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1. Geography of symplectic Lefschetz fibrations and rational blowdowns

   https://doi.org/10.1090/tran/8968  

   Baykur, R; Korkmaz, Mustafa; Simone, Jonathan (July 2024, Transactions of the American Mathematical Society)

   We produce simply-connected, minimal, symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line. This provides asymplecticextension of the classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Our examples are obtained by rationally blowing down Lefschetz fibrations with clustered nodal fibers, the total spaces of which are potentially new homotopy elliptic surfaces. Similarly, clustering nodal fibers on higher genera Lefschetz fibrations on standard rational surfaces, we get rational blowdown configurations that yield new constructions of small symplectic exotic $4$–manifolds. We present an example of a construction of a minimal symplectic exotic $\mathbb{C}\mathbb{P}{}^2\mathrm{\#<\#comment/>}5\stackrel{¯<\#comment/>}{\mathbb{C}\mathbb{P}}{}^2$through this procedure applied to a genus– $3$fibration. 

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2. Rational homotopy type and nilpotency of mapping spaces between Quaternionic projective spaces

   https://doi.org/10.15673/pigc.v16i2.2648  

   Abebaw, Tilahun; Gatsinzi, Jean-Baptiste; Yeruk, Smegnsh Demelash (June 2024, Proceedings of the International Geometry Center)

   The rational homotopy type of a mapping space is a way to describe the structure of the space using the algebra of its homotopy groups and the differential graded algebra of its cochains. An L∞-model is a graded Lie algebra with a family of higher-order brackets satisfying the generalized Jacobi identity and antisymmetry. It can be used to study the rational homotopy type of a space. The nilpotency index of an L∞-model is useful in understanding a space's algebraic structure. In this paper, we compute the rational homotopy type of the component of some mapping spaces between projective spaces and determine the nilpotency index of corresponding L∞-models. 

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3. Bounds on cohomological support varieties

   https://doi.org/10.1090/btran/182  

   Briggs, Benjamin; Grifo, Eloísa; Pollitz, Josh (March 2024, Transactions of the American Mathematical Society, Series B)

   Over a local ring $R$, the theory of cohomological support varieties attaches to any bounded complex $M$of finitely generated $R$-modules an algebraic variety ${\mathrm{V}}_R\left(M\right)$that encodes homological properties of $M$. We give lower bounds for the dimension of ${\mathrm{V}}_R\left(M\right)$in terms of classical invariants of $R$. In particular, when $R$is Cohen–Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When $M$has finite projective dimension, we also give an upper bound for $\dim <\#comment/>{\mathrm{V}}_R\left(M\right)$in terms of the dimension of the radical of the homotopy Lie algebra of $R$. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of $R$. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring. 

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4. Koszul modules with vanishing resonance in algebraic geometry

   https://doi.org/10.1007/s00029-023-00912-4  

   Aprodu, Marian; Farkas, Gavril; Raicu, Claudiu; Weyman, Jerzy (April 2024, Selecta Mathematica)

   Abstract We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace$$K\subseteq \bigwedge ^2 V$$ $K\subseteq {\bigwedge }^{2}V$, whereVis a vector space. Previously Koszul modules of finite length have been used to give a proof of Green’s Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves onK3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion. 

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5. Algebraic cobordism and a Conner–Floyd isomorphism for algebraic K-theory

   https://doi.org/10.1090/jams/1045  

   Annala, Toni; Hoyois, Marc; Iwasa, Ryomei (March 2024, Journal of the American Mathematical Society)

   We formulate and prove a Conner–Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable $\mathrm{\infty <\#comment/>}$-category of non- ${\mathbb{A}}^1$-invariant motivic spectra, which turns out to be equivalent to the $\mathrm{\infty <\#comment/>}$-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this $\mathrm{\infty <\#comment/>}$-category satisfies ${\mathbb{P}}^1$-homotopy invariance and weighted ${\mathbb{A}}^1$-homotopy invariance, which we use in place of ${\mathbb{A}}^1$-homotopy invariance to obtain analogues of several key results from ${\mathbb{A}}^1$-homotopy theory. These allow us in particular to define a universal oriented motivic ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring spectrum $\mathrm{M}\mathrm{G}\mathrm{L}$. We then prove that the algebraic K-theory of a qcqs derived scheme $X$can be recovered from its $\mathrm{M}\mathrm{G}\mathrm{L}$-cohomology via a Conner–Floyd isomorphism\[ ${\mathrm{M}\mathrm{G}\mathrm{L}}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right){\otimes <\#comment/>}_{\mathrm{L}}\mathbb{Z}\left[{\mathrm{\beta <\#comment/>}}^{\pm <\#comment/>1}\right]\simeq <\#comment/>\mathrm{K}{}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right),$\]where $\mathrm{L}$is the Lazard ring and $\mathrm{K}{}^{p,q}\left(X\right)=\mathrm{K}{}_{2q-<\#comment/>p}\left(X\right)$. Finally, we prove a Snaith theorem for the periodized version of $\mathrm{M}\mathrm{G}\mathrm{L}$. 

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