Davesh Maulik
YOU?
Author Swipe
View article: Gromov-Witten theory, degenerations, and the tautological ring
Gromov-Witten theory, degenerations, and the tautological ring Open
Gromov-Witten (GW) theory produces Chow and cohomology classes on the moduli of curves, and there are several conjectures/speculations about their relation to the tautological ring. We develop new degeneration techniques to address these. …
View article: The intrinsic cohomology ring of the universal compactified Jacobian over the moduli space of stable curves
The intrinsic cohomology ring of the universal compactified Jacobian over the moduli space of stable curves Open
The purpose of this paper is to study the cohomology rings of universal compactified Jacobians. Over the moduli space $\overline{\mathcal{M}}_{g,n}$ of Deligne-Mumford stable marked curves with $n\geq 1$, on the one hand we show that the c…
View article: The period-index problem for hyper-Kähler varieties via hyperholomorphic bundles
The period-index problem for hyper-Kähler varieties via hyperholomorphic bundles Open
We prove new bounds for the period-index problem for hyper-Kähler varieties of $K3^{[n]}$-type using projectively hyperholomorphic bundles constructed by Markman. We show that $\mathrm{dim}(X)$ is a bound for any $X$ of $K3^{[n]}$-type. We…
View article: Arbeitsgemeinschaft: Geometry and Representation Theory around the P=W Conjecture
Arbeitsgemeinschaft: Geometry and Representation Theory around the P=W Conjecture Open
Given a smooth projective curve C , nonabelian Hodge theory gives a diffeomorphism between two different moduli spaces associated to C . The first is the moduli space of Higgs bundles on C of rank n , which is equipped with the structure o…
View article: The D-equivalence conjecture for hyper-Kähler varieties via hyperholomorphic bundles
The D-equivalence conjecture for hyper-Kähler varieties via hyperholomorphic bundles Open
We show that birational hyper-Kähler varieties of $K3^{[n]}$-type are derived equivalent, establishing the D-equivalence conjecture in these cases. The Fourier-Mukai kernels of our derived equivalences are constructed from projectively hyp…
View article: Algebraic cycles and Hitchin systems
Algebraic cycles and Hitchin systems Open
The purpose of this paper is to study motivic aspects of the Hitchin system for $\mathrm{GL}_n$. Our results include the following. (a) We prove the motivic decomposition conjecture of Corti-Hanamura for the Hitchin system; in particular, …
View article: On generalized Beauville decompositions
On generalized Beauville decompositions Open
Motivated by the Beauville decomposition of an abelian scheme and the "Perverse = Chern" phenomenon for a compactified Jacobian fibration, we study in this paper splittings of the perverse filtration for compactified Jacobian fibrations. O…
View article: Logarithmic Donaldson–Thomas theory
Logarithmic Donaldson–Thomas theory Open
Let X be a smooth and projective threefold with a simple normal crossings divisor D. We construct the Donaldson–Thomas theory of the pair $(X|D)$ enumerating ideal sheaves on X relative to D. These moduli spaces are compactified by studyin…
View article: Cohomology of the moduli of Higgs bundles on a curve via positive characteristic
Cohomology of the moduli of Higgs bundles on a curve via positive characteristic Open
For a curve of genus g and any two degrees coprime to the rank, we construct a family of ring isomorphisms parameterized by the complex Lie group \mathrm{GSp}(2g), between the cohomology of the moduli spaces of stable Higgs bundles which p…
View article: Logarithmic enumerative geometry for curves and sheaves
Logarithmic enumerative geometry for curves and sheaves Open
We propose a logarithmic enhancement of the Gromov-Witten/Donaldson-Thomas correspondence, with descendants, and study its behavior under simple normal crossings degenerations. The formulation of the logarithmic correspondence requires a m…
View article: Perverse filtrations and Fourier transforms
Perverse filtrations and Fourier transforms Open
We study the interaction between Fourier-Mukai transforms and perverse filtrations for a certain class of dualizable abelian fibrations. Multiplicativity of the perverse filtration and the "Perverse $\supset$ Chern" phenomenon for these ab…
