Using the degeneration formula for Donaldson-Thomas invariants [L-W,MNOP2], we proved formulae for blowing up a point, simple flops, and extremal transitions.
We prove that generalized conifolds and orbifolded conifolds are mirror symmetric under the SYZ program with quantum corrections. Our work mathematically confirms the gaugetheoretic assertion of Aganagic–Karch–Lust–Miemiec, and also provides a supportive evidence to Morrison’s conjecture that geometric transitions are reversed under mirror symmetry.
Guichard O, Wienhard A. Topological Invariants of Anosov Representations[J]. Journal of Topology, 2009, 3(3): 578-642.
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William M Goldman. Locally homogeneous geometric manifolds. 2010.
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Labourie F. Cross ratios, surface groups, PSL(n,ℝ) and diffeomorphisms of the circle[J]. Publications Mathématiques de l\u0027IHÉS, 2007, 106(1): 139-213.
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Bridgeman M, Canary R D, Labourie F, et al. The pressure metric for convex representations[C]., 2013.
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Dreyer G. Length functions of Hitchin representations[J]. Algebraic \u0026 Geometric Topology, 2011, 13(6): 3153-3173.
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Guichard O, Wienhard A. Domains of discontinuity for surface groups[J]. Comptes Rendus Mathematique, 2009: 1057-1060.
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Zhe Sun. Rank n Swapping Algebra for the PSL(n, ℝ) Hitchin Component. 2016.
We show that the notion of hyperconvex representation due to F. Labourie gives a geometric characterization of the representations
of a surface group in PSLn(R) that belong to the Hitchin component.
On a compact K¨ahler manifold we introduce a cohomological obstruction to the solvability of the constant scalar curvature
(cscK) equation twisted by a semipositive form, appearing in works of Fine and Song-Tian.
As a special case we find an obstruction for a manifold to be the base of a holomorphic submersion carrying a cscK metric in
certain “adiabatic” classes. We apply this to find new examples of general type threefolds with classes which do not admit a cscK representative.
When the twist vanishes our obstruction extends the slope stability of Ross-Thomas to effective divisors on a K¨ahler manifold.
Thus we find examples of non-projective slope unstable manifolds.