arXiv subjects

Artan Sheshmani

Publications and source records attributed to Artan Sheshmani.

58 records · Page 4Linked to original sources

arXiv2015

Intersection numbers on the relative Hilbert schemes of points on surfaces

We study certain top intersection products on the Hilbert scheme of points on a nonsingular surface relative to an effective smooth divisor. We find a formula relating these numbers to the corresponding intersection numbers on the non-relative Hilbert schemes. In particular, we obtain a relative version of the explicit formula found by Carlsson-Okounkov for the Euler class of the twisted tangent bundle of the Hilbert schemes.

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arXiv2015

Counting curves on surfaces in Calabi-Yau 3-folds

Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of 2-dimensional torsion sheaves, enumerating pairs $Z\subset H$ in a Calabi-Yau threefold X. Here H is a member of a sufficiently positive linear system and Z is a 1-dimensional subscheme of it. The associated sheaf is the ideal sheaf of $Z\subset H$, pushed forward to X and considered as a certain Joyce-Song pair in the derived category of X. We express these invariants in terms of the MNOP invariants of X.

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arXiv2013

Invariants of pure 2-dimensional sheaves inside threefolds and modular forms

Motivated by S-duality modularity conjectures in string theory, we study the Donaldson-Thomas type invariants of pure 2-dimensional sheaves inside a nonsingular threefold X in three different situations: (1). X is a K3 fibration over a curve. We study the Donaldson-Thomas invariants of the 2 dimensional Gieseker stable sheaves in X supported on the fibers. Analogous to the Gromov-Witten theory formula established in the work of Maulik-Pandharipande, we express these invariants in terms of the Euler characteristic of the Hilbert scheme of points on the K3 surface and the Noether-Lefschetz numbers of the fibration, and prove that the invariants have modular properties. (2). X is the total space of the canonical bundle of P^2. We study the generalized Donaldson-Thomas invariants defined by Joyce-Song of the moduli spaces of the 2-dimensional Gieseker semistable sheaves on X with first Chern class equal to k times the class of the zero section of X. When k=1,2 or 3, and semistability implies stability, we express the invariants in terms of known modular forms. We prove a combinatorial formula for the invariants when k=2 in the presence of the strictly semistable sheaves, and verify the BPS integrality conjecture of Joyce-Song in some cases. (3). (Joint with Richard Thomas) X is a Calabi-Yau threefold and L is a sufficiently positive line bundle. We define new invariants counting a restricted class of 2-dimensional torsion sheaves, enumerating pairs (Z,H) in X where H is a member of the linear system |L| and Z is a 1-dimensional subscheme of H. The associated sheaf is the ideal sheaf of Z in H, pushed forward to X and considered as a certain Joyce-Song pair in the derived category of X. We express these invariants in terms of the MNOP invariants of X.

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arXiv2012

An introduction to the theory of Higher rank stable pairs and Virtual localization

This article is an attempt to briefly introduce some of the results from arXiv:1011.6342 on development of a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold X. More precisely, we develop a moduli theory for highly frozen triples given by the data O^r-->F for r>1 where F is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory. Instead, we build a zero-dimensional virtual fundamental class by truncating a deformation-obstruction theory coming from the moduli of objects in the derived category of X. We briefly include the results of calculations in this enumerative theory for local P^1 using the Graber-Pandharipande virtual localization technique. We emphasize that in this article we merely include the statement of our theorems and illustrate the final result of some of the computations. The proofs and more detailed calculations in arXiv:1011.6342 will appear elsewhere.

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