View article: Cohomological χ–independence for moduli ofone-dimensional sheaves and moduli of Higgs bundles
Cohomological χ–independence for moduli ofone-dimensional sheaves and moduli of Higgs bundles Open
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 th…
View article: Vanishing of Brauer classes on K3 surfaces under reduction
Vanishing of Brauer classes on K3 surfaces under reduction Open
Given a Brauer class on a K3 surface defined over a number field, we prove that there exists infinitely many reductions where the Brauer class vanishes, under certain technical hypotheses, answering a question of Frei--Hassett--Várilly-Alv…
View article: Fourier-Mukai transforms and the decomposition theorem for integrable systems
Fourier-Mukai transforms and the decomposition theorem for integrable systems Open
We study the interplay between the Fourier-Mukai transform and the decomposition theorem for an integrable system $π: M \rightarrow B$. Our main conjecture is that the Fourier-Mukai transform of sheaves of Kähler differentials, after restr…
View article: The $P=W$ conjecture for $\mathrm{GL}_n$
The $P=W$ conjecture for $\mathrm{GL}_n$ Open
We prove the $P=W$ conjecture for $\mathrm{GL}_n$ for all ranks $n$ and curves of arbitrary genus $g\geq 2$. The proof combines a strong perversity result on tautological classes with the curious Hard Lefschetz theorem of Mellit. For the p…
View article: Reductions of abelian surfaces over global function fields
Reductions of abelian surfaces over global function fields Open
Let $A$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a mu…
View article: Cohomology of the moduli of Higgs bundles via positive characteristic
Cohomology of the moduli of Higgs bundles via positive characteristic Open
For any two degrees coprime to the rank, we construct a family of ring isomorphisms parameterized by GSp(2g) between the cohomology of the moduli spaces of stable Higgs bundles which preserve the perverse filtrations. As consequences, we p…
View article: On the intersection cohomology of the moduli of $\mathrm{SL}_n$-Higgs bundles on a curve
On the intersection cohomology of the moduli of $\mathrm{SL}_n$-Higgs bundles on a curve Open
We explore the cohomological structure for the (possibly singular) moduli of $\mathrm{SL}_n$-Higgs bundles for arbitrary degree on a genus g curve with respect to an effective divisor of degree >2g-2. We prove a support theorem for the $\m…
View article: Endoscopic decompositions and the Hausel–Thaddeus conjecture
Endoscopic decompositions and the Hausel–Thaddeus conjecture Open
We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and T…
View article: Endoscopic decompositions and the Hausel–Thaddeus conjecture
Endoscopic decompositions and the Hausel–Thaddeus conjecture Open
We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and T…
View article: Cohomological $χ$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles
Cohomological $χ$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles Open
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 th…
View article: Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture
Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture Open
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 $C$.…
View article: Endoscopic decompositions and the Hausel-Thaddeus conjecture
Endoscopic decompositions and the Hausel-Thaddeus conjecture Open
We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialization, recover the topological mirror symmetry conjecture of Hausel-Thadd…
View article: LEHN’S FORMULA IN CHOW AND CONJECTURES OF BEAUVILLE AND VOISIN
LEHN’S FORMULA IN CHOW AND CONJECTURES OF BEAUVILLE AND VOISIN Open
The Beauville–Voisin conjecture for a hyperkähler manifold $X$ states that the subring of the Chow ring $A^{\ast }(X)$ generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of $X$ . We pro…
View article: Logarithmic Donaldson-Thomas theory
Logarithmic Donaldson-Thomas theory Open
Let $X$ be a smooth threefold with a simple normal crossings divisor $D$. We construct the Donaldson-Thomas theory of the pair $(X|D)$ enumerating ideal sheaves on $X$ relative to $D$. These moduli spaces are compactified by studying subsc…
View article: On the P=W conjecture for $\mathrm{SL}_n$
On the P=W conjecture for $\mathrm{SL}_n$ Open
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, w…