We prove a generalization to the context of real geometry of an intersection formula for the vanishing cycle functor, which in the complex context is due to Dubson, Ginsburg, Le and Sabbah (after a conjecture of Deligne). It is also a generalization of similar results of Kashiwara-Schapira, where these authors work with a suitable assumption about the micro-support of the corresponding constructible complex of sheaves. We only use a similar assumption about the support of the corresponding characteristic cycle so that our result can be formulated in the language of constructible functions and Lagrangian cycles.
Given a reduced analytic space $Y$ we introduce a class of {\it nice} cycles, including all effective $\mathbb{Q}$-Cartier divisors. Equidimensional nice cycles that intersect properly allow for a natural intersection product. Using $\bar{\partial}$-potentials and residue calculus we provide an intrinsic way of defining this product. The intrinsic definition makes it possible to prove global formulas. In case $Y$ is smooth all cycles are differences of nice cycles, and so we get a new way to define classical proper intersections.
In this article we prove a rigidity theorem for lagrangian singularities by studying the local cohomology of the lagrangian de Rham complex that was introduced in math.AG/0002083. The result can be applied to show the rigidity of all open swallowtails of dimension greater or equal than two. In the case of lagrangian complete intersection singularities the lagrangian de Rham complex turns out to be perverse. We also show that lagrangian complete intersection in dimension greater than two cannot be regular in codimension one.
We study intersections of complex Lagrangian in complex symplectic manifolds, proving two main results. First, we construct global canonical perverse sheaves on complex Lagrangian intersections in complex symplectic manifolds for any pair of {\it oriented} Lagrangian submanifolds in the complex analytic topology. Our method uses classical results in complex symplectic geometry and some results from arXiv:1304.4508. The algebraic version of our result has already been obtained by the author et al. in arXiv:1211.3259 using different methods, where we used, in particular, the recent new theory of…
We give a new way to produce examples of Lagrangians in shifted symplectic derived stacks, based on multiple intersections. Specifically, we show that an m-fold fiber product of Lagrangians in a shifted symplectic derived stack its itself Lagrangian in a certain cyclic product of pairwise homotopy fiber products of the Lagrangians.
In this article we develop intersection theory in terms of the $\mathcal{B}$-group of a reduced analytic space. This group was introduced in a previous work as an analogue of the Chow group; it is generated by currents that are direct images of Chern forms and it contains all usual cycles. However, contrary to Chow classes, the $\mathcal{B}$-classes have well-defined multiplicities at each point. We focus on a $\mathcal{B}$-analogue of the intersection theory based on the Stückrad-Vogel procedure and the join construction in projective space. Our approach provides global $\mathcal{B}$-classes …
For complete intersection Calabi-Yau manifolds in toric varieties, Gross and Haase-Zharkov have given a conjectural combinatorial description of the special Lagrangian torus fibrations whose existence was predicted by Strominger, Yau and Zaslow. We present a geometric version of this construction, generalizing an earlier conjecture of the first author.
We prove several results concerning the intersection cohomology and the perverse filtration associated with a Lagrangian fibration of an irreducible symplectic variety. We first show that the perverse numbers only depend on the deformation equivalence class of the ambient variety. Then we compute the border of the perverse diamond, which further yields a complete description of the intersection cohomology of the Lagrangian base and the invariant cohomology classes of the fibers. Lastly, we identify the perverse and Hodge numbers of intersection cohomology when the irreducible symplectic variet…
We introduce a class of normal complex spaces having only mild sin-gularities (close to quotient singularities) for which we generalize the notion of a (analytic) fundamental class for an analytic cycle and also the notion of a relative fundamental class for an analytic family of cycles. We also generalize to these spaces the geometric intersection theory for analytic cycles with rational positive coefficients and show that it behaves well with respect to analytic families of cycles. We prove that this intersection theory has most of the usual properties of the standard geometric intersection …
Given pure-dimensional (generalized) cycles $μ_1$ and $μ_2$ on a complex manifold $Y$ we introduce a product $μ_1\diamond_{Y} μ_2$ that is a generalized cycle whose multiplicities at each point are the local intersection numbers at the point. % If $Y$ is projective, then given a very ample line bundle $L\to Y$ we define a product $μ_1\bl μ_2$ whose multiplicities at each point also coincide with the local intersection numbers. In addition, provided that $μ_1$ and $μ_2$ are effective, this product satisfies a Bézout inequality. If $i\colon Y\to \Pk^N$ is an embedding such that $i^*\Ok(1)=L$, th…
We establish a close connection between the intersection multiplicity of three arithmetic Hirzebruch-Zagier cycles and the Fourier coefficients of the derivative of a certain Siegel-Eisenstein series at its center of symmetry. Our main result proves a conjecture of Kudla and Rapoport.
