The limits of Kahler manifolds under holomorphic deformations
With some mild assumptions on metric and topology of the central fiber, we prove that the limit of Kahler manifolds under holomorphic deformation is still Kahler.
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arXiv subjects
Wanmin Liu
Publications and source records attributed to Wanmin Liu.
Fourier-Mukai transforms and stable sheaves on Weierstrass elliptic surfaces
On a Weierstraß elliptic surface $X$, we define a `limit' of Bridgeland stability conditions, denoted as $Z^l$-stability, by moving the polarisation towards the fiber direction in the ample cone while keeping the volume of the polarisation fixed. We describe conditions under which a slope stable torsion-free sheaf is taken by a Fourier-Mukai transform to a $Z^l$-stable object, and describe a modification upon which a $Z^l$-semistable object is taken by the inverse Fourier-Mukai transform to a slope semistable torsion-free sheaf. We also study wall-crossing for Bridgeland stability, and show that 1-dimensional twisted Gieseker semistable sheaves are taken by a Fourier-Mukai transform to Bridgeland semistable objects.
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Contractibility of space of stability conditions on the projective plane via global dimension function
We compute the global dimension function $\mathrm{gldim}$ on the principal component $\mathrm{Stab}^{\dag}(\mathbb{P}^2)$ of the space of Bridgeland stability conditions on $\mathbb{P}^2$. It admits $2$ as the minimum value and the preimage $\mathrm{gldim}^{-1}(2)$ is contained in the closure $\bar{\mathrm{Stab}^{\mathrm{Geo}}(\mathbb{P}^2)}$ of the subspace consisting of geometric stability conditions. We show that $\mathrm{gldim}^{-1}[2,x)$ contracts to $\mathrm{gldim}^{-1}(2)$ for any real number $x\geq 2$ and that $\mathrm{gldim}^{-1}(2)$ is contractible.
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Classification of full exceptional collections of line bundles on three blow-ups of $\mathbb{P}^{3}$
A fullness conjecture of Kuznetsov says that if a smooth projective variety $X$ admits a full exceptional collection of line bundles of length $l$, then any exceptional collection of line bundles of length $l$ is full. In this paper, we show that this conjecture holds for $X$ as the blow-up of $\mathbb{P}^{3}$ at a point, a line, or a twisted cubic curve, i.e. any exceptional collection of line bundles of length 6 on $X$ is full. Moreover, we obtain an explicit classification of full exceptional collections of line bundles on such $X$.
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Bayer-Macr\`i decomposition on Bridgeland moduli spaces over surfaces
We find a decomposition formula of the local Bayer-Macr\`i map for the nef line bundle theory on the Bridgeland moduli space over surface. If there is a global Bayer-Macr\`i map, such decomposition gives a precise correspondence from Bridgeland walls to Mori walls. As an application, we compute the nef cone of the Hilbert scheme $S^{[n]}$ of $n$-points over special kinds of fibered surface $S$ of Picard rank two.
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