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Register a new userDonaldson-Thomas invariants, linear systems and punctual Hilbert schemes
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We study certain DT invariants arising from stable coherent sheaves in a nonsingular projective threefold supported on the members of a linear system of a fixed line bundle. When the canonical bundle of the threefold satisfies certain positivity conditions, we relate the DT invariants to Carlsson-Okounkov formulas for the twisted Eulers number of the punctual Hilbert schemes of nonsingular surfaces, and conclude they have a modular property.
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This article provides a summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of nested Hilbert schemes of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and Seiberg-Witten theories.
We study higher rank Donaldson-Thomas invariants of a Calabi-Yau 3-fold using Joyce-Songs wall-crossing formula. We construct quivers whose counting invariants coincide with the Donaldson-Thomas invariants. As a corollary, we prove the integrality and a certain symmetry for the higher rank invariants.
We study motivic Donaldson-Thomas invariants in the sense of Behrend-Bryan-Szendroi. A wall-crossing formula under a mutation is proved for a certain class of quivers with potentials.
We compute the motivic Donaldson-Thomas theory of small crepant resolutions of toric Calabi-Yau 3-folds.
We study Hilbert schemes of points on a smooth projective Calabi-Yau 4-fold $X$. We define $mathrm{DT}_4$ invariants by integrating the Euler class of a tautological vector bundle $L^{[n]}$ against the virtual class. We conjecture a formula for their generating series, which we prove in certain cases when $L$ corresponds to a smooth divisor on $X$. A parallel equivariant conjecture for toric Calabi-Yau 4-folds is proposed. This conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples. Combining the equivariant conjecture with a vertex calculation, we find explicit positive rational weights, which can be assigned to solid partitions. The weighted generating function of solid partitions is given by $exp(M(q)-1)$, where $M(q)$ denotes the MacMahon function.
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