For every even number $n$, and every $n$-dimensional smooth hypersurface of $\mathbb{P}^{n+1}$ of degree $d$, we compute the periods of all its $\frac{n}{2}$-dimensional complete intersection algebraic cycles. Furthermore, we determine the image of the given algebraic cycle under the cycle class map inside the De Rham cohomology group of the corresponding hypersurface in terms of its Griffiths basis and the polarization. As an application, we use this information to address variational Hodge conjecture for a non complete intersection algebraic cycle. We prove that the locus of general hypersur…
We define an intersection product of tropical cycles on matroid varieties (via cutting out the diagonal) and show that it is well-behaved. In particular, this enables us to intersect cycles on moduli spaces of tropical rational marked curves $M_{0,n}$ and $M_{0,n}(\R^r, d)$. This intersection product can be extended to smooth varieties (whose local models are matroid varieties). We also study pull-backs of cycles and rational equivalence.
We define a new family of open Gromov-Witten type invariants based on intersection theory on the moduli space of pseudoholomorphic curves of arbitrary genus with boundary in a Lagrangian submanifold. We assume the Lagrangian submanifold arises as the fixed points of an anti-symplectic involution and has dimension 2 or 3. In the strongly semi-positive genus 0 case, the new invariants coincide with Welschinger's invariant counts of real pseudoholomorphic curves. Furthermore, we calculate the new invariant for the real quintic threefold in genus 0 and degree 1 to be 30.
We study the relative Hilbert scheme of a family of nodal (or smooth) curves via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We derive an intersection calculus for Chern classes of tautological bundles on the relative Hilbert scheme, which has applications to enumerative geometry.
We evaluate the intersection numbers of loaded cycles and twisted forms associated with an n-fold Selberg-type integral. The result is deeply related with the geometry of the configuration space of n+3 points in the projective line.
This paper addresses the problem of constructing a cycle-level intersection theory for toric varieties. We show that by making one global choice, we can determine a cycle representative for the intersection of an equivariant Cartier divisor with an invariant cycle on a toric variety. For a toric variety defined by a fan in N, the choice consists of giving an inner product or a complete flag for M_Q=Hom(N,Q), or more generally giving for each cone sigma in the fan a linear subspace of M_Q complementary to the subspace of M_Q perpendicular to sigma, satisfying certain compatibility conditions. W…
We introduce conjectures relating the Chow ring of a smooth Artin stack $\mathcal{X}$ to the Chow groups of its possibly singular good moduli space $X$. In particular, we conjecture the existence of an intersection product on a subgroup of the full Chow group $A^*(X)$ coming from strong cycles on $\mathcal{X}$.
We show that the image of a dominant meromorphic map from an irreducible compact Calabi-Yau manifold $X$ whose general fiber is of dimension strictly between $0$ and $\dim X$ is rationally connected. Using this result, we construct for any hyper-Kähler manifold $X$ admitting a Lagrangian fibration a Lagrangian constant cycle subvariety $Σ_H$ in $X$ which depends on a divisor class $H$ whose restriction to some smooth Lagrangian fiber is ample. If $\dim X = 4$, we also show that up to a scalar multiple, the class of a zero-cycle supported on $Σ_H$ in $\mathrm{CH}_0(X)$ depend neither on $H$ nor…
Given a cycle module M with a ring structure we show that the cycle complex with coefficients in M of a smooth scheme of finite type over a field has a A-infinity algebra structure. In the case of Milnor K-theory this gives a homotopy model for the classical intersection theory of algebraic cycles.
We explain a general construction of double covers of quadratic degeneracy loci and Lagrangian intersection loci based on reflexive sheaves. We relate the double covers of quadratic degeneracy loci to the Stein factorizations of the relative Hilbert schemes of linear spaces of the corresponding quadric fibrations. We give a criterion for these double covers to be nonsingular. As applications of these results, we show that the double covers of the EPW sextics obtained by our construction give O'Grady's double EPW sextics and that an analogous construction gives Iliev-Kapustka-Kapustka-Ranestad'…
We study an analogue of the Noether-Lefschetz theorem in the context of Beilinson's Hodge conjecture. We estimate the dimension of the Noether-Lefschetz locus by the infinitesimal methods in Hodge theory.
We define the notion of antispecial cycles on the Drinfeld upper half plane in analogy to the notion of special cycles defined by Kudla and Rapoport in their Inventiones paper. We determine equations for antispecial cycles and calculate the intersection multiplicity of two antispecial cycles. The result is applied to calculate the intersection multiplicity of certain degenerate Hirzebruch-Zagier cycles. Finally we compare this intersection multiplicity to certain representation densities.
