ترغب بنشر مسار تعليمي؟ اضغط هنا

Pillowcase covers: Counting Feynman-like graphs associated with quadratic differentials

142   0   0.0 ( 0 )
 نشر من قبل Elise Goujard
 تاريخ النشر 2018
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

We prove the quasimodularity of generating functions for counting pillowcase covers, with and without Siegel-Veech weight. Similar to prior work on torus covers, the proof is based on analyzing decompositions of half-translation surfaces into horizontal cylinders. It provides an alternative proof of the quasimodularity results of Eskin-Okounkov and a practical method to compute area Siegel-Veech constants. A main new technical tool is a quasi-polynomiality result for 2-orbifold Hurwitz numbers with completed cycles.



قيم البحث

اقرأ أيضاً

We prove the quasimodularity of generating functions for counting torus covers, with and without Siegel-Veech weight. Our proof is based on analyzing decompositions of flat surfaces into horizontal cylinders. The quasimodularity arise as contour inte gral of quasi-elliptic functions. It provides an alternative proof of the quasimodularity results of Bloch-Okounkov, Eskin-Okounkov and Chen-Moeller-Zagier, and generalizes the results of Boehm-Bringmann-Buchholz-Markwig for simple ramification covers.
A $k$-differential on a Riemann surface is a section of the $k$-th power of the canonical bundle. Loci of $k$-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification for the moduli space of $k$-differe ntials. The classification of connected components of the strata of $k$-differentials was known for holomorphic differentials, meromorphic differentials and quadratic differentials with at worst simple poles by Kontsevich--Zorich, Boissy and Lanneau, respectively. Built on their work we develop new techniques to study connected components of the strata of $k$-differentials for general $k$. As an application, we give a complete classification of connected components of the strata of quadratic differentials with arbitrary poles. Moreover, we distinguish certain components of the strata of $k$-differentials by generalizing the hyperelliptic structure and spin parity for higher $k$. We also describe an approach to determine explicitly parities of $k$-differentials in genus zero and one, which inspires an amusing conjecture in number theory. A key viewpoint we use is the notion of multi-scale $k$-differentials introduced by Bainbridge--Chen--Gendron--Grushevsky--Moller for $k = 1$ and extended by Costantini--Moller--Zachhuber for all $k$.
The local invariants of a meromorphic Abelian differential on a Riemann surface of genus $g$ are the orders of zeros and poles, and the residues at the poles. The main result of this paper is that with few exceptions, every pattern of orders and resi dues can be obtain by an Abelian differential. These exceptions are two families in genus zero when the orders of the poles are either all simple or all nonsimple. Moreover, we even show that the pattern can be realized in each connected component of strata. Finally we give consequences of these results in algebraic and flat geometry. The main ingredient of the proof is the flat representation of the Abelian differentials.
Recent important and powerful frameworks for the study of differential forms by Huber-Joerder and Huber-Kebekus-Kelly based on Voevodskys h-topology have greatly simplified and unified many approaches. This article builds towards the goal of putting Illusies de Rham-Witt complex in the same framework by exploring the h-sheafification of the rational de Rham-Witt differentials. Assuming resolution of singularities in positive characteristic one recovers a complete cohomological h-descent for all terms of the complex. We also provide unconditional h-descent for the global sections and draw the expected conclusions. The approach is to realize that a certain right Kan extension introduced by Huber-Kebekus-Kelly takes the sheaf of rational de Rham-Witt forms to a qfh-sheaf. As such, we state and prove many results about qfh-sheaves which are of independent interest.
Let S be a closed topological surface. Haupts theorem provides necessary and sufficient conditions for a complex-valued character of the first integer homology group of S to be realized by integration against a complex-valued 1-form that is holomorph ic with respect to some complex structure on S. We prove a refinement of this theorem that takes into account the divisor data of the 1-form.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا