Zero Section Research Articles

On maximal totally real embeddings

Abstract We consider complex structures with totally real zero section of the tangent bundle. We assume that the complex structure tensor is real-analytic along the fibers of the tangent bundle. This assumption is quite natural in view of a well-known result by Bruhat and Whitney. We provide explicit integrability equations for such complex structures in terms of the fiberwise Taylor expansion. In a particular geometric case considered in the literature, we explicit much further the fiberwise Taylor expansion of the complex structure as well as the integrability equations.

Quantum speed limit and categorical energy relative to microlocal projector

Inspired by recent developments of quantum speed limit we introduce a categorical energy of sheaves in the derived category over a manifold relative to a microlocal projector. We utilize the tool of algebraic microlocal analysis to show that with regard to the microsupports of sheaves, our categorical energy gives a lower bound of the Hofer displacement energy. We also prove that on the other hand our categorical energy obeys a relative energy-capacity type inequality. As a by-product this provides a sheaf-theoretic proof of the positivity of the Hofer displacement energy for disjointing the zero section [Formula: see text] from an open subset [Formula: see text] in [Formula: see text], given that [Formula: see text].

Symplectic realizations of holomorphic Poisson manifolds

Symplectic realization is a longstanding problem which can be traced back to Sophus Lie. In this paper, we present an explicit solution to this problem for an arbitrary holomorphic Poisson manifold. More precisely, for any holomorphic Poisson manifold $(X, \pi)$, we prove that there exists a holomorphic symplectic structure in a neighborhood $Y$ of the zero section of $T^*X$ such that the projection map is a symplectic realization of the given Poisson manifold, and moreover the zero section is a holomorphic Lagrangian submanifold. We describe an explicit construction for such a new holomorphic symplectic structure on $Y \subseteq T^*X$.

Linking of Lagrangian Tori and Embedding Obstructions in Symplectic 4-Manifolds

Abstract We classify weakly exact, rational Lagrangian tori in $T^* \mathbb{T}^2- 0_{\mathbb{T}^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on $\mathbb{T}^n$ and also has applications to symplectic topology. As a 1st corollary, we strengthen a result due independently to Eliashberg–Polterovich and to Giroux describing Lagrangian tori in $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$, which are homologous to the zero section. As a 2nd corollary, we exhibit pairs of disjoint totally real tori $K_1, K_2 \subset T^*\mathbb{T}^2$, each of which is isotopic through totally real tori to the zero section, but such that the union $K_1 \cup K_2$ is not even smoothly isotopic to a Lagrangian. In the 2nd part of the paper, we study linking of Lagrangian tori in $({\mathbb{R}}^4, \omega )$ and in rational symplectic $4$-manifolds. We prove that the linking properties of such tori are determined by purely algebro-topological data, which can often be deduced from enumerative disk counts in the monotone case. We also use this result to describe certain Lagrangian embedding obstructions.

Irregular Eguchi–Hanson type metrics and their soliton analogues

We verify the extension to the zero section of momentum construction of Kaehler-Einstein metrics and Kaehler-Ricci solitons on the total space Y of positive rational powers of the canonical line bundle of toric Fano manifolds with possibly irregular Sasaki-Einstein metrics. More precisely, we show that the extended metric along the zero section has an expression which can be extended to Y, restricts to the associated unit circle bundle as a transversely Kaehler-Einstein (Sasakian eta-Einstein) metric scaled in the Reeb flow direction, and that there is a Riemannian submersion from the scaled Sasakian eta-Einstein metric to the induced metric of the zero section.

Transversalité quantitative en géométrie symplectique : sous-variétés et hypersurfaces

The main result of this paper is an extension of a transversality theorem due to Donaldson and Auroux: we construct approximately holomorphic sections whose restriction to some given compact submanifold is quantitatively transverse to the zero section.

An application of Cartan's equivalence method to Hirschowitz's conjecture on the formal principle

A conjecture of Hirschowitz's predicts that a globally generated vector bundle $W$ on a compact complex manifold $A$ satisfies the formal principle, i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle $W$. By applying Cartan's equivalence method to a suitable differential system on the universal family of the Douady space of the complex manifold, we prove that this conjecture is true if $A$ is a Fano manifold, or if the global sections of $W$ separate points of $A$. Our method shows more generally that for any unobstructed compact submanifold $A$ in a complex manifold, if the normal bundle is globally generated and its sections separate points of $A$, then a sufficiently general deformation of $A$ satisfies the formal principle. In particular, a sufficiently general smooth free rational curve on a complex manifold satisfies the formal principle.

Parametrized Borsuk–Ulam theorems for free involutions on [formula omitted

Let X be a finitistic space with mod 2 cohomology algebra isomorphic to that of FPm×S3, where F=R,C or H. Let (X,E,π,B) be a fibre bundle and (Rk,E′,π′,B) be a k-dimensional real vector bundle with fibre preserving G=Z2 action such that G acts freely on E and E′−{0}, where {0} is the zero section of the vector bundle. We determine lower bounds for the cohomological dimension of the zero set f−1({0}) of a fibre preserving G-equivariant map f:E→E′. As an application of this result, we determine a lower bound for the cohomological dimension of the coincidence sets of continuous maps f:X→Rn. In particular, we estimate the size of the coincidence sets of continuous maps f:Si×S3→Rk relative to any free involution on Si×S3, (i=1,2,4).

