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Davesh Maulik

Publications and source records attributed to Davesh Maulik.

At least 37 records · Page 2Linked to original sources
arXiv2021

Cohomological $χ$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles

We prove that the intersection cohomology (together with the perverse and the Hodge filtrations) for the moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent of the Euler characteristic of the sheaves. We also prove an analogous result for the moduli space of semistable Higgs bundles with respect to an effective divisor $D$ of degree $\mathrm{deg}(D)>2g-2$. Our results confirm the cohomological $χ$-independence conjecture by Bousseau for $\mathbb{P}^2$, and verify Toda's conjecture for Gopakumar-Vafa invariants for certain local curves and local surfaces. For the proof, we combine a generalized version of Ngô's support theorem, a dimension estimate for the stacky Hilbert-Chow morphism, and a splitting theorem for the morphism from the moduli stack to the good GIT quotient.

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arXiv2021

Hitchin fibrations, abelian surfaces, and the P=W conjecture

We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus $2$ curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert-Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman's monodromy operators play a crucial role.

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arXiv2020

Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture

Let $\mathscr{X} \rightarrow C$ be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve $C$ in characteristic $p \geq 5$. We prove that the geometric Picard rank jumps at infinitely many closed points of $C$. More generally, suppose that we are given the canonical model of a Shimura variety $\mathcal{S}$ of orthogonal type, associated to a lattice of signature $(b,2)$ that is self-dual at $p$. We prove that any generically ordinary proper curve $C$ in $\mathcal{S}_{\overline{\mathbb{F}}_p}$ intersects special divisors of $\mathcal{S}_{\overline{\mathbb{F}}_p}$ at infinitely many points. As an application, we prove the ordinary Hecke orbit conjecture of Chai--Oort in this setting; that is, we show that ordinary points in $\mathcal{S}_{\overline{\mathbb{F}}_p}$ have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties.

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arXiv2020

Reductions of abelian surfaces over global function fields

Let $A$ be a non-isotrivial ordinary abelian surface over a global function field with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. We prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves.

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arXiv2020

Lehn's formula in Chow and Conjectures of Beauville and Voisin

The Beauville-Voisin conjecture for a hyperkähler manifold X states that the subring of the Chow ring A^*(X) generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of X. We prove a weak version of this conjecture when X is the Hilbert scheme of points on a K3 surface, for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn's formula and the Li-Qin-Wang W_{1+infinity} algebra action from cohomology to Chow groups, for the Hilbert scheme of an arbitrary smooth projective surface S

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arXiv2020

On the P=W conjecture for $\mathrm{SL}_n$

Let $p$ be a prime number. We prove that the $P=W$ conjecture for $\mathrm{SL}_p$ is equivalent to the $P=W$ conjecture for $\mathrm{GL}_p$. As a consequence, we verify the $P=W$ conjecture for genus 2 and $\mathrm{SL}_p$. For the proof, we compute the perverse filtration and the weight filtration for the variant cohomology associated with the $\mathrm{SL}_p$-Hitchin moduli space and the $\mathrm{SL}_p$-twisted character variety, relying on Gröchenig-Wyss-Ziegler's recent proof of the topological mirror conjecture by Hausel-Thaddeus. Finally we discuss obstructions of studying the cohomology of the $\mathrm{SL}_n$-Hitchin moduli space via compact hyper-Kähler manifolds.

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arXiv2019

Sheaf counting on local K3 surfaces

There are two natural ways to count stable pairs or Joyce-Song pairs on $X=\mathrm{K3}\times\mathbb C$; one via weighted Euler characteristic and the other by virtual localisation of the reduced virtual class. Since $X$ is noncompact these need not be the same. We show their generating series are related by an exponential. As applications we prove two conjectures of Toda, and a conjecture of Tanaka-Thomas defining Vafa-Witten invariants in the semistable case.

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arXiv2019

A note on the cone conjecture for K3 surfaces in positive characteristic

We prove that, for a K3 surface in characteristic p > 2, the automorphism group acts on the nef cone with a rational polyhedral fundamental domain and on the nodal classes with finitely many orbits. As a consequence, for any non-negative integer g, there are only finitely many linear systems of irreducible curves on the surface of arithmetic genus g, up to the action of the automorphism group.

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arXiv2019

Stable pairs and Gopakumar-Vafa type invariants for Calabi-Yau 4-folds

As an analogy to Gopakumar-Vafa conjecture on CY 3-folds, Klemm-Pandharipande defined GV type invariants on CY 4-folds using GW theory and conjectured their integrality. In this paper, we define stable pair type invariants on CY 4-folds and use them to interpret these GV type invariants. Examples are computed for both compact and non-compact CY 4-folds to support our conjectures.

