The Gromov-Witten invariants of a smooth, projective variety $V$, when twisted by the tautological classes on the moduli space of stable maps, give rise to a family of cohomological field theories and endow the base of the family with coordinates. We prove that the potential functions associated to the tautological $ψ$ classes (the large phase space) and the $κ$ classes are related by a change of coordinates which generalizes a change of basis on the ring of symmetric functions. Our result is a generalization of the work of Manin--Zograf who studied the case where $V$ is a point. We utilize this change of variables to derive the topological recursion relations associated to the $κ$ classes from those associated to the $ψ$ classes.
We prove some base change theorems for coherent cohomology in the setting of Berkovich spaces. In this setting, we get a flat base change theorem, and some proper base change theorems that are analogue to theorems from scheme theory.
We compute the mixed Hodge structure on the cohomology ring of complements of complex coordinate subspace arrangements. The mixed Hodge structure can be described in terms of the special bigrading on the cohomology ring of complements of complex coordinate subspace arrangements. Originally this bigrading was introduced in the setting of toric topology by V.M. Buchstaber and T.E. Panov.
Using the homogeneous coordinate ring construction of a toric variety IP defined by a complete simplicial fan and the methods of local cohomology theory we develop a framework for the calculation of cohomology groups H^{*}(IP, O(p)) of twisting sheaves O(p) on IP.
We prove that cohomology and base change holds for algebraic stacks, generalizing work of Brochard in the tame case. We also show that Hom-spaces on algebraic stacks are represented by abelian cones, generalizing results of Grothendieck, Brochard, Olsson, Lieblich, and Roth--Starr. To accomplish all of this, we prove that a wide class of Ext-functors in algebraic geometry are coherent (in the sense of M. Auslander).
We describe the standard and Leray filtrations on the cohomology groups with compact supports of a quasi projective variety with coefficients in a constructible complex using flags of hyperplane sections on a partial compactification of a related variety. One of the key ingredients of the proof is the Lefschetz hyperplane theorem for perverse sheaves and, in an appendix, we discuss the base change maps for constructible sheaves on algebraic varieties and their role in a proof, due to Beilinson, of the Lefschetz hyperplane theorem.
We consider the conormal bundle of a Schubert variety $S_I$ in the cotangent bundle $T^* Gr$ of the Grassmannian $Gr$ of $k$-planes in $C^n$. This conormal bundle has a fundamental class ${κ_I}$ in the equivariant cohomology $H^*_{T}(T^* Gr)$. Here $T=(C^*)^n\times C^*$. The torus $(C^*)^n$ acts on $T^* Gr$ in the standard way and the last factor $C^*$ acts by multiplication on fibers of the bundle. We express this fundamental class as a sum $Y_I$ of the Yangian $Y(gl_2)$ weight functions $(W_J)_J$. We describe a relation of $Y_I$ with the double Schur polynomial $[S_I]$. A modified version of…
We give a proof of a result of D. Peterson's identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of $GL_n$. The totally positive part of this subvariety is then constructed and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of Vafa and Intriligator and Bertram for the structure constants (Gromov--Witten invariants). Finally, we use the positivity of these Gromov--Witten invariants to prove certain inequalities f…
We compute the Hodge filtration on cohomology groups of complements of complex coordinate subspace arrangements. By means of this result we construct integral representations of holomorphic functions such that kernels of these representations have singularities on complex coordinate subspace arrangements.
In this paper, we explicitly derive the generalized mirror transformation of quantum cohomology of general type projective hypersurfaces, proposed in our previous article, as an effect of coordinate change of the virtual Gauss-Manin system.
The paper is dedicated to the study of algebraic manifolds whose quantum cohomology or a part of it is a semisimple Frobenius manifold. Theorem 1.8.1 says, roughly speaking, that the sum of $(p,p)$--cohomology spaces is a maximal Frobenius submanifold that has chances to be semisimple. Theorem 1.8.3 provides a version of the Reconstruction theorem, assuming semisimplicity but not $H^2$--generation. Theorem 3.6.1 establishes the semisimplicity for all del Pezzo surfaces, providing an evidence for the conjecture that semisimplicity is related to the existence of a full system of exceptional shea…
Let $X \overset{f}\longrightarrow S$ be a morphism of Noetherian schemes, with $S$ reduced. For any closed subscheme $Z$ of $X$ finite over $S$, let $j$ denote the open immersion $X\setminus Z \hookrightarrow X$. Kollár asked whether for any coherent sheaf $\mathcal F$ on $X\setminus Z$ and any index $r\geq 1$, the sheaf $f_*(R^rj_*\mathcal F)$ is generically free on $S$ and commutes with base change. We answer this affirmatively, by proving a related statement about local cohomology: Let $ R$ be Noetherian algebra over a Noetherian domain $A$, and let $I \subset R$ be an ideal such that $ R/I…
We show that the cohomology algebra of the complement of a coordinate subspace arrangement in m-dimensional complex space is isomorphic to the cohomology algebra of Stanley-Reisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) Then we calculate the latter cohomology algebra by means of the standard Koszul resolution of polynomial ring. To prove these facts we construct an equivariant with respect to the torus action homotopy equivalence between the complement of a coordinate subspace arrangement and th…
The Hikita conjecture relates the coordinate ring of a conical symplectic singularity to the cohomology ring of a symplectic resolution of the dual conical symplectic singularity. We formulate a quantum version of this conjecture, which relates the quantized coordinate ring of the first variety to the quantum cohomology of a symplectic resolution of the dual variety. We prove this conjecture for hypertoric varieties and for the Springer resolution. Our paper includes an appendix, written by Ben Webster, which studies highest weights for quantizations of symplectic resolutions with isolated tor…
An old result of the first author and David Lieberman says that if a compact Kaehler manifold X admits a holomorphic vector field V having at least one zero, then the Dolbeault cohomology algebra H^*(X, Ω^*) of X is isomorphic with the bigraded algebra associated to a certain filtered graded ring. If V has isolated zeros this is the graded algebra associated to a certain filtration of the coordinate ring of the scheme Z defined by zero(V). The purpose of this note is to show that if V is generated by a torus action then equivariant Dolbeault cohomology can be used to give direct geometric proo…
We study $A_{\infty}$-structures extending the natural algebra structure on the cohomology of $\oplus_n L^n$, where $L$ is a very ample line bundle on a projective $d$-dimensional variety $X$ such that $H^i(X,L^n)=0$ for $0<i<d$ and all $n$. We prove that there exists a unique such nontrivial $A_{\infty}$-structure up to homotopy and rescaling. In the case when $X$ is a curve we also compute the group of self-homotopies of this $A_{\infty}$-structure.
