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Vivek Shende

Publications and source records attributed to Vivek Shende.

At least 55 records · Page 3Linked to original sources
arXiv2021

Colored HOMFLYPT counts holomorphic curves

We compute the contribution of all multiple covers of an isolated rigid embedded holomorphic annulus, stretching between Lagrangians, to the skein-valued count of open holomorphic curves in a Calabi-Yau 3-fold. The result agrees with the predictions from topological string theory and we use it to prove the Ooguri-Vafa formula that identifies the colored HOMFLYPT invariants of a link with a count of holomorphic curves ending on the conormal Lagrangian of the link in the resolved conifold. This generalizes our previous work which proved the result for the fundamental color.

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arXiv2020

Skein recursion for holomorphic curves and invariants of the unknot

We determine the skein-valued Gromov-Witten partition function for a single toric Lagrangian brane in $\mathbb{C}^3$ or the resolved conifold. We first show geometrically they must satisfy a certain skein-theoretic recursion, and then solve this equation. The recursion is a skein-valued quantization of the equation of the mirror curve. The solution is the expected hook-content formula.

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arXiv2019

Augmentations are Sheaves

We show that the set of augmentations of the Chekanov-Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC], but is related to it in the same way that cohomology is related to compactly supported cohomology. The existence of such a category was predicted by [STZ], who moreover conjectured its equivalence to a category of sheaves on the front plane with singular support meeting infinity in the knot. After showing that the augmentation category forms a sheaf over the x-line, we are able to prove this conjecture by calculating both categories on thin slices of the front plane. In particular, we conclude that every augmentation comes from geometry.

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arXiv2019

Covariantly functorial wrapped Floer theory on Liouville sectors

We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the open-closed map for Liouville sectors, and we show that these invariants are covariantly functorial with respect to inclusions of Liouville sectors. From this foundational setup, a local-to-global principle for Abouzaid's generation criterion follows.

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arXiv2018

Arboreal singularities from Lefschetz fibrations

Nadler introduced certain Lagrangian singularities indexed by trees, and determined their microlocal sheaves to be the category of modules over the corresponding tree quiver. Another family of spaces indexed by trees: the tree plumbings of spheres. The Fukaya-Seidel category of the Lefschetz fibration with this plumbing as fiber and all spheres as vanishing cycles is well known to also be modules over the tree quiver. Here we upgrade this matching of categories to a matching of geometry.

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arXiv2018

Generating families and constructible sheaves

Let $Λ$ be a Legendrian in the jet space of some manifold $X$. To a generating family presentation of $Λ$, we associate a constructible sheaf on $X \times \mathbb{R}$ whose singular support at infinity is $Λ$, and such that the generating family homology is canonically isomorphic to the endomorphism algebra of this sheaf. That is, the theory of generating family homology embeds in sheaf theory, and more specifically in the category studied in [STZ]. When $X = \mathbb{R}$, i.e., for the theory of Legendrian knots and links in the standard contact $\mathbb{R}^3$, we use ideas from the proof of the h-cobordism theorem to show this embedding is an equivalence. Combined with the results of [NRSSZ], this implies in particular that the generating family homologies of a knot are the same as its linearized Legendrian contact homologies.

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arXiv2018

A support theorem for Hilbert schemes of planar curves, II

We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration.

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arXiv2018

Cluster varieties from Legendrian knots

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.

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arXiv2017

A complete knot invariant from contact homology

We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphic-curve proof of a result of the third author that the Legendrian isotopy class of the conormal torus is a complete knot invariant.

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arXiv2017

Microlocal category for Weinstein manifolds via h-principle

On a Weinstein manifold, we define a constructible co/sheaf of categories on the skeleton. The construction works with arbitrary coefficients, and depends only on the homotopy class of a section of the Lagrangian Grassmannian of the stable symplectic normal bundle. The definition is as follows. Take any, possibly high codimension, exact embedding into a cosphere bundle. Thicken to a hypersurface, and consider the Kashiwara-Schapira stack along the thickened skeleton. Pull back along the inclusion of the original skeleton. Gromov's h-principle for contact embeddings guarantees existence and uniqueness up to isotopy of such an embedding. Invariance of microlocal sheaves along such isotopy is well known. We expect, but do not prove here, invariance of the global sections of this co/sheaf of categories under Liouville deformation.

