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1. On finite time Type I singularities of the Kähler–Ricci flow on compact Kähler surfaces

   https://doi.org/10.4171/JEMS/1485  

   Cifarelli, Charles; Conlon, Ronan J.; Deruelle, Alix (July 2024, Journal of the European Mathematical Society)

   We show that the underlying complex manifold of a complete non-compact two-dimensional shrinking gradient Kähler-Ricci soliton (M,g,X) with soliton metric g with bounded scalar curvature Rg whose soliton vector field X has an integral curve along which Rg↛0 is biholomorphic to either C×P1 or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension. 

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2. Gromov-Witten Invariants in Complex and Morava-Local K-Theories

   https://doi.org/10.1007/s00039-024-00697-4  

   Abouzaid, Mohammed; McLean, Mark; Smith, Ivan (December 2024, Geometric and Functional Analysis)

   Abstract Given a closed symplectic manifoldX, we construct Gromov-Witten-type invariants valued both in (complex)K-theory and in any complex-oriented cohomology theory$$\mathbb{K}$$which isK_p(n)-local for some MoravaK-theoryK_p(n). We show that these invariants satisfy a version of the Kontsevich-Manin axioms, extending Givental and Lee’s work for the quantumK-theory of complex projective algebraic varieties. In particular, we prove a Gromov-Witten type splitting axiom, and hence define quantumK-theory and quantum$$\mathbb{K}$$-theory as commutative deformations of the corresponding (generalised) cohomology rings ofX; the definition of the quantum product involves the formal group of the underlying cohomology theory. The key geometric input of these results is a construction of global Kuranishi charts for moduli spaces of stable maps of arbitrary genus toX. On the algebraic side, in order to establish a common framework covering both ordinaryK-theory andK_p(n)-local theories, we introduce a formalism of ‘counting theories’ for enumerative invariants on a category of global Kuranishi charts. 

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3. The integral cohomology of the Hilbert scheme of points on a surface

   https://doi.org/10.1017/fms.2020.35  

   Totaro, Burt (January 2020, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $$X^{[n]}$$ has torsion-free cohomology for every natural number n . This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces. 

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4. Stabilization of the cohomology of thickenings

   Bhatt, Bhargav; Blickle, Manuel; Lyubeznik, Gennady; Singh, Anurag; Zhang, Wenliang. (April 2019, American journal of mathematics)

   For a local complete intersection subvariety $X = V (I)$ in $P^n$ over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of $$X$$, the cohomology of vector bundles on the formal completion of $P^n$ along $$X$$ can be effectively computed as the cohomology on any sufficiently high thickening $$X_t = V (I^t)$$; the main ingredient here is a positivity result for the normal bundle of $$X$$. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings $$X_t$$ in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on $$X$$, and the main new ingredient is a version of the Kodaira- Akizuki-Nakano vanishing theorem for $$X$$, formulated in terms of the cotangent complex. 

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5. Product decompositions of moment-angle manifolds and B -rigidity

   https://doi.org/10.4153/S0008439523000383  

   Amelotte, Steven; Briggs, Benjamin (December 2023, Canadian Mathematical Bulletin)

   Abstract A simple polytopePis calledB-rigidif its combinatorial type is determined by the cohomology ring of the moment-angle manifold$$\mathcal {Z}_P$$overP. We show that any tensor product decomposition of this cohomology ring is geometrically realized by a product decomposition of the moment-angle manifold up to equivariant diffeomorphism. As an application, we find thatB-rigid polytopes are closed under products, generalizing some recent results in the toric topology literature. Algebraically, our proof establishes that the Koszul homology of a Gorenstein Stanley–Reisner ring admits a nontrivial tensor product decomposition if and only if the underlying simplicial complex decomposes as a join of full subcomplexes. 

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