arXiv subjects

Filip Živanović

Publications and source records attributed to Filip Živanović.

8 recordsLinked to original sources

arXiv2026

Perverse filtration on Hilbert schemes via upward flow

We explicitly compute the perverse Leray filtration on the top cohomology of the Hilbert scheme of points on $Σ\times\mathbb{C}$, for any connected smooth projective curve $Σ$. The computation is carried out in the natural basis given by the $\mathbb{C}^*$-upward-flow cycles. The result is described by a simple symmetric-function dictionary: upward-flow classes correspond to products of complete homogeneous symmetric functions, while the perverse-homogeneous basis corresponds to products of power-sum symmetric functions. This gives an explicit triangular change-of-basis between the two bases.

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arXiv2026

Equivariant multiplicities and mirror symmetry for Hilbert schemes

Following Hausel-Hitchin, we investigate core Lagrangians and upward flows in Hilbert schemes of points on elliptic surfaces. We compute the scheme-theoretic multiplicities of core Lagrangians, as well as the equivariant multiplicities of the very stable ones. Furthermore, we extend the notion of equivariant multiplicity to wobbly components and compute it for Hilbert schemes of two points. Inspired by Eisenstein series functor in Dolbeault Langlands correspondence, we propose that upward flows of very stable ideals are mirror dual to modified Procesi bundles, and justify this claim through numerical checks. Finally, we make some conjectures about extending this picture to wobbly upward flows.

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arXiv2025

Filtrations on quantum cohomology from the Floer theory of $\mathbb{C}^*$-actions

We construct a filtration by ideals on quantum cohomology for symplectic manifolds with a Hamiltonian $S^1$-action that extends to a pseudoholomorphic $\mathbb{C}^*$-action. These spaces include all Conical Symplectic Resolutions, in particular all Quiver Varieties. In particular, we obtain a family of filtrations on singular cohomology for any Conical Symplectic Resolution, that is sensitive to the choice of $\mathbb{C}^*$-action. The symplectic form is rarely exact at infinity for these spaces, so substantial foundational work is carried out to rigorously define Floer theory, in particular symplectic cohomology. Using Floer theory, we construct a periodic persistence module, giving rise to a graded periodic barcode associated to the $\mathbb{C}^*$-action. This encodes birth-death phenomena of Floer invariants. Our filtrations can be viewed as a Floer-theoretic analogue of Atiyah-Bott filtrations, arising from stratifying a manifold by gradient flowlines of a Morse-Bott function, but they are distinct from those and they can detect non-topological properties of the quantum product.

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arXiv2025

Filtrations on quantum cohomology via Morse-Bott-Floer Spectral Sequences

Using Morse-Bott-Floer spectral sequences, we describe a filtration by ideals on quantum cohomology for symplectic manifolds with a Hamiltonian $S^1$-action that extends to a pseudoholomorphic $\mathbb{C}^*$-action. These spaces include all Conical Symplectic Resolutions, in particular all Quiver Varieties. Our spectral sequences give explicit descriptions of birth-death phenomena of the barcode of the persistence module associated to the $\mathbb{C}^*$-action. This paper contains the foundational work to rigorously construct a filtration on Floer complexes from the $\mathbb{C}^*$-action, announced in our earlier paper. A substantial appendix on Morse-Bott-Floer theory deals with several of the technical difficulties of the paper. We compute a plethora of explicit examples, each highlighting various features, for Springer resolutions, ADE resolutions, and several Slodowy varieties of type A. We also consider certain Higgs moduli spaces, for which we compare our filtration with the famous P=W filtration.

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arXiv2025

Floer-theoretic filtration on Painlevé Hitchin systems

We classify equivariant $\mathbb{C}^*$-actions on moduli spaces of Higgs bundles corresponding to the Painlevé equations. Using this, we compute the Floer-theoretic filtrations on the cohomology of these spaces, introduced by Ritter and the second author in arXiv:2304.13026. We compare it with the ``$P=W$'' and the filtration obtained by multiplicities of the irreducible components of the nilpotent cone, ultimately deducing that the Floer-theoretic filtration coincides with the multiplicity filtration, for all 2-dimensional Higgs moduli.

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arXiv2025

Quantum cohomology and Floer invariants of semiprojective toric manifolds

We use Floer theory to describe invariants of symplectic $\mathbb{C}^*$-manifolds admitting several commuting $\mathbb{C}^*$-actions. The $\mathbb{C}^*$-actions induce filtrations by ideals on quantum cohomology, as well as filtrations on Hamiltonian Floer cohomologies, and we prove relationships between these filtrations. We also carry this out in the equivariant setting, in particular $\mathbb{C}^*$-actions then give rise to Hilbert-Poincaré polynomials on ordinary cohomology that depend on Floer theory. For semiprojective toric manifolds, we obtain an explicit presentation for quantum and symplectic cohomology in the Fano and CY setting, both in the equivariant and non-equivariant setting.

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arXiv2024

Filtrations on equivariant quantum cohomology and Hilbert-Poincaré series

We prove that Floer theory induces a filtration by ideals on equivariant quantum cohomology of symplectic manifolds equipped with a $\mathbb{C}^*$-action. In particular, this gives rise to Hilbert-Poincaré polynomials on ordinary cohomology that depend on Floer theory. En route, the paper develops structural properties of filtrations on three versions of equivariant Floer cohomology. We obtain an explicit presentation for equivariant symplectic cohomology in the Calabi-Yau and Fano settings.

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arXiv2022

Exact Lagrangians from contracting $\mathbb{C}^*$-actions

We obtain families of non-isotopic closed exact Lagrangian submanifolds in quasi-projective holomorphic symplectic manifolds that admit contracting $\mathbb{C}^*$-actions. We show that the Floer cohomologies of these Lagrangians are topological in nature, recovering the ordinary cohomologies of their intersection. Moreover, by using these Lagrangians and a version of Carrell-Goresky's integral decomposition theorem, we obtain degree-wise lower bounds on the symplectic cohomology of these spaces.

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