Mathematical programs with vanishing constraints: critical point theory

We study mathematical programs with vanishing constraints (MPVCs) from a topological point of view. We introduce the new concept of a T-stationary point for MPVC. Under the Linear Independence Constraint Qualification we derive an equivariant Morse Lemma at nondegenerate T-stationary points. Then, two basic theorems from Morse Theory (deformation theorem and cell-attachment theorem) are proved. Outside the T-stationary point set, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. However, when passing a T-stationary level, the topology of the lower level set changes via the attachment of a q-dimensional cell. The dimension q equals the stationary T-index of the (nondegenerate) T-stationary point. The stationary T-index depends on both the restricted Hessian of the Lagrangian and the number of bi-active vanishing constraints. Further, we prove that all T-stationary points are generically nondegenerate. The latter property is shown to be stable under C ²-perturbations of the defining functions. Finally, some relations with other stationarity concepts, such as strong, weak, and M-stationarity, are discussed.     

Genus-decreasing relation of Gromov-Witten invariants for surfaces under blow-up

Using the degeneration formula, we study the change of Gromov-Witten invariants under blow-up for symplectic 4-manifolds and obtain a genus-decreasing relation of Gromov-Witten invariant of symplectic four manifold under blow-up.     

Vanishing theorems on toric varieties in positive characteristic

We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata–Viehweg vanishing theorem for log pairs on toric varieties in positive characteristic. These results generalize those results of Danilov, Buch–Thomsen–Lauritzen–Mehta, Musta?ǎ and Fujino to the case where concerned Weil divisors are not necessarily torus invariant.     

Hamiltonian Gromov-Witten invariants

In this paper we introduce invariants of semi-free Hamiltonian actions of S¹ on compact symplectic manifolds using the space of solutions to certain gauge theoretical equations. These equations generalise both the vortex equations and the holomorphicity equation used in Gromov-Witten theory. In the definition of the invariants we combine ideas coming from gauge theory and the ideas underlying the construction of Gromov-Witten invariants.     

Generalized bochner theorems and the spectrum of complete manifolds

Bochner's theorem that a compact Riemannian manifold with positive Ricci curvature has vanishing first cohomology group has various extensions to complete noncompact manifolds with Ricci possibly negative. One still has a vanishing theorem for L ² harmonic one-forms if the infimum of the spectrum of the Laplacian on functions is greater than minus the infimum of the Ricci curvature. This result and its analogues for p-forms yield vanishing results for certain infinite volume hyperbolic manifolds. This spectral condition also imposes topological restrictions on the ends of the manifold. More refined results are obtained by taking a certain Brownian motion average of the Ricci curvature; if this average is positive, one has a vanishing theorem for the first cohomology group with compact supports on the universal cover of a compact manifold. There are corresponding results for L ² harmonic spinors on spin manifolds.     

Remarks on the Defining Equations of Smooth Threefolds in P5

In this paper, we prove the vanishing of the first cohomology groups of ideal sheaves twisted by small integers for smooth threefolds in P⁵ defined over the complex number field C by using Le Potier vanishing theorem and Castelnuovo regularity for vector bundles, which contributes Peskine–Van de Ven conjecture for 2-normality.     

Barth type vanishing theorem

In this paper we give a vanishing result for cohomology groups of symmetric powers of the co-normal bundle of a non-degenerate smooth subvariety X of projective space, then we use this theorem to give a Barth type vanishing theorem.      

Stabilization of 4-manifolds

The complex surface X obtained by 8 points blown up on CP2 and Barlow’s surface Y are homeomorphic,but not diffeomorphic.Using the Gromov-Witten invariant Ruan showed that the stabilized manifolds X×S2and Y×S2are not deformation equivalent.In this note,we show that the stabilized manifolds X×S1and Y×S1are diffeomorphic and non-deformation equivalent in cosymplectic sense.     

Orbifold Gromov-Witten theory of weighted blowups

Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifold groupoid (X, ω). Let X_a be the weight-a blowup of X along S, and D_a = PN_a be the exceptional divisor, where N is the normal bundle of S in X. In this paper we show that the absolute orbifold Gromov-Witten theory of X_α can be effectively and uniquely reconstructed from the absolute orbifold Gromov-Witten theories of X, S and D_α, the natural restriction homomorphism H*_CR(X) → H*_CR(S) and the first Chern class of the tautological line bundle over D_α. To achieve this we first prove similar results for the relative orbifold Gromov-Witten theories of (X_α | D_α) and (N_α | D_α). As applications of these results, we prove an orbifold version of a conjecture of Maulik and Pandharipande (Topology, 2006) on the Gromov-Witten theory of blowups along complete intersections, a conjecture on the Gromov-Witten theory of root constructions and a conjecture on the Leray-Hirsch result for the orbifold Gromov-Witten theory of Tseng and You (J Pure Appl Algebra, 2016).