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arXiv2016

Legendrian knots and constructible sheaves

We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of the Chekanov-Eliashberg differential graded algebra. We also find two connections to topological knot theory. First, drawing a positive braid closure on the annulus, the moduli space of rank-n objects maps to the space of local systems on a circle. The second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the (colored-by-n) triply-graded Khovanov-Rozansky homology. Second, drawing a positive braid closure in the plane, the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in 'a' of the HOMFLY polynomial of the braid closure.

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arXiv2016

Higher discriminants and the topology of algebraic maps

We show that the way in which Betti cohomology varies in a proper family of complex algebraic varieties is controlled by certain "higher discriminants" in the base. These discriminants are defined in terms of transversality conditions, which in the case of a morphism between smooth varieties can be checked by a tangent space calculation. They control the variation of cohomology in the following two senses: (1) the support of any summand of the pushforward of the IC sheaf along a projective map is a component of a higher discriminant, and (2) any component of the characteristic cycle of the proper pushforward of the constant function is a conormal variety to a component of a higher discriminant. The same would hold for the Whitney stratification of the family, but there are vastly fewer higher discriminants than Whitney strata. For example, in the case of the Hitchin fibration, the stratification by higher discriminants gives exactly the δ stratification introduced by Ngo.

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arXiv2016

On the combinatorics of exact Lagrangian surfaces

We study Weinstein 4-manifolds which admit Lagrangian skeleta given by attaching disks to a surface along a collection of simple closed curves. In terms of the curves describing one such skeleton, we describe surgeries that preserve the ambient Weinstein manifold, but change the skeleton. The surgeries can be iterated to produce more such skeleta --- in many cases, infinitely many more. Each skeleton is built around a Lagrangian surface. Passing to the Fukaya category, the skeletal surgeries induce cluster transformations on the spaces of rank one local systems on these surfaces, and noncommutative analogues of cluster transformations on the spaces of higher rank local systems. In particular, the problem of producing and distinguishing such Lagrangians maps to a combination of combinatorial-geometric questions about curve configurations on surfaces and algebraic questions about exchange graphs of cluster algebras. Conversely, this expands the dictionary relating the cluster theory of character varieties, positroid strata, and related spaces to the symplectic geometry of Lagrangian fillings of Legendrian knots, by incorporating cluster charts more general than those associated to bicolored surface graphs.

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arXiv2015

The cardinality of the augmentation category of a Legendrian link

We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R^3. This `homotopy cardinality' is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We present an application to the augmentation category of doubly Lagrangian slice knots.

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arXiv2015

The chi-y genera of relative Hilbert schemes for linear systems on Abelian and K3 surfaces

For an ample line bundle on an Abelian or K3 surface, minimal with respect to the polarization, the relative Hilbert scheme of points on the complete linear system is known to be smooth. We give an explicit expression in quasi-Jacobi forms for the chi-y genus of the restriction of the Hilbert scheme to a general linear subsystem. This generalizes a result of Yoshioka and Kawai for the complete linear system on the K3 surface, a result of Maulik, Pandharipande, and Thomas on the Euler characteristics of linear subsystems on the K3 surface, and a conjecture of the authors.

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arXiv2015

Refined curve counting on complex surfaces

We define refined invariants which "count" nodal curves in sufficiently ample linear systems on surfaces, conjecture that their generating function is multiplicative, and conjecture explicit formulas in the case of K3 and abelian surfaces. We also give a refinement of the Caporaso-Harris recursion, and conjecture that it produces the same invariants in the sufficiently ample setting. The refined recursion specializes at y = -1 to the Itenberg-Kharlamov-Shustin recursion for Welschinger invariants. We find similar interactions between refined invariants of individual curves and real invariants of their versal families.

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