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We consider elliptic fibrations with arbitrary base dimensions, and generalise previous work by the second author. In particular, we check universal closedness for the moduli of semistable objects with respect to a polynomial stability that reduces to PT-stability on threefolds. We also show openness of this polynomial stability. On the other hand, we write down criteria under which certain 2-term polynomial semistable complexes are mapped to torsion-free semistable sheaves under a Fourier-Mukai transform. As an application, we construct an open immersion from a moduli of complexes to a moduli of Gieseker stable sheaves on higher dimensional elliptic fibrations.
On a Weierstra{ss} 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. W
We show that the adjunction counits of a Fourier-Mukai transform $Phi$ from $D(X_1)$ to $D(X_2)$ arise from maps of the kernels of the corresponding Fourier-Mukai transforms. In a very general setting of proper separable schemes of finite type over a
A theorem by Orlov states that any equivalence between the bounded derived categories of coherent sheaves of two smooth projective varieties, X and Y, is isomorphic to a Fourier-Mukai transform with kernel in the bounded derived category of coherent
Given a Fourier-Mukai functor $Phi$ in the general setting of singular schemes, under various hypotheses we provide both left and a right adjoints to $Phi$, and also give explicit formulas for them. These formulas are simple and natural, and recover
In this paper we prove that any smooth projective variety of dimension $ge 3$ equipped with a tilting bundle can serve as the source variety of a non-Fourier-Mukai functor between smooth projective schemes.