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arXiv2018

Quantum Groups and Quantum Cohomology

In this paper, we study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver Q. Using a geometric R-matrix formalism, we construct a Hopf algebra Y_Q, the Yangian of Q, acting on the cohomology of these varieties, and show several results about their basic structure theory. We prove a formula for quantum multiplication by divisors in terms of this Yangian action. The quantum connection can be identified with the trigonometric Casimir connection for Y_Q; equivalently, the divisor operators correspond to certain elements of Baxter subalgebras of Y_Q. A key role is played by geometric shift operators which can be identified with the quantum KZ difference connection. In the second part, we give an extended example of the general theory for moduli spaces of sheaves on C^2, framed at infinity. Here, the Yangian action is analyzed explicitly in terms of a free field realization; the corresponding R-matrix is closely related to the reflection operator in Liouville field theory. We show that divisor operators generate the quantum ring, which is identified with the full Baxter subalgebras. As a corollary of our construction, we obtain an action of the W-algebra W(gl(r)) on the equivariant cohomology of rank $r$ moduli spaces, which implies certain conjectures of Alday, Gaiotto, and Tachikawa.

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arXiv2018

Gopakumar-Vafa invariants via vanishing cycles

In this paper, we propose an ansatz for defining Gopakumar-Vafa invariants of Calabi-Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal is a modification of a recent approach of Kiem-Li, which is itself based on earlier ideas of Hosono-Saito-Takahashi. We conjecture that these invariants are equivalent to other curve-counting theories such as Gromov-Witten theory and Pandharipande-Thomas theory. Our main theorem is that, for local surfaces, our invariants agree with PT invariants for irreducible one-cycles. We also give a counter-example to the Kiem-Li conjectures, where our invariants match the predicted answer. Finally, we give examples where our invariant matches the expected answer in cases where the cycle is non-reduced, non-planar, or non-primitive.

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arXiv2018

Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds

In analogy with the Gopakumar-Vafa conjecture on CY 3-folds, Klemm and Pandharipande defined GV type invariants on Calabi-Yau 4-folds using Gromov-Witten theory and conjectured their integrality. In this paper, we propose a sheaf-theoretic interpretation of their genus zero invariants using Donaldson-Thomas theory on CY 4-folds. More specifically, we conjecture genus zero GV type invariants are $\mathrm{DT_{4}}$ invariants for one-dimensional stable sheaves on CY 4-folds. Some examples are computed for both compact and non-compact CY 4-folds to support our conjectures. We also propose an equivariant version of the conjectures for local curves and verify them in certain cases.

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arXiv2014

Purity for graded potentials and quantum cluster positivity

Consider a smooth quasiprojective variety X equipped with a C*-action, and a regular function f: X -> C which is C*-equivariant with respect to a positive weight action on the base. We prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of f on proper components of the critical locus of f, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work of Kontsevich-Soibelman, Nagao and Efimov, we use this result to prove the quantum positivity conjecture for cluster mutations for all quivers admitting a positively graded nondegenerate potential. We deduce quantum positivity for all quivers of rank at most 4; quivers with nondegenerate potential admitting a cut; and quivers with potential associated to triangulations of surfaces with marked points and nonempty boundary.

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arXiv2014

Finiteness of K3 surfaces and the Tate conjecture

Given a finite field k of characteristic at least 5, we show that the Tate conjecture holds for K3 surfaces defined over the algebraic closure of k if and only if there are only finitely many K3 surfaces over each finite extension of k.

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arXiv2012

Stable pairs and the HOMFLY polynomial

Given a planar curve singularity, we prove a conjecture of Oblomkov-Shende, relating the geometry of its Hilbert scheme of points to the HOMFLY polynomial of the associated algebraic link. More generally, we prove an extension of this conjecture, due to Diaconescu-Hua-Soibelman, relating stable pair invariants on the conifold to the colored HOMFLY polynomial of the algebraic link. Our proof uses wall-crossing techniques to prove a blowup identity on the algebro-geometric side. We prove a matching identity for the colored HOMFLY polynomials of a link using skein-theoretic techniques.

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arXiv2012

Constructible functions and Lagrangian cycles on orbifolds

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.

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arXiv2012

Supersingular K3 surfaces for large primes

Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height) then its Picard rank is 22. Along with work of Nygaard-Ogus, this conjecture implies the Tate conjecture for K3 surfaces over finite fields with p \geq 5. We prove Artin's conjecture under the additional assumption that X has a polarization of degree 2d with p > 2d+4. Assuming semistable reduction for surfaces in characteristic p, we can improve the main result to K3 surfaces which admit a polarization of degree prime-to-p when p \geq 5. The argument uses Borcherds' construction of automorphic forms on O(2,n) to construct ample divisors on the moduli space. We also establish finite-characteristic versions of the positivity of the Hodge bundle and the Kulikov-Pinkham-Persson classification of K3 degenerations. In the appendix by A. Snowden, a compatibility statement is proven between Clifford constructions and integral p-adic comparison functors.

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