We describe a theory of logarithmic Chow rings and tautological subrings for logarithmically smooth algebraic stacks, via a generalisation of the notion of piecewise-polynomial functions. Using this machinery we prove that the double-double ramification cycle lies in the tautological subring of the (classical) Chow ring of the moduli space of curves, and that the logarithmic double ramification cycle is divisorial (as conjectured by Molcho, Pandharipande, and Schmitt).
On a smooth manifold M, the Kashiwara index formula expresses the weighted Euler characteristic of a constructible function in terms of its characteristic cycle. We generalize this formula to the case when M is a smooth orbifold, answering a question of Behrend.
We establish a close connection between intersection multiplicities of special cycles on arithmetic models of the Shimura variety for GU(1,2) and Fourier coefficients of derivatives of certain incoherent Eisenstein series, confirming a conjecture of Kudla and Rapoport. The paper is divided into two parts. The first part is the bulk of the paper and investigates intersections of local special cycles which are defined on a certain moduli space of p-divisible groups. This moduli space of p-divisible groups can be used for the uniformization of the completion of the Shimura variety along the super…
A smooth intersection $Y$ of two quadrics in $\mathbb{P}^{2g+1}$ has Hodge level 1. We show that such varieties $Y$ have a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.
In this note, we would like to propose a suitable extension of the arithmetic Chow group of codimension one, in which the Hodge index theorem holds. We also prove an arithmetic analogue of Bogomolov's instability theorem for rank 2 vector bundles on arbitrary regular projective arithmetic varieties.
We study the intersection numbers defined on twisted homology or cohomology groups that are associated with hypergeometric integrals corresponding to degenerate hyperplane arrangements in the projective $k$-space. We present formulas to evaluate the intersection numbers in the case when exactly one $(k+1)$-tuple of the hyperplanes intersects at a point. As an application, we discuss the contiguity relations of hypergeometric functions in terms of the intersection numbers on twisted cohomology groups.
This paper generalizes Manin's approach towards a geometrical interpretation of Arakelov theory at infinity to linear cycles on projective spaces. We show how to interpret certain non-Archimedean Arakelov intersection numbers of linear cycles on $P^{n-1}$ with the combinatorial geometry of the Bruhat-Tits building associated to PGL(n). This geometric setting has an Archimedean analogue, namely the Riemannian symmetric space associated to SL(n,C), which we use to interpret analogous Archimedean intersection numbers of linear cycles in a similar way.
We study the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension, via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We work out the action of the blowup or 'discriminant' polarization on some natural cycles in the Hilbert scheme, including generalized diagonals and cycles, called 'node scrolls', parametrizing schemes supported on singular points. We derive an intersection calculus for Chern classes of tautologic…
We give examples of Calabi-Yau threefolds containing Lagrangian spheres which are not vanishing cycles of nodal degenerations, answering a question of Donaldson in the negative.
Let $Y$ be a smooth complete intersection of three quadrics, and assume the dimension of $Y$ is even. We show that $Y$ has a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of (powers of) $Y$ displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow-Künneth decomposition for double planes.
We give an explicit formula for the arithmetic intersection number of CM cycles on Lubin-Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions $K_1,K_2/F$ of non-Archimedean local fields $F$. Our formula works for all cases, $K_1$ and $K_2$ can be either the same or different, ramify or unramified. As applications, this formula translate the linear Arithmetic Fundamental Lemma (linear AFL) into a comparison of integrals. This formula can also be used to recover …
In these notes we survey the tropical intersection theory on R^n by deriving the properties for tropical cycles from the corresponding properties in Chow cohomology. For this we review the stable intersection product introduced by Mikhalkin and the push forward of tropical cycles defined by Allermann and Rau. Furthermore we define a pull back for tropical cycles based on the pull back of Minkowski weights. This pull back commutes with the tropical intersection product and satisfies the projection formula. Our main result is to deduce the latter from the corresponding projection formula in Chow…
Given a symplectic cohomology class of degree 1, we define the notion of an equivariant Lagrangian submanifold. The Floer cohomology of equivariant Lagrangian submanifolds has a natural endomorphism, which induces a grading by generalized eigenspaces. Taking Euler characteristics with respect to the induced grading yields a deformation of the intersection number. Dehn twists act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz fibrations give fully computable examples. A key step in computations is to impose the "dilation" condition stipulating that the BV operator applied…
This paper defines new intersection homology groups. The basic idea is this. Ordinary homology is locally trivial. Intersection homology is not. It may have significant local cycles. A local-global cycle is defined to be a family of such local cycles that is, at the same time, a global cycle. The motivating problem is the numerical characterisation of the flag vectors of convex polytopes. Central is a study of the cycles on a cone and a cylinder, in terms of those on the base. This leads to the topological definition of local-global intersection homology, and a formula for the expected Betti n…
We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the theory of Lagrangian intersection. We also introduce and study extensively a normalized Legendre transformation of Lagrangian subvarieties under a birational transformation of projective hyperkahler manifolds. We give a Plucker type formula for Lagrangian intersections under thi…
We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in R^n and then for "abstract" cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in R^n.