The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of Small Quantum Cohomology of complex Grassmannians is studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann Hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequenc…
We show that crystals with the properties of crystalline cohomology of ordinary Calabi-Yau threefolds in characteristic p>0, exhibit a remarkable similarity with the well known structure on the cohomology of complex Calabi-Yau threefolds near a boundary point of the moduli space with maximal unipotent local monodromy. In particular, there are canonical coordinates and an analogue of the prepotential of the Yukawa coupling. Moreover we show p-adic analogues of the integrality properties for the canonical coordinates and the prepotential of the Yukawa coupling, which have been observed in the ex…
We prove the finiteness and compatibility with base change of the (phi,Gamma)-cohomology and the Iwasawa cohomology of arithmetic families of (phi,Gamma)-modules. Using this finiteness theorem, we show that a family of Galois representations that is densely pointwise refined in the sense of Mazur is actually trianguline as a family over a large subspace. In the case of the Coleman-Mazur eigencurve, we determine the behavior at all points.
We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fields and $\mathcal{N} = (0, 1)$ supersymmetry. We also consider a theory of free fermions valued in a representation whose partition function is a section of a determinant line bundle. We identify this section with a cocycle representative of the (twisted) equivariant elliptic Euler class of the representation. Finally, we show that the moduli stack of $U(1)$-g…
We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace. Part of the proof uses the fact that subvarieties of positive codimension are stably displaceable.
Let Z_3 act on C^2 by non-trivial opposite characters. Let X =[C^2/Z_3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 Gromov-Witten potentials of X and Y are equal after a change of variables -- verifying the Crepant Resolution Conjecture for the pair (X,Y). Our computations involve Hodge integrals on trigonal Hurwitz spaces which are of independent interest. In a self contained Appendix, we derive closed formulas for these Hurwitz-Hodge integrals.
Given a flag variety $Fl(n;r_1, \dots , r_ρ)$, there is natural ring morphism from the symmetric polynomial ring in $r_1$ variables to the quantum cohomology of the flag variety. In this paper, we show that for a large class of partitions $λ$, the image of $s_λ$ under the ring homomorphism is a Schubert class which is described by partitioning $λ$ into a quantum hook (or $q$-hook) and a tuple of smaller partitions. We use this result to show that the Plücker coordinate mirror of the flag variety describes quantum cohomology relations. This gives new insight into the structure of this superpote…
We use group cohomology and the de Rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2-manifolds. These generators are constructed using the de Rham representatives for the cohomology of classifying spaces $BK$ where $K$ is a compact Lie group; such representatives (universal characteristic classes) were found by Bott and Shulman. Thus our representatives for the generators of the cohomology of moduli spaces are given explicitly in terms of the Maurer-Cartan form.…
The Seidel maps are two maps associated to a Hamiltonian circle action on a convex symplectic manifold, one on Floer cohomology and one on quantum cohomology. We extend their definitions to $S^1$-equivariant Floer cohomology and $S^1$-equivariant quantum cohomology based on a construction of Maulik and Okounkov. The $S^1$-action used to construct $S^1$-equivariant Floer cohomology changes after applying the equivariant Seidel map (a similar phenomenon occurs for $S^1$-equivariant quantum cohomology). We show the equivariant Seidel map on $S^1$-equivariant quantum cohomology does not commute wi…
Let $S$ be the coordinate ring of the space of $n\times n$ complex skew-symmetric matrices. This ring has an action of the group $\textrm{GL}_n(\mathbb{C})$ induced by the action on the space of matrices. For every invariant ideal $I\subseteq S$, we provide an explicit description of the modules $\textrm{Ext}^{\bullet}_S(S/I,S)$ in terms of irreducible representations. This allows us to give formulas for the regularity of basic invariant ideals and (symbolic) powers of ideals of Pfaffians, as well as to characterize when these ideals have a linear free resolution. In addition, given an inclusi…
We develop methods for computing the restriction map from the cohomology of the automorphism group of a height $dn$ formal group law (i.e., the height $dn$ Morava stabilizer group) to the cohomology of the automorphism group of an $A$-height $n$ formal $A$-module, where $A$ is the ring of integers in a degree $d$ field extension of $\mathbb{Q}_p$. We then compute this map for the quadratic extensions of $\mathbb{Q}_p$ and the height $2$ Morava stabilizer group at primes $p>3$. We show that the these automorphism groups of formal modules are closed subgroups of the Morava stabilizer groups, and…
