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Yukinobu Toda

Publications and source records attributed to Yukinobu Toda.

At least 55 records · Page 3Linked to original sources
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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arXiv2017

Stable pairs on nodal K3 fibrations

We study Pandharipande-Thomas's stable pair theory on $K3$ fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka's formula for the Euler characteristics of moduli spaces of stable pairs on $K3$ surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the $K3$ fibration. In the case that the $K3$ fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.

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arXiv2016

Hall algebras in the derived category and higher rank DT invariants

We remark that the combination of the works of Ben-Bassat-Brav-Bussi-Joyce and Alper-Hall-Rydh imply the conjectured local description of the moduli stacks of semi-Schur objects in the derived category of coherent sheaves on projective Calabi-Yau 3-folds. This result was assumed in the author's previous papers to apply wall-crossing formulas of DT type invariants in the derived category, e.g. DT/PT correspondence, rationality, etc. We also show that the above result is applied to prove the higher rank version of DT/PT correspondence and rationality.

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arXiv2016

Flops and the S-duality conjecture

We prove the transformation formula of Donaldson-Thomas (DT) invariants counting two dimensional torsion sheaves on Calabi-Yau 3-folds under flops. The error term is described by the Dedekind eta function and the Jacobi theta function, and our result gives evidence of a 3-fold version of Vafa-Witten's S-duality conjecture. As an application, we prove a blow-up formula of DT type invariants on the total spaces of canonical line bundles on smooth projective surfaces. It gives an analogue of the similar blow-up formula in the original S-duality conjecture by Yoshioka, Li-Qin and Göttsche.

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arXiv2016

Moduli of Bridgeland semistable objects on 3-folds and Donaldson-Thomas invariants

We show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are proper algebraic stacks of finite type, if they satisfy the Bogomolov-Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson-Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi-Yau 3-folds satisfying the BG inequality conjecture, for example on étale quotients of abelian 3-folds.

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arXiv2016

Contraction algebra and invariants of singularities

In [7], Donovan and Wemyss introduced the contraction algebra of flop- ping curves in 3-folds. When the flopping curve is smooth and irreducible, we prove that the contraction algebra together with its A_\infty-structure recovers various invariants associated to the underlying singularity and its small resolution, including the derived category of singularities and the genus zero Gopakuma-Vafa invariants.

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arXiv2015

Non-commutative virtual structure sheaves

The moduli spaces of stable sheaves on projective schemes admit certain gluing data of Kapranov's NC structures, which we call quasi NC structures. The formal completion of the quasi NC structure at a closed point coincides with the pro-representable hull of the non-commutative deformation functor of the corresponding sheaf. In this paper, we show the existence of smooth non-commutative dg-resolutions of the above quasi NC structures, and call them quasi NCDG structures. When there are no higher obstruction spaces, the quasi NCDG structures define the notion of NC virtual structure sheaves, the non-commutative analogue of virtual structure sheaves. We show that the NC virtual structure sheaves are described in terms of usual virtual structure sheaves together with Schur complexes of the perfect obstruction theories.

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arXiv2015

Non-commutative thickening of moduli spaces of stable sheaves

We show that the moduli spaces of stable sheaves on projective schemes admit certain non-commutative structures, which we call quasi NC structures, generalizing Kapranov's NC structures. The completion of our quasi NC structure at a closed point of the moduli space gives a pro-representable hull of the non-commutative deformation functor of the corresponding sheaf developed by Laudal, Eriksen, Segal and Efimov-Lunts-Orlov. We also show that the framed stable moduli spaces of sheaves have canonical NC structures.

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arXiv2014

Stable pair invariants on Calabi-Yau threefolds containing P^2

We relate Pandharipande-Thomas stable pair invariants on Calabi-Yau 3-folds containing the projective plane with those on the derived equivalent orbifolds via wall-crossing method. The difference is described by generalized Donaldson-Thomas invariants counting semistable sheaves on the local projective plane, whose generating series form theta type series for indefinite lattices. Our result also derives non-trivial constraints among stable pair invariants on such Calabi-Yau 3-folds caused by Seidel-Thomas twist.

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arXiv2014

Derived category of coherent sheaves and counting invariants

We survey recent developments on Donaldson-Thomas theory, Bridgeland stability conditions and wall-crossing formula. We emphasize the importance of the counting theory of Bridgeland semistable objects in the derived category of coherent sheaves to find a hidden property of the generating series of Donaldson-Thomas invariants.

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arXiv2014

Generalized Donaldson-Thomas invariants on the local projective plane

We show that the generating series of generalized Donaldson-Thomas invariants on the local projective plane with any positive rank is described in terms of modular forms and theta type series for indefinite lattices. In particular it absolutely converges to give a holomorphic function on the upper half plane.

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arXiv2013

S-duality for surfaces with $A_n$-type singularities

We show that the generating series of Euler characteristics of Hilbert schemes of points on any algebraic surface with at worst $A_n$-type singularities is described by the theta series determined by integer valued positive definite quadratic forms and the Dedekind eta function. In particular it is a Fourier development of a meromorphic modular form with possibly half integer weight. The key ingredient is to apply the flop transformation formula of Donaldson-Thomas type invariants counting two dimensional torsion sheaves on 3-folds proved in the author's previous paper.

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arXiv2013

Gepner type stability condition via Orlov/Kuznetsov equivalence

We show the existence of Gepner type Bridgeland stability conditions on the triangulated categories of graded matrix factorizations associated with homogeneous polynomials which define general cubic fourfolds containing a plane. The key ingredient is to describe the grade shift functor of matrix factorizations in terms of sheaves of Clifford algebras on the projective plane under Orlov/Kuznetsov equivalence.

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arXiv2013

Gepner point and strong Bogomolov-Gieseker inequality for quintic 3-folds

We propose a conjectural stronger version of Bogomolov-Gieseker inequality for stable sheaves on quintic 3-folds. Our conjecture is derived from an attempt to construct a Bridgeland stability condition on graded matrix factorizations, which should correspond to the Gepner point via mirror symmetry and Orlov equivalence. We prove our conjecture in the rank two case.

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arXiv2013

Gepner type stability conditions on graded matrix factorizations

We introduce the notion of Gepner type Bridgeland stability conditions on triangulated categories, which depends on a choice of an autoequivalence and a complex number. We conjecture the existence of Gepner type stability conditions on the triangulated categories of graded matrix factorizations of weighted homogeneous polynomials. Such a stability condition may give a natural stability condition for Landau-Ginzburg B-branes, and correspond to the Gepner point of the stringy Kahler moduli space of a quintic 3-fold. The main result is to show our conjecture when the variety defined by the weighted homogeneous polynomial is a complete intersection of hyperplanes in a Calabi-Yau manifold with dimension less than or equal to two.

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