Let $Y$ be a smooth complete intersection of a quadric and a cubic in $\mathbb{P}^n$, with $n$ even. We show that $Y$ has a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of (powers of) $Y$ displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow-Künneth decomposition for the resolution of singularities of a general nodal cubic hypersurface of even dimension.
We consider the problem of estimating the intersection multiplicity between an algebraic variety and a Pfaffian foliation, at every point of the variety. We show that this multiplicity can be majorized at every point $p$ by the local algebraic multiplicity at $p$ of a suitably constructed algebraic cycle. The construction is based on Gabrielov's complex analog of the Rolle-Khovanskii lemma. We illustrate the main result by deriving similar uniform estimates for the complexity of the Milnor fiber of a deformation (under a smoothness assumption) and for the order of contact between an algebraic …
The paper generalizes a result of Behrend-Fantechi and Baranovsky-Ginzburg to the 1-shifted cotangent bundle $T^*X[1]$ of a smooth scheme $X$ over the field of complex numbers. We show how one can obtain twisted cotangent bundles as derived intersections of Lagrangians in $T^*X[1]$, moreover and we show that the derived intersection of the quantized Lagrangians coincide with the canonical quantization of the twisted cotangent bundles.
We investigate the viability of defining an intersection product on algebraic cycles on a singular algebraic variety by pushing forward intersection products formed on a resolution of singularities. For varieties with resolutions having a certain structure (including all varieties over a field of characteristic zero), we obtain a stratification which reflects the geometry of the centers and the exceptional divisors. This stratification is sufficiently fine that divisors can be intersected with $r$-cycles (for $r \geq 1$), and 2-cycles can be intersected on a fourfold, provided their incidences…
In analogy to chapter 9 of arXiv:0709.3705 we define an intersection product of tropical cycles on tropical linear spaces L^n_k, i.e. on tropical fans of the type max{0,x_1,...,x_n}^(n-k)*R^n. Afterwards we use this result to obtain an intersection product of cycles on every smooth tropical variety, i.e. on every tropical variety that arises from gluing such tropical linear spaces. In contrast to classical algebraic geometry these products always yield well-defined cycles, not cycle classes only. Using these intersection products we are able to define the pull-back of a tropical cycle along a …
We define special cycles on arithmetic models of twisted Hilbert-Blumenthal surfaces at primes of good reduction. These are arithmetic versions of these cycles. In particular, we characterize the non-degenerate intersections and partially determine the generating series formed from the intersection numbers of them relating it to the value at the center of symmetry of the derivative of a certain metaplectic Eisenstein series in 6 variables. These results are analogous to those obtained by us in the case of Siegel threefolds (alg-geom/9711025). We also study the case of degenerate intersections …
We define the rigid homology. The trace morphism in rigid cohomology define by duality the cycle class in rigid homology. We verify the compatibility of this classes with rationnal equivalence and intersection theory. We deduce some formal consequences such as the Riemman-Roch-Grothendieck theorem in rigid cohomology and the self-intersection formula.
We consider knot invariants in the context of large $N$ transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicity constructed in the case of algebraic knots. We use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with HOMFLY polynomials. Furthermore, for torus knots, using the explicit construction of the Lagrangian cycle, we also give a direct A-model computation and recover the HOMFLY polynomial for this case.
We analyze the intersection properties of projective planes embedded in generalized Kummer fourfolds, with a view toward classifying the homology classes represented by these submanifolds.
We define the characteristic cycle of a constructible sheaf on a smooth surface in the cotangent bundle. We prove that the intersection number with the 0-section equals the Euler number and that the total dimension of vanishing cycles at an isolated characteristic point is also computed as an intersection number.
A. Mironov proposed a construction of lagrangian submanifolds in $\mathbb{C}^n$ and $\mathbb{C} \mathbb{P}^n$; there he was mostly motivated by the fact that these lagrangian submanifolds (which can have in general self intersections, therefore below we call them lagrangian cycles) present new example of minimal or Hamiltonian minimal lagrangian submanifolds. However the Mironov construction of lagrangian cycles itself can be directly extended to much wider class of compact algrebraic varieties: namely it works in the case when algebraic variety $X$ of complex dimension $n$ admits $T^k$ - acti…
