimize ∑n+1i=1|ai| are said to provide “minimal error difference formulas”. For very small h, close approximations to them are obtainable from similar derivative formulas. For other combinations (X0, h), non-minimal formulas for equally spaced Xi's, with ai's precomputed to higher accuracy than that in f(X), greatly reduce ∑n+1i=1|ai| from 2r, ensure its approach to zero with h, and in many cases also yield more decimals and significant figures than the direct differencing of f(X). For r = 1, simple conditions for the non-existence of any expression ∑n+1i=1 ai f(Xi), which improves ∑n+1i=1|ai| to be <2, are given for (X0, h), expressed as h ≥ h0 which depends upon X0 and extrema of Chebyshev polynomials.      15. L''interpretation de la classe de Maslov dans la K-theorie Hermitienne et dans la theorie relative de Chern-Weil      Maria Luiza Lapa de Souza 《K-Theory》1998,13(4):347-361 The purpose of this paper is, first to define the Maslov invariant in the U-theory, the intermediate theory between real K-theory and complex K-theory, whose vanishing is a necessary and sufficient condition for the stable transversality of two Lagrangian sub-bundles of a symplectic fiber bundle. The second purpose is to show, in conformity with Bott periodicity, that Chern, Pontryand Stiefel-Whitney classes are precise enough to play the same role as the Maslov classes when one replaces the base space X of the symplectic bundle, by S 7 X +the seventh topological suspension of X +the compactification of X.      16. 1-Cocycles on the group of contactomorphisms on the supercircle S1|3 generalizing the Schwarzian derivative      Boujemaa Agrebaoui  Raja Hattab 《Czechoslovak Mathematical Journal》2016,66(4):1143-1163 The relative cohomology Hdiff1(K(1|3), osp(2, 3);Dγ,µ(S1|3)) of the contact Lie superalgebra K(1|3) with coefficients in the space of differential operators Dγ,µ(S1|3) acting on tensor densities on S1|3, is calculated in N.Ben Fraj, I. Laraied, S. Omri (2013) and the generating 1-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative 1-cocycle s(Xf) = D1D2D3(f)α31/2, Xf ∈ K(1|3) which is invariant with respect to the conformal subsuperalgebra osp(2, 3) of K(1|3).      17. The Mukai conjecture for log Fano manifolds      Kento Fujita 《Central European Journal of Mathematics》2014,12(1):14-27 For a log Fano manifold (X,D) with D ≠ 0 and of the log Fano pseudoindex ≥2, we prove that the restriction homomorphism Pic(X) → Pic(D 1) of Picard groups is injective for any irreducible component D 1 ? D. The strategy of our proof is to run a certain minimal model program and is similar to Casagrande’s argument. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).      18. On Sectional Genus of Quasi—Polarized Manifolds with Non-Negative Kodaira Dimension      Yoshiaki Fukuma 《Mathematische Nachrichten》1996,180(1):75-84 Let X be a smooth projective variety over ? and L be a nef-big divisor on X. Then (X, L) is called a quasi - polarized manifold. Then we conjecture that g(L) ≥ q(X), where g(L) is the sectional genus of L and q(X) = dim H1(Ox) is the irregularity of X. In general it is unknown that this conjecture is true or not even in the case of dim X = 2. For example, this conjecture is true if dim X = 2 and dim H≥(L) > 0. But it is unknown if dim X ≥ 3 and dim H0(L) > 0. In this paper, we consider a lower bound for g(L) if dim X = 2, dim H0(L) ≥ 2, and k(X) ≥ 0. We obtain a stronger result than the above conjecture if dim Bs|L| ≤ 0 by a new method which can be applied to higher dimensional cases. Next we apply this method to the case in which dim X = n ≥ 3 and we obtain a lower bound for g(L) if dim X = 3, dim H0(L) ≥ 2, and k(X) ≥ 0.      19. On cardinality bounds involving the weak Lindelöf degree      A. Bella 《Quaestiones Mathematicae》2018,41(1):99-113 We give a general closing-off argument in Theorem 2.3 from which several corollaries follow, including (1) if X is a locally compact Hausdorff space then |X| ≤ 2wL(X)ψ(X), and (2) if X is a locally compact power homogeneous Hausdorff space then |X| ≤ 2wL(X)t(X). The first extends the well-known cardinality bound 2ψ(X) for a compactum X in a new direction. As |X| ≤ 2wL(X)χ(X) for a normal spaceX[4], this enlarges the class of known Tychonoff spaces for which this bound holds. In 2.12 we give a short, direct proof of (1) that does not use 2.3. Yet 2.3 is broad enough to establish results much more general than (1), such as if X is a regular space with a π-base ? such that |B| ≤ 2wL(X)χ(X) for all B ∈ ?, then |X| ≤ 2wL(X)χ(X). Separately, it is shown that if X is a regular space with a π-base whose elements have compact closure, then |X| ≤ 2wL(X)ψ(X)t(X). This partially answers a question from [4] and gives a third, separate proof of (1). We also show that if X is a weakly Lindelöf, normal, sequential space with χ(X) ≤ 2?0, then |X| ≤ 2?0. Result (2) above is a new generalization of the cardinality bound 2t(X) for a power homogeneous compactum X (Arhangel'skii, van Mill, and Ridderbos [3], De la Vega in the homogeneous case [10]). To this end we show that if U ? clD ? X, where X is power homogeneous and U is open, then |U| ≤ |D|πχ(X). This is a strengthening of a result of Ridderbos [19].      20. Calcul differentiel exterieur de degre arbitraire      Constantino M. de Barros 《manuscripta mathematica》1971,5(1):59-89 LetX and V be F-modules. For every integer t, it is developed an algebraic formalism which generates, by an analogous procedure of construction of the exterior de rivation in the usual sense, couples formed by a deriva tion of degree t of AF(X,F), an endomorphism of degree t of the graded additive group AF(X,V), and satisfying a rule of derivation of degree t. This formalism is the most general ifX is projective of finite type. If t is odd, the curvature, and ifX - V, the torsion, are discussed at some lenght. If t=1, the differen tial calculus associated to Lie modules, linear connexions and fields of endomorphisms, are special cases. Ce travail a été financé par le Conselho Nacio nal de Pesquisas (Brésil), contract TC 8233.