     

Nonexistence of stable currents

In this paper, we investigate nonexistence of stable integral currents in a compact hypersurface with positive Ricci curvature in a Euclidean spaceR ^m+1 and a vanishing theorem concerning the homology group is obtained.The project was supported by NNSFC, FECC and CPSF     

Virasoro constraints for target curves

We prove generalized Virasoro constraints for the relative Gromov-Witten theories of all nonsingular target curves. Descendents of the even cohomology classes are studied first by localization, degeneration, and completed cycle methods. Descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations. As an outcome of our results, the relative theories of target curves are completely and explicitly determined.     

New recursions for genus-zero Gromov-Witten invariants

New relations among the genus-zero Gromov-Witten invariants of a complex projective manifold X are exhibited. When the cohomology of X is generated by divisor classes and classes “with vanishing one-point invariants,” the relations determine many-point invariants in terms of one-point invariants.     

Vanishing theorems for ach Kähler manifolds and harmonic maps

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L ² cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincaré inequality case and establish a vanishing theorem provided that the weighted function ρ is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincaré inequality.     

A topological view of Gromov-Witten theory

We study relative Gromov-Witten theory via universal relations provided by the degeneration and localization formulas. We find relative Gromov-Witten theory is completely determined by absolute Gromov-Witten theory. The relationship between the relative and absolute theories is guided by a strong analogy to classical topology.As an outcome, we present a mathematical determination of the Gromov-Witten invariants (in all genera) of the Calabi-Yau quintic 3-fold in terms of known theories.     

Courbure sectiounelle et intégrales premières symétriques du flot géodésique: généralisation d'un théorème de S. Bochner

Let (M, g) be a Riemannian manifold. We prove that the space of symmetric tensors invariant under the geodesic flow, is a Lie algebra which contains, as a subalgebra, the Lie algebra of Killing vector fields, and which also contains the space of parallel symmetric tensors as an Abelian subalgebra. Morever, we give a Weitzenböck decomposition of some Laplace—Beltrami operator on symmetric tensors and prove a vanishing theorem which generalizes a theorem due to S. Bochner [2].     

Insensitizing controls with one vanishing component for the Navier–Stokes system

In this paper we prove the existence of insensitizing controls, having one vanishing component, for the local L²$L^2$-norm of the solutions of the Navier–Stokes system. This problem can be recast as a null controllability problem for a nonlinear cascade system. We first prove a controllability result, with controls having one vanishing component, for a linear problem. Then, by means of an inverse mapping theorem, we deduce the controllability for the cascade system.     

Moebius geometry of submanifolds in ? n

In this paper we define a Moebius invariant metric and a Moebius invariant second fundamental form for submanifolds in ? ⁿ and show that in case of a hypersurface with n≥ 4 they determine the hypersurface up to Moebius transformations. Using these Moebius invariants we calculate the first variation of the moebius volume functional. We show that any minimal surface in ? ⁿ is also Moebius minimal and that the image in ? ⁿ of any minimal surface in ℝ ⁿ unter the inverse of a stereographic projection is also Moebius minimal. Finally we use the relations between Moebius invariants to classify all surfaces in ?³ with vanishing Moebius form. Received: 18 November 1997     

An extension of Biran's Lagrangian barrier theorem

We use the Gromov-Witten invariants and a nonsqueezing theorem by the author to affirm a conjecture by P. Biran on the Lagrangian barriers.

     

Gromov-Witten invariants of blow-ups along points and curves

Abstract. In this paper, using the gluing formula of Gromov-Witten invariants under symplectic cutting, due to Li and Ruan, we studied the Gromov-Witten invariants of blow-ups at a smooth point or along a smooth curve. We established some relations between Gromov-Witten invariants of M and its blow-ups at a smooth point or along a smooth curve. Received February 4, 1999     

A Generalization of Takegoshi’s Relative Vanishing Theorem

We present a generalization of Takegoshi’s relative version of the Grauert–Riemenschneider vanishing theorem. Under some natural assumptions, we extend Takegoshi’s vanishing theorem to the case of Nakano semi-positive coherent analytic sheaves on singular complex spaces. We also obtain some new results about proper modifications of torsion-free coherent analytic sheaves.