Chern–Gauss–Bonnet and Lefschetz duality from a currential point of view

We use the mapping cone for the relative deRham cohomology of a manifold with boundary in order to show that the Chern–Gauss–Bonnet Theorem for oriented Riemannian vector bundles over such manifolds is a manifestation of Lefschetz Duality in any of the two embodiments of the latter. We explain how Thom isomorphism fits into this picture, complementing thus the classical results about Thom forms with compact support. When the rank is odd, we construct, by using secondary transgression forms introduced here, a new closed pair of forms on the disk bundle associated to a vector bundle, pair which is Lefschetz dual to the zero section.

A multidimensional Birkhoff theorem for time-dependent Tonelli Hamiltonians

Let M be a closed and connected manifold, $$H:T^*M\times {{\mathbb {R}}}/\mathbb {Z}\rightarrow \mathbb {R}$$ a Tonelli 1-periodic Hamiltonian and $${\mathscr {L}}\subset T^*M$$ a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if $${\mathscr {L}}$$ is invariant by the time-one map of H, then $${\mathscr {L}}$$ is a graph over M. An interesting consequence in the autonomous case is that in this case, $${\mathscr {L}}$$ is invariant by all the time t maps of the Hamiltonian flow of H.

Compactified Universal Jacobian and the Double Ramification Cycle

Using the compactified universal jacobian over the moduli space of stable marked curves, we give an expression in terms of natural classes of the zero section of the compactified universal jacobian the (rational) Chow ring. After extending variants of the Abel-Jacobi map to a locus containing curves of treelike type we give a formula for the pullback of the said zero section along these extensions. The same approach is also applied to recover known formulas for the pullback of theta divisors to the moduli space of marked stable curves.

Virtual signed Euler characteristics

Roughly speaking, to any space $M$ with perfect obstruction theory we associate a space $N$ with symmetric perfect obstruction theory. It is a cone over $M$ given by the dual of the obstruction sheaf of $M$, and contains $M$ as its zero section. It is locally the critical locus of a function. More precisely, in the language of derived algebraic geometry, to any quasi-smooth space $M$ we associate its $(-1)$-shifted cotangent bundle $N$. By localising from $N$ to its $\mathbb C^*$-fixed locus $M$ this gives five notions of virtual signed Euler characteristic of $M$: (1) The Ciocan-Fontanine-Kapranov/Fantechi-G\"ottsche signed virtual Euler characteristic of $M$ defined using its own obstruction theory, (2) Graber-Pandharipande's virtual Atiyah-Bott localisation of the virtual cycle of $N$ to $M$, (3) Behrend's Kai-weighted Euler characteristic localisation of the virtual cycle of $N$ to $M$, (4) Kiem-Li's cosection localisation of the virtual cycle of $N$ to $M$, (5) $(-1)^{vd}$ times by the topological Euler characteristic of $M$. Our main result is that (1)=(2) and (3)=(4)=(5). The first two are deformation invariant while the last three are not.

A note on the symplectic topology of $b$-manifolds

Poisson manifold (M2n; ) is b-symplectic if Vn is transverse to the zero section. In this paper we apply techniques native to Symplectic Topology to address questions pertaining to b-symplectic manifolds. We pro- vide constructions of b-symplectic structures on open manifolds by Gromov's h-principle, and of b-symplectic manifolds with a prescribed singular locus, by means of surgeries.

Equivariant Maps from Stiefel Bundles to Vector Bundles

AbstractLet E → B be a fibre bundle and let Eʹ → B be a vector bundle. Let G be a compact Lie group acting fibre preservingly and freely on both E and Eʹ – 0, where 0 is the zero section of Eʹ → B. Let f : E → Eʹ be a fibre-preserving G-equivariant map and let Zf = {x ∈ E | f(x) = 0} be the zero set of f. In this paper we give a lower bound for the cohomological dimension of the zero set Zf when a fibre of E → B is a real Stiefel manifold with a free ℤ/2-action or a complex Stiefel manifold with a free 𝕊1-action. This generalizes a well-known result of Dold for sphere bundles equipped with free involutions.

On Newton equations which are totally integrable at infinity

In this paper Hamiltonian system of time dependent periodic Newton equations is studied. It is shown that for dimensions 3 and higher the following rigidity results holds true: if all the orbits in a neighborhood of infinity are action minimizing then the potential must be constant. This gives a generalization of the previous result Bialy and Polterovich (Math Res Lett 2(6):695–700, 1995), where it was required all the orbits to be minimal. As a result we have the following application: suppose that for the time-1 map of the Hamiltonian flow there exists a neighborhood of infinity which is filled by invariant Lagrangian tori homologous to the zero section. Then the potential must be constant. Remarkably, the statement is false for \(n=1\) case and remains unknown to the author for \(n=2\).