Minimal error difference formulas

The usual formula for the r^th difference of f(X), at intervals of h, may introduce an error of 2^rε, where ε is the |error| in f(X). When f(X) is either an exact polynomial of the n^th degree, or very closely approximated by one within a finite interval, say [?1, 1], the r^th difference, at X = X₀, is expressible as ∑ⁿ⁺¹_i=1 a_i f(X_i), where for certain points X_i within [?1, 1], depending upon (X₀, h), ∑ⁿ⁺¹_i=1 |a_i| may be very much less than 2^r. Nodes X_i that minimize ∑ⁿ⁺¹_i=1|a_i| are said to provide “minimal error difference formulas”. For very small h, close approximations to them are obtainable from similar derivative formulas. For other combinations (X₀, h), non-minimal formulas for equally spaced X_i's, with a_i's precomputed to higher accuracy than that in f(X), greatly reduce ∑ⁿ⁺¹_i=1|a_i| from 2^r, ensure its approach to zero with h, and in many cases also yield more decimals and significant figures than the direct differencing of f(X). For r = 1, simple conditions for the non-existence of any expression ∑ⁿ⁺¹_i=1 a_i f(X_i), which improves ∑ⁿ⁺¹_i=1|a_i| to be <2, are given for (X₀, h), expressed as h ≥ h₀ which depends upon X₀ and extrema of Chebyshev polynomials.     

L''interpretation de la classe de Maslov dans la K-theorie Hermitienne et dans la theorie relative de Chern-Weil

The purpose of this paper is, first to define the Maslov invariant in the U-theory, the intermediate theory between real K-theory and complex K-theory, whose vanishing is a necessary and sufficient condition for the stable transversality of two Lagrangian sub-bundles of a symplectic fiber bundle. The second purpose is to show, in conformity with Bott periodicity, that Chern, Pontryand Stiefel-Whitney classes are precise enough to play the same role as the Maslov classes when one replaces the base space X of the symplectic bundle, by S ⁷ X ⁺the seventh topological suspension of X ⁺the compactification of X.     

1-Cocycles on the group of contactomorphisms on the supercircle S1|3 generalizing the Schwarzian derivative

The relative cohomology H_diff¹(K(1|3), osp(2, 3);D_γ,µ(S^1|3)) of the contact Lie superalgebra K(1|3) with coefficients in the space of differential operators D_γ,µ(S^1|3) acting on tensor densities on S^1|3, is calculated in N.Ben Fraj, I. Laraied, S. Omri (2013) and the generating 1-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative 1-cocycle s(X_f) = D₁D₂D₃(_f)α₃^1/2, X^f ∈ K(1|3) which is invariant with respect to the conformal subsuperalgebra osp(2, 3) of K(1|3).     

The Mukai conjecture for log Fano manifolds

For a log Fano manifold (X,D) with D ≠ 0 and of the log Fano pseudoindex ≥2, we prove that the restriction homomorphism Pic(X) → Pic(D ₁) of Picard groups is injective for any irreducible component D ₁ ? D. The strategy of our proof is to run a certain minimal model program and is similar to Casagrande’s argument. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).     

On Sectional Genus of Quasi—Polarized Manifolds with Non-Negative Kodaira Dimension

Let X be a smooth projective variety over ? and L be a nef-big divisor on X. Then (X, L) is called a quasi - polarized manifold. Then we conjecture that g(L) ≥ q(X), where g(L) is the sectional genus of L and q(X) = dim H¹(O_x) is the irregularity of X. In general it is unknown that this conjecture is true or not even in the case of dim X = 2. For example, this conjecture is true if dim X = 2 and dim H≥(L) > 0. But it is unknown if dim X ≥ 3 and dim H⁰(L) > 0. In this paper, we consider a lower bound for g(L) if dim X = 2, dim H⁰(L) ≥ 2, and k(X) ≥ 0. We obtain a stronger result than the above conjecture if dim Bs|L| ≤ 0 by a new method which can be applied to higher dimensional cases. Next we apply this method to the case in which dim X = n ≥ 3 and we obtain a lower bound for g(L) if dim X = 3, dim H⁰(L) ≥ 2, and k(X) ≥ 0.     

On cardinality bounds involving the weak Lindelöf degree

We give a general closing-off argument in Theorem 2.3 from which several corollaries follow, including (1) if X is a locally compact Hausdorff space then |X| ≤ 2^wL(X)ψ(X), and (2) if X is a locally compact power homogeneous Hausdorff space then |X| ≤ 2^wL(X)t(X). The first extends the well-known cardinality bound 2^ψ(X) for a compactum X in a new direction. As |X| ≤ 2^wL(X)χ(X) for a normal spaceX[4], this enlarges the class of known Tychonoff spaces for which this bound holds. In 2.12 we give a short, direct proof of (1) that does not use 2.3. Yet 2.3 is broad enough to establish results much more general than (1), such as if X is a regular space with a π-base ? such that |B| ≤ 2^wL(X)χ(X) for all B ∈ ?, then |X| ≤ 2^wL(X)χ(X).

Separately, it is shown that if X is a regular space with a π-base whose elements have compact closure, then |X| ≤ 2^{wL(X)ψ(X)t(X)}. This partially answers a question from [4] and gives a third, separate proof of (1). We also show that if X is a weakly Lindelöf, normal, sequential space with χ(X) ≤ 2^?0, then |X| ≤ 2^?0.

Result (2) above is a new generalization of the cardinality bound 2^t(X) for a power homogeneous compactum X (Arhangel'skii, van Mill, and Ridderbos [3], De la Vega in the homogeneous case [10]). To this end we show that if U ? clD ? X, where X is power homogeneous and U is open, then |U| ≤ |D|^πχ(X). This is a strengthening of a result of Ridderbos [19].     

Calcul differentiel exterieur de degre arbitraire

LetX and V be F-modules. For every integer t, it is developed an algebraic formalism which generates, by an analogous procedure of construction of the exterior de rivation in the usual sense, couples formed by a deriva tion of degree t of A_F(X,F), an endomorphism of degree t of the graded additive group A_F(X,V), and satisfying a rule of derivation of degree t. This formalism is the most general ifX is projective of finite type. If t is odd, the curvature, and ifX - V, the torsion, are discussed at some lenght. If t=1, the differen tial calculus associated to Lie modules, linear connexions and fields of endomorphisms, are special cases.

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Ce travail a été financé par le Conselho Nacio nal de Pesquisas (Brésil), contract TC 8233.