We study the logrank conjecture from the perspective of pointhyperplane incidence geometry. We formulate the following conjecture: Given a point set in ℝ d that is covered by constantsized sets of parallel hyperplanes, there exists an affine subspace that accounts for a large (i.e., 2 –polylog( d ) ) fraction of the incidences, in the sense of containing a large fraction of the points and being contained in a large fraction of the hyperplanes. In other words, the pointhyperplane incidence graph for such configurations has a large complete bipartite subgraph. Alternatively, our conjecture may be interpreted linearalgebraically as follows: Any rank d matrix containing at most O (1) distinct entries in each column contains a submatrix of fractional size 2 –polylog( d ) , in which each column is constant. We prove that our conjecture is equivalent to the logrank conjecture; the crucial ingredient of this proof is a reduction from bounds for parallel k partitions to bounds for parallel ( k 1)partitions. We also introduce an (apparent) strengthening of the conjecture, which relaxes the requirements that the sets of hyperplanes be parallel. Motivated by the connections above, we revisit wellstudied questions in pointhyperplane incidence geometry without structural assumptions (i.e.,more »
Algebraicity of the metric tangent cones and equivariant Kstability
We prove two new results on the K K polystability of Q \mathbb {Q} Fano varieties based on purely algebrogeometric arguments. The first one says that any K K semistable log Fano cone has a special degeneration to a uniquely determined K K polystable log Fano cone. As a corollary, we combine it with the differentialgeometric results to complete the proof of DonaldsonSun’s conjecture which says that the metric tangent cone of any point appearing on a GromovHausdorff limit of KählerEinstein Fano manifolds depends only on the algebraic structure of the singularity. The second result says that for any log Fano variety with the torus action, K K polystability is equivalent to equivariant K K polystability, that is, to check K K polystability, it is sufficient to check special test configurations which are equivariant under the torus action.
 Award ID(s):
 2153115
 Publication Date:
 NSFPAR ID:
 10330263
 Journal Name:
 Journal of the American Mathematical Society
 Volume:
 34
 Issue:
 4
 Page Range or eLocationID:
 1175 to 1214
 ISSN:
 08940347
 Sponsoring Org:
 National Science Foundation
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