Zero sets of equivariant maps from products of spheres to Euclidean spaces

Let E→B be a fiber bundle and E′→B be a vector bundle. Let G be a compact group acting fiber preservingly and freely on both E and E′−0, where 0 is the zero section of E′→B. Let f:E→E′ be a fiber preserving G-equivariant map, and let Zf={x∈E | f(x)=0} be the zero set of f. It is an interesting problem to estimate the dimension of the set Zf. In 1988, Dold [5] obtained a lower bound for the cohomological dimension of the zero set Zf when E→B is the sphere bundle associated with a vector bundle which is equipped with the antipodal action of G=Z/2. In this paper, we generalize this result to products of finitely many spheres equipped with the diagonal antipodal action of Z/2. We also prove a Bourgin–Yang type theorem for products of spheres equipped with the diagonal antipodal action of Z/2.

The special linear version of the projective bundle theorem

AbstractA special linear Grassmann variety$\text{SGr}(k,n)$is the complement to the zero section of the determinant of the tautological vector bundle over$\text{Gr}(k,n)$. For an$SL$-oriented representable ring cohomology theory$A^{\ast }(-)$with invertible stable Hopf map${\it\eta}$, including Witt groups and$\text{MSL}_{{\it\eta}}^{\ast ,\ast }$, we have$A^{\ast }(\text{SGr}(2,2n+1))\cong A^{\ast }(pt)[e]/(e^{2n})$, and$A^{\ast }(\text{SGr}(k,n))$is a truncated polynomial algebra over$A^{\ast }(pt)$whenever$k(n-k)$is even. A splitting principle for such theories is established. Using the computations for the special linear Grassmann varieties, we obtain a description of$A^{\ast }(\text{BSL}_{n})$in terms of homogeneous power series in certain characteristic classes of tautological bundles.

Symplectic and Poisson geometry on b-manifolds

Let M2n be a Poisson manifold with Poisson bivector field Π. We say that M is b-Poisson if the map Πn:M→Λ2n(TM) intersects the zero section transversally on a codimension one submanifold Z⊂M. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of (M,Π) in the neighborhood of Z and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology.

Floer theory for negative line bundles via Gromov–Witten invariants

We prove that the GW theory of negative line bundles M=Tot(L→B) determines the symplectic cohomology: indeed SH⁎(M) is the quotient of QH⁎(M) by the kernel of a power of quantum cup product by c1(L). We prove this also for negative vector bundles and the top Chern class.We calculate SH⁎ and QH⁎ for O(−n)→CPm. For example: for O(−1), M is the blow-up of Cm+1 at the origin and SH⁎(M) has rank m.We prove Kodaira vanishing: for very negative L, SH⁎=0; and Serre vanishing: if we twist a complex vector bundle by a large power of L, SH⁎=0.Observe SH⁎(M)=0 iff c1(L) is nilpotent in QH⁎(M). This implies Oancea's result: ωB(π2(B))=0⇒SH⁎(M)=0.We prove the Weinstein conjecture for any contact hypersurface surrounding the zero section of a negative line bundle.For symplectic manifolds X conical at infinity, we build a homomorphism from π1(Hamℓ(X,ω)) to invertibles in SH⁎(X,ω). This is similar to Seidel's representation for closed X, except now they are not invertibles in QH⁎(X,ω).

Chiral four-dimensional F-theory compactifications with SU(5) and multiple U(1)-factors

We develop geometric techniques to determine the spectrum and the chiral indices of matter multiplets for four-dimensional F-theory compactifications on elliptic Calabi-Yau fourfolds with rank two Mordell-Weil group. The general elliptic fiber is the Calabi-Yau onefold in dP_2. We classify its resolved elliptic fibrations over a general base B. The study of singularities of these fibrations leads to explicit matter representations, that we determine both for U(1)xU(1) and SU(5)xU(1)xU(1) constructions. We determine for the first time certain matter curves and surfaces using techniques involving prime ideals. The vertical cohomology ring of these fourfolds is calculated for both cases and general formulas for the Euler numbers are derived. Explicit calculations are presented for a specific base B=P^3. We determine the general G_4-flux that belongs to H^{(2,2)}_V of the resolved Calabi-Yau fourfolds. As a by-product, we derive for the first time all conditions on G_4-flux in general F-theory compactifications with a non-holomorphic zero section. These conditions have to be formulated after a circle reduction in terms of Chern-Simons terms on the 3D Coulomb branch and invoke M-theory/F-theory duality. New Chern-Simons terms are generated by Kaluza-Klein states of the circle compactification. We explicitly perform the relevant field theory computations, that yield non-vanishing results precisely for fourfolds with a non-holomorphic zero section. Taking into account the new Chern-Simons terms, all 4D matter chiralities are determined via 3D M-theory/F-theory duality. We independently check these chiralities using the subset of matter surfaces we determined. The presented techniques are general and do not